origins of chaos in nonlinear oscillatory hamiltonian systems a … · 2016-05-19 · origins of...

Origins of Chaos in Nonlinear Oscillatory Hamiltonian Systems

A Thesis

Presented to

The Division of Mathematics and Natural Sciences

Reed College

In Partial Fulfillment

of the Requirements for the Degree

Bachelor of Arts

Mateo R. Ochoa Coloma

May 2016

Approved for the Division(Physics)

Lucas Illing

Acknowledgements

This thesis is the culmination of a journey that started when, at the age of 15, Idecided I was going to study physics. I would not have been able to make it to thispoint without many people.

Thank you Lucas for your guidance, patience, and for putting up with my desireto learn every minute thing that I thought was relevant to my thesis.

Mamita Luly, gracias por todo tu amor y carino. Los anos que pasamos juntos losguardo en mi corazon. Sin tu paciencia ni amor incondicional este tu hijo no podrıahaber llegado a ser fısico. Gracias por dejarme levantar vuelo tan temprano.

Papito Eddy, tus consejos a traves de los anos me dan vida. Gracias por todo elamor, por todo el apoyo que me das. Y mas que todo, gracias por ser mi amigo.

A mis hermanitos. Marcelo, gracias por haber plantado en mı la curiosidad in-telectual que ha guiado mis decisiones de vida. Gabriel, gracias por el apoyo quesiempre me has dado.

Sabita, tanto que hemos reıdo juntos! Contigo he crecido, y gracias a ti he llegadoa ser quien soy. Sin tus comidas riquitas ni los sandwiches de huevo que te pedıacuando ya estabas en la puerta ya para irte, no hubiera podido estudiar para llegardonde estoy. Gracias por ser mi segunda mama.

A mis profes del cole: Humberto, contigo di mis primeros pasos en la fısica, esospasos que fueron claves para que esta tesis se haga realidad. Tincho, gracias porsiempre alentar mi deseo por aprender mas de lo requerido y por lo dedicado quefuiste conmigo.

Thanks to all the Reed people! Steph, you have been the light that has kept mesane and alive this year. Than you for the baaa’s, the yay’s, the bube’s, the British(Boston?) accent, and all the other silly stuff we do. Tanner you crazy mofo, youbrought the wackiness I needed during my time at Reed. It’s time for us to fuckshit up in the real world now. Paloma, we have been through a lot the past 4 years,and I am glad we got each other’s backs along the way. Sidney, I am still betterthan you at FIFA. Alex and Sarah, those late night math sessions are amongst themost memorable times of the past 4 years. Cat, thanks for laughing with me atthe dumbest things. Camila, my twin from (almost) the same country, thank youfor all the support and love. Max, please do not chase me again. Brian, you’re sotall. Carbone, your back&butt are the sexiest things ever. Zahra and Sasha, thelibrary and stim table would not have been the same without you. Emma, pleasestop signing me up for meetings with our HA. Jossef, you were the best HA. Sandesh,you have guided me since I was a baby freshman through all physics-related stuff, you

are my Jesus. Enzo, thanks for showing me how a shy kid from La Paz can turn intoa badass motherfucker, without knowing you I wouldn’t be half of who I am now.Andre, you were the senior I looked up to my freshman year, thank you for being myfriend. Dana, thank you for putting up with all my stupid questions and for being sopositive all the time.

Thanks to the Physics Department. Joel, your love for physics inspired me count-less times. John, you believed in me and made me feel like I was good at physicsfor the first time during my time at Reed. Darrell, the topics you cover in class areamazing. Nelia, thanks for kicking my butt with the 8-hour problem sets.

It has been a wild ride Reed, thank you!

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 1: Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . 31.1 Symplectic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Canonical Transformations . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 Symplectic Condition & Invariance of Phase Space Volume . . 8

1.4 Hamilton-Jacobi Theory and Action-Angle Variables . . . . . . . . . 101.5 Integrable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5.1 Example: The Pendulum . . . . . . . . . . . . . . . . . . . . . 161.5.2 The Solution to the Pendulum in terms of Action-Angle Variables 21

Chapter 2: Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.1 Surface of Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.1 The twist map, resonant tori, and nonresonant tori . . . . . . 282.3 Nonintegrable systems . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.1 Mappings for time-periodic systems . . . . . . . . . . . . . . . 322.4 Introducing the Standard Map . . . . . . . . . . . . . . . . . . . . . . 33

2.4.1 Hamiltonian Formulation of the Standard Map . . . . . . . . . 35

Chapter 3: Phase Space Topology of Nonintegrable Systems . . . . . 413.1 KAM theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 Linear Independence Condition . . . . . . . . . . . . . . . . . 433.1.2 Sufficiently Far From Resonance . . . . . . . . . . . . . . . . . 43

3.2 Fixed Points and their Linear Stability . . . . . . . . . . . . . . . . . 453.2.1 Application: The Standard Map . . . . . . . . . . . . . . . . . 51

3.3 Poincare-Birkhoff Theorem . . . . . . . . . . . . . . . . . . . . . . . . 513.3.1 Motion in the vicinity of an elliptic point . . . . . . . . . . . . 533.3.2 Homoclinic and Heteroclinic Points . . . . . . . . . . . . . . . 63

3.4 Motion in the Vicinity of the Separatrix . . . . . . . . . . . . . . . . 683.4.1 Application: The Standard Map . . . . . . . . . . . . . . . . . 72

3.5 Self-Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Chapter 4: Global Chaos and Diffusion in the Standard Map . . . . . 774.1 Overlapping Resonances . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Global Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.3 Diffusion in the Standard Map . . . . . . . . . . . . . . . . . . . . . . 81

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Appendix A: Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . . 85

Appendix B: Dirac Delta Properties . . . . . . . . . . . . . . . . . . . . 87B.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87B.2 Rewriting the Perturbation Term of the Standard Map Hamiltonian . 88

Appendix C: Defining Chaos . . . . . . . . . . . . . . . . . . . . . . . . . 89C.1 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

List of Figures

1.1 Ensemble of points moving in phase in accordance with Hamilton’sequations of motion from t1 to t2. . . . . . . . . . . . . . . . . . . . . 5

1.2 An integrable system with two degrees of freedom represented as a torus. 15

1.3 Motion along the torus for rational r. The trajectory closes on itselfand thus it does not densely cover the torus. In (a) an integrablesystem with r = 4 is shown. In (b) an integrable system with r = 1/4is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 An integrable system with r = 2π. The motion is quasiperiodic andthus densely covers the torus. The system has done: (a) three revolu-tions, (b) ten revolutions, (c) sixty revolutions in θ1. . . . . . . . . . . 16

1.5 The pendulum system. The coordinate variable q can be clearly seen. 19

1.6 Phase space of the pendulum Hamiltonian for various values of E. Thearrows on the curves indicate the direction of the flow. . . . . . . . . 19

2.1 By setting our variable q2 as a constant we define the surface of sectionin the (p1, q1) plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 The surface of section for the Hamiltonian flow in Fig. 2.1. . . . . . . 26

2.3 Surface of section with θ1 constant for an integrable system with (a)r = 4 for only one initial condition, and (b) r = 1/4 for two initialconditions denoted by • and N. . . . . . . . . . . . . . . . . . . . . . 27

2.4 Surface of section with θ1 constant for an integrable system with r =2π. The system has done (a) three revolutions, (b) ten revolutions,and (c) six hundred revolutions in θ1. . . . . . . . . . . . . . . . . . . 27

2.5 The twist map for different values of r. For r = 1/4 two trajectories areplotted, each one with different initial conditions from the other. Forr = 1/6 only one trajectory is plotted. For r irrational, the trajectoriescorrespond to only one initial condition. The green arrows indicate thelocation of the invariant resonant tori and the blue arrows indicate thelocation of the nonresonant tori. As indicated, the nonresonant toriare densely covered by a single trajectory. . . . . . . . . . . . . . . . 29

2.6 The kicked rotor. A delta function kicks the rotating mass every timet = n in the direction of the arrows, regardless of where the mass is. . 33

2.7 1000 iterations of the standard map for only one pair of initial condi-tions, (I0 = π, θ0 = π + .3). The value of K was: for (a) K = 0, for(b) K = 0.5, and for (c) K = 2.5. . . . . . . . . . . . . . . . . . . . . 37

2.8 10000 iterations of the standard map for four pairs of initial conditionsall with θ0 = π+.3 and each individual pair with I0 = 2π/5, 4π/5, 6π/5, 8π/5.Each initial condition is marked in the plot. The value of K was: for(a) K = 0, for (b) K = 0.5, and for (c) K = 2.5. . . . . . . . . . . . . 37

2.9 1000 iterations of the standard map for thirty pairs of initial conditionsall with θ0 = π+ .3 and K = 2.5 and each individual pair with I0 = 2πn

31

where n ∈ [1, 30]. Each initial condition is marked in the plot. Thevalue of K was: for (a) K = 0, for (b) K = 0.5, and for (c) K = 2.5. . 38

2.10 Two different options for plotting the standard map. The preferencefor a visualization as in (a) is evident by noticing that the islands andinvariant curves are easier to see in (a). . . . . . . . . . . . . . . . . . 39

3.1 The shaded regions are those for which a KAM torus cannot exist. Theshaded regions correspond to those r that satisfy

∣∣r − ab

∣∣ < γb2.5

, wherea/b corresponds to each rational number shown on the axis. The valuesof γ used were: for (a) γ = 0.12 and for (b) γ = 2. Notice that as γincreases, the region where the KAM tori can exist decreases. . . . . . 45

3.2 Same as Fig. 3.1 but with γ = 4. For this value of γ the regions whereKAM tori can live is very small compared to the region of Fig. 3.1a. . 45

3.3 The mapping given in Eq. (3.48) with δσ0 = 2, δν0 = 2 and: for (a)λ = 1.3 and for (b) λ = −1.3. . . . . . . . . . . . . . . . . . . . . . . 50

3.4 Behavior of the invariant curves C, C+, C− under the map T b. In (b)we can see one iteration of the twist map for five pairs of initial values(J0, θ0) all with θ0 = π/2. The stationary point (green point) corre-sponds to J0 = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.5 The new shape of the invariant curves of the unperturbed system underperturbation. (a) shows that the (small) perturbation barely affectsthe nonresonant curve, whereas the resonant curve changes its shapedrastically. In (b) we can see that by iterating the map T b on thepoints lying on Cp we obtain C ′p. The intersections of Cp and C ′p arethe fixed points of the system. . . . . . . . . . . . . . . . . . . . . . 54

3.6 The flow around the fixed points depends both on the direction themapping moves the points on the curve Cp and on the direction themapping moves the points on the curves Cp+, Cp−. This flow indicatesto us what kind the fixed points are, if elliptic or hyperbolic. . . . . 54

3.7 Iterations of initial conditions close to resonance. The phase spaceis very similar to that of the pendulum, which is shown in Fig. 1.6.The map was iterated for perturbation strength (a) K = 0.4 and (b)K = 0.95. A zoom of the region inside the rectangle is shown in Fig.3.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.8 Blow up of the rectangle in Fig. 3.7a. This is a secondary resonance,with its own pendulum-like motion in its vicinity. (a) was iteratedwith the same initial conditions as those in Fig. 3.7a, whereas for (b)we picked initial conditions so that the structure of phase space in thevicinity of this point is clearer. . . . . . . . . . . . . . . . . . . . . . . 62

3.9 The curves H+ and H− are the same. The intersection is at the hy-perbolic point H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.10 A homoclinic point and the confusion regarding its position after ap-pying the map T to it. In (a) the curves H+ and H− originate from thesame hyperbolic point and intersect each other at the homoclinic pointq. Notice how the curve H+ does not intersect itself, and neither doesH−. In (b) The nearby points q, q′ are mapped to Tq, Tq′ respectively.It is unclear where we should put Tq. . . . . . . . . . . . . . . . . . . 64

3.11 The problem of where to put Tq is solved by folding the curve H− sothat q follows both flows after one iteration of the map T . The newcrossing of H+ and H− corresponds to a new homoclinic point. . . . 65

3.12 A second iteration of the map T on q. In (a) we see that for a seconditeration of the map T on q, the curve H− has to fold a second time.The distance between Tq and T 2q is smaller than the distance betweenq and Tq and thus, since the mapping is area-preserving, the foldingthis time is thinner and longer. (b) shows an easier visualization ofFig. 3.12a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.13 As we iterate the map backwards in time, the curve H+ also undergoesthe same folding that we described for H−. . . . . . . . . . . . . . . . 66

3.14 The problem of where to put Tq is solved by folding the curve H− sothat q follows both flows after one iteration of the map T . The newcrossing of H+ and H− corresponds to a new homoclinic point. . . . 66

3.15 Both stable and unstable curves are shown. They both behave in awild manner as they approach the hyperbolic point. . . . . . . . . . . 66

3.16 Visualization of the motion close to the hyperbolic point (2π, 0) forK = 0.5. We picked (J0 = 0.01, θ0 = 0.01) as the initial point fromwhich we obtained the line segment, and from the line segment a setof 1000 initial conditions. Each initial condition was iterated 100 times. 67

3.17 The standard map iterated with I0 = 10−8, θ0 = 0, initial conditionsthat are very close to the hyperbolic point and are very likely to bewithin the chaotic layer close to the separatrix. For (a) K = 0.6, andfor (b) K = 0.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.18 K = 0.1 and with 106 number of map iterations. In (a) the range of theplot does not let us appreciate the chaotic layer, making the trajectoryseem regular. In (b), however, we zoom in, and we find that the regionof chaotic motion is clearly bounded. Islands of stable motion, theregions inside the chaotic sea that are blank, are unexplored by thechaotic trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.19 Amplification of the highlighted region in Fig. 3.18b. In (a) we cansee the islands of stability corresponding to primary resonances of veryhigh rotation number. (b) is the same as (a) but iterated 107 times. 74

3.20 Same conditions as Fig. 3.19 but with initial conditions that fall insidethe islands. (a) The motion can be seen to be stable inside theseislands. (b) We amplify the highlighted island in (a). . . . . . . . . . 75

3.21 Blow up of the highlighted region in Fig. 3.20b. (a) was plotted withthe same initial conditions as Fig. 3.20b, whereas for (b) we pickeddifferent initial conditions so that the islands of stability could be vis-ible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1 The standard map iterated with K = 0.9. Chaotic regions have finitewidths and are bounded by KAM curves. . . . . . . . . . . . . . . . . 78

4.2 The separatrices of the resonances corresponding to In = 0 and In = 2πare seen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3 The standard map is iterated for various initial conditions with K ≈0.971635. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4 The standard map is iterated for various initial conditions with K = 1.Even though the last KAM torus has been destroyed, there are stillregions where the motion is regular. . . . . . . . . . . . . . . . . . . . 81

C.1 In (a) we can see chaotic trajectories only as layers, whereas in (b)chaotic trajectories can take any value of I. . . . . . . . . . . . . . . 91

Abstract

In this thesis we provide a formal exposition of the origins of chaos in perturbed,oscillatory Hamiltonian systems. We start by understanding the fundamentals of theHamiltonian formalism and phase space, followed by the construction of Hamiltonianmaps from continuous Hamiltonian systems. By making use of the KAM theoremand the Poincare-Birkhoff theorem, we identify the origin of chaos in the vicinity ofthe separatrix for arbitrarily small perturbation strengths. Once the perturbationstrength is beyond a certain threshold value, chaotic trajectories overtake almost allphase space and it becomes possible to describe the dynamics of the system withstatistical mechanics. Thus a connection between classical mechanics and statisticalmechanics is found.

Introduction

The predictable behavior of mechanical systems was advocated by people such asNewton, Euler, Lagrange, and Laplace, leading natural philosophers (those whom wecurrently call “scientists”) to believe that predictability was an inherent property ofnature. This was due to the high success with which the then universally acceptedNewtonian laws predicted motion. It was believed that all the possible informationof a system in time past and time future could be fully known if one knew the currentstate of the system. There was no room for chance, probability, or unpredictability.The mechanical systems that formed the basis of this view were mathematically char-acterized by the fact that their solutions could be written in a nice closed form. Suchsystems are now known as integrable systems.

This view was challenged by Poincare, in his study of the stability of the threebody problem. He discovered that minimal differences in initial conditions led toradical differences in the long term behavior of the system. He stated that [1]

If we knew exactly the laws of nature and the situation of the universe atthe initial moment, we could predict exactly the situation of that sameuniverse at a succeeding moment. But even if it were the case that thenatural laws had no longer any secret for us, we could still only know theinitial situation approximately. If that enabled us to predict the succeed-ing situation with the same approximation, that is all we require, and weshould say that the phenomenon had been predicted, that it is governedby laws. But it is not always so; it may happen that small differences inthe initial conditions produce very great ones in the final phenomenon.A small error in the former will produce an enormous error in the latter.Prediction becomes impossible.

On top of predicting what we now know is a basic indicator of chaotic motion, thatis, high sensitivity to initial conditions, he also noticed the complex motion close tohyperbolic points of integrable systems with a perturbation (also known as nonin-tegrable systems) of the stable and unstable manifolds emanating from hyperbolicpoints. Of this he wrote that [2]

The intersections form a kind of lattice, web or network with infinitelytight loops; neither of the two curves must ever intersect itself but itmust bend in such a complex fashion that it intersects all the loops ofthe network infinitely many times. One is struck by the complexity ofthis figure which I am not even attempting to draw. Nothing can give us

2 Introduction

a better idea of the complexity of the three body problem and of all theproblems in dynamics where there is no holomorphic integral and Bohlin’sseries diverge.

These discoveries hinted at the falseness of the then held belief that all mechanicalsystems were fully predictable. Poincare made one final contribution: he assertedthat solutions describing the long term behavior of nonintegrable systems were notpossible due to the presence of internal resonances that made the solutions diverge.

It was then believed that solutions describing regular motion of nonintegrablesystems did not exist, until, in 1954, Kolmogorov sketched a proof that showed that,for small enough perturbations to the integrable system, stable solutions exist as longas the system is far enough from resonance. This was later formalized by Arnold andMoser, and is now known as the KAM theorem. This was a revolutionary result, asit proves that stable trajectories do exist in nonintegrable systems, under a few not-so-restrictive conditions. Even more surprisingly, a system, depending on its initialconditions, can exhibit chaotic or regular motion.

The KAM theorem indicates that the existence of regular trajectories is dependenton the strength of the perturbation. Once the perturbation is very strong (strong withrespect to some other parameters of the system) the initial conditions for which themotion is chaotic is comprised of almost all the possible initial conditions. Once allpossible trajectories are chaotic, it becomes advantageous and mathematically per-missible to utilize the tools of statistical mechanics, and start studying the statisticalproperties of the system.

In a similar fashion, when a physics student is learning elementary classical me-chanics, the student only encounters integrable systems; one can say that this isexpected due to didactic reasons. However, this leads to the false belief that me-chanical systems can only exhibit predictable trajectories described by solutions thatare well-behaved and can be written in closed form. Additionally, most of the litera-ture in chaos is not undergraduate-friendly, and often requires knowledge of advancedtopics in physics. This thesis has as its aim to provide an easy-to-follow expositionof the origins of chaos in nonintegrable Hamiltonian systems, so that the physicsstudent, with only an elementary course in classical mechanics in their toolbox, canappreciate the intricacy of chaos. This will be done as follows: we introduce conceptsof Hamiltonian mechanics that are not covered in an elementary classical mechanicscourse. Then, we introduce the formulation of Hamiltonian mappings as a simplertool to study the dynamics of continuous Hamiltonian systems. Next, we proceed tothe study of perturbations of integrable systems. We will see that chaos is presenteven for arbitrarily small perturbations and that its origin is near the separatrices.Finally, we increase the strength of the perturbation such that almost all the solutionsto the system exhibit chaotic behavior, and make the connection between classicalmechanics and statistical mechanics.

For most topics covered, I give an example using the standard map, a very simplenonlinear two dimensional map derived from a Hamiltonian that exhibits chaoticbehavior, in the hope that the reader can grasp the concepts treated more easily.

Chapter 1

Hamiltonian Formalism

The Hamiltonian formulation of classical mechanics provides us with the dynamics ofa system in phase space. Working in phase space is convenient because every point inthis space specifies a unique state of the system. Additionally, we will be looking intothe integrability and non-integrability of dynamical systems. This, in turn, followsfrom using perturbation theory, whose understanding will be essential in our studyof chaotic behavior.

Before being introduced to Hamiltonian mechanics, the student of physics usu-ally learns to think about dynamical systems in terms of the position variable andits higher (usually first and second) derivatives in time. This is done by using theLagrangian formalism. The Lagrangian of a system is a function of the position, itsfirst derivative, and time L = L(x, x, t). The Hamiltonian formalism changes thingsby making use of the Hamiltonian, which depends on the position and momentumH = H(x, p, t). As it is noted in an elementary course in classical mechanics, one ofthe advantages of using the Hamiltonian formalism is that it provides us with two firstorder differential equations, describing the evolution in time of the position and themomentum in phase space, as opposed to the Lagrangian formalism, which providesus with one second order differential equation describing the time evolution of theposition only in coordinate space.

There are several properties of the Hamiltonian formulation of classical mechanicsthat are necessary to introduce before delving into the particular case of the standardmap. Once we have these tools in place, it will be easier to appreciate the differentparticularities of the standard map.

1.1 Symplectic Formulation

Hamiltonian systems can be studied from a mathematically rigorous point of view.For our purposes, it will suffice to simply quote and explain some of the results thatare obtained by such an approach to Hamiltonian systems.

The study of a system using the Hamiltonian formalism is entirely given by theHamiltonian, H(p(t),q(t), t). The Hamiltonian depends on a scalar and two vectors:the conjugate coordinates q, the conjugate momenta p, and time t. The position and

4 Chapter 1. Hamiltonian Formalism

momenta vectors are of dimension N .The dynamics of a system are described by Hamilton’s equations of motion. These

equations can be written in a compact manner as:

η = J∂H

∂η, (1.1)

where η is a vector of dimension 2N defined as

η =

(qp

)(1.2)

and J is a 2N × 2N matrix defined as

J =

[ON IN−IN ON

], (1.3)

where ON is an N ×N matrix with all of its entries equal to zero and IN an N ×Nidentity matrix. The way of representing Hamilton’s equations of motion as shown inEq. (1.1) is referred to as symplectic notation [3]. Eq. (1.1) is a set of 2N equationsthat completely determines the trajectory of the system in phase space. Additionally,for a given time t = t, the state of the system is given by the vector η evaluated att. We will make use of this notation after introducing canonical transformations, butbefore we do that, let us delve further into what the phase space is.

1.2 Phase Space

As mentioned in the introduction to the Hamiltonian formalism, one of the advantagesHamiltonian systems have is that a point in phase space at any time t uniquelydetermines the state of the system. How is this true? Well, a system’s state is fullyspecified by its position and momentum at time t1. Let us have a system with Ndegrees of freedom. Since Hamilton’s equations of motion (cf. Eq. (1.1)) provideus with the time evolution of the position and momentum variables (q(t),p(t)), wecan construct a space of 2N dimensions with p as N of the coordinates and q as theremaining N coordinates. This space is called the phase space. Since every point inthe phase space corresponds to a pair of positions and momenta, then every pointfully describes the state of a system.

We already mentioned this, but it is worth mentioning it again: the motion inphase space of a system is fully described by Hamilton’s equations of motion. Remem-ber that for a system with N degrees of freedom, Hamilton’s equations of motion forsuch system are a set of 2N first order differential equations. This means that if wesolve them, we get 2N constants of integration. These constants of integration haveto be specified if we want to describe a specific system. This can be done by specify-ing the initial conditions of the system. Then, once we specify the initial conditions,

1Note that if we have a rotating system, the position would correspond to the angular displace-ment and the momentum would be the angular momentum.

1.3. Canonical Transformations 5

Figure 1.1: Ensemble of points moving in phase in accordance with Hamilton’s equa-tions of motion from t1 to t2.

we fix the constants of integration. Different initial conditions will give us differenttrajectories in phase space. This leads to one of the most interesting properties ofphase space, which can be phrased as: no two trajectories can intersect each otherbecause the initial conditions make every trajectory unique. This property allows usto confirm the already known fact that Hamiltonian systems are deterministic2. Ifthis property would not hold, that is, if two trajectories in phase space were allowed tointersect, then these trajectories would share one point P in phase space. If we thendecided to study the time evolution of a system in the state described by P , we wouldnot know which of the two trajectories to follow, since the point P is a valid stateof both trajectories. This ambiguity is not compatible with Hamilton’s equations ofmotion, which uniquely describe a trajectory given a set of initial conditions.

Another important property of Hamiltonian systems is that they preserve phasespace volume. If we take the set of points evaluated at t1 inside a boundary Ω1 thatencloses a volume V1, and we evolve the system to a time t2, then there will be a newboundary Ω2 that encloses a volume V2. These two volumes will be the same, thatis, V1 = V2. Inside V2 are all the points from Ω1 evaluated at t2. In other words,phase space volume is conserved. We will prove this property in section 1.3.2 once weintroduce canonical transformations. Fig. 1.1 shows the preservation of phase spacevolume for the 1-degree-of-freedom case.

1.3 Canonical Transformations

For a given Hamiltonian H(p,q, t), it is sometimes convenient to perform a changeof variables to simplify the problem. The new variables (Q,P) are a function of theold variables, that is, Q = Q(p,q) and P = P(p,q). This change of variables will

2A deterministic system is one whose state at a later or previous time is uniquely determined byits state at an arbitrary time [4].


change the form of the Hamiltonian H(p,q, t) → K(P,Q, t) 6= H(p,q, t). However,not any change of variables will do. If we perform an arbitrary change of variables,the equations of motion will not retain the symplectic form of Hamilton’s equations(cf. Eq. (1.1)). This is not to say that no transformation will do. In fact, there existtransformations called canonical transformations that preserve the form of Hamilton’sequations of motion. A transformation from (p,q) to (P,Q) is canonical if [5]

ξ = J∂K

∂ξ, (1.4)

where ξ =

(QP

). In order to understand the procedure to make this kind of trans-

formation, we need to introduce generating functions.

1.3.1 Generating Functions

Satisfying Hamilton’s equations of motion is equivalent to saying that the followingcondition must hold:

δ

∫ tf

t0

( N∑i=1

piqi −H(q,p, t)

)dt = δ

∫ tf

t0

( N∑i=1

PiQi −K(Q,P, t)

)dt = 0, (1.5)

where the expression in the very left of Eq. (1.5) is equal to zero by virtue of thePrinciple of Least Action3 and from writing the Lagrangian in terms of the Hamil-tonian as L(q, q, t) =

∑Ni=1 piqi −H(q,p, t). What happens if we add the total time

derivative of a function F to any of the terms in Eq. (1.5)? The integral would looklike ∫ tf

t0

dF

dtdt = F (tf )− F (t0). (1.6)

The integral is equal to the difference of F (t) evaluated at tf and t0. Since tf andt0 are the end points, they will remain fixed under any variation4, and this impliesthat Eq. (1.6) is constant. Upon variation of the Action, this constant term willcontribute nothing [6]. Then, we can freely add dF

dtto the action S, or equivalently

to any of the terms in Eq. (1.5) because the variation of a constant term is zero. Wecan finally relate the integrands of the terms in Eq. (1.5) thanks to the newly foundfunction F . This condition can be expressed as

N∑i=1

piqi −H(q,p, t) =N∑i=1

PiQi −K(Q,P, t) +dF

dt(1.7)

3In Lagrangian mechanics we obtain the Euler-Lagrange equations of motion by setting thevariation of the Action S =

∫Ldt equal to zero. After writing the Lagragian L in terms of the

Hamiltonian, the same procedure can be followed to obtain Hamilton’s equations of motion.4Think of a loose string nailed to the ground at its ends. We can wave the string all we want but

the end points will remain fixed.


We will call the function F the generating function. To properly relate the old vari-ables (q,p) with the new variables (Q,P), we have to pick one old and one newvariable in order to “couple” them.

Let the generating function be a function of the old and new coordinates5

F (q,Q, t) = F2(q,P, t)−N∑i=1

QiPi. (1.10)

Plugging in Eq. (1.10) into Eq. (1.7) and expanding, we have

N∑i=1

piqi−H(q,p, t) =N∑i=1

PiQi−K(Q,P, t)+∂F2

∂t+

N∑i=1

(∂F2

∂qiqi+

∂F2

∂PiPi−QiPi−PiQi

).

(1.11)The equality in Eq. (1.11) holds if

pi =∂F2

∂qi(1.12)

Qi =∂F2

∂Pi(1.13)

K(Q,P, t) = H(q(Q,P, t),p(Q,P, t), t) +∂F2(q,P, t)

∂t. (1.14)

These relations provide us with the canonical transformation we were lookingfor. If somebody hands us a generating function, then we can generate a new setof canonical variables using Eqs. (1.12) and (1.13) that obey Hamilton’s equationsof motion with the Hamiltonian given by Eq. (1.14). This is a remarkable propertyof the Hamiltonian formalism. All this derivation originated from the fact that thevariation of the Action is insensitive to adding the time derivative of a function tothe Lagrangian.

To get a sense of how a generating function is used, let us see an example. I willpick the most simple form for F2 possible. Let F2 = q P. Using Eqs. (1.12), (1.13),and (1.14) we get that the new variables are pi = Pi and Qi = qi and that the Hamil-tonian does not change. The form of F2 we just used is called the identity generating

5Notice that F is a function of (q,Q, t) and the right hand side of Eq. (1.10) has P as a variable.That is because F2 is the Legendre transform of F . The Legendre transform works as follows.Suppose we have a function f(x) that satisfies df(x) = g(x)dx, and we instead want a function g(p)with dg(p) = x(p)dp. This can be done if we define g(p) as [5]

g(p) = px− f(p) (1.8)

and we can see that, even though g is a function only of p, on the right hand side we also have x,which is the variable we want to get rid of. For two functions, if we have F (x, y) and want a functionG(x, z), we can obtain G from F by defining G as

G(x, z) = yz − F (x, y). (1.9)

Again, we can see that, on the right hand side the variable y is present even though G is not afunction of y.


function, since the new and old variables are the same. That was entertaining butnot insightful at all. Let us see another example. Let F2 be

F2 = q P +Hdt, (1.15)

where H is a Hamiltonian and dt is small. Plugging this new form into Eqs. (1.12)and (1.13) the new variables are

pi = Pi +∂H

∂qidt (1.16)

Qi = qi +∂H

∂Pidt. (1.17)

We are going to do some massaging, remembering that we assumed that dt is small.If we want to keep things only to first order in dt (which we do) we can replace Pi bypi in the denominator of Eq. (1.17)6. After this replacement we get

pi = Pi +∂H

∂qidt (1.18)

Qi = qi +∂H

∂pidt. (1.19)

Hamilton’s equations of motion appear at the very right of both equations. InsertingEq. (1.1) we obtain

Pi = pi + pidt ≈ qi(t+ dt) (1.20)

Qi = qi + qidt ≈ pi(t+ dt). (1.21)

What is this result? Well, the new variables are just the old variables evaluated att = t+dt and they were obtained with help of the Hamiltonian. Thus the Hamiltonianis the generator of time evolution of the system, or, stating it more provocatively, thetime evolution of a Hamiltonian system is nothing more than a successive infinitesimalcanonical change of variables. Now this is quite a result!

1.3.2 Symplectic Condition & Invariance of Phase Space Vol-ume

The treatment of canonical transformations does not have to be limited to the use ofgenerating functions. In this section we will approach these transformations from adifferent perspective. An important property of canonical transformations is that theysatisfy what is called the symplectic condition. This condition is defined as follows.Let us define a symplectic matrix A. This matrix satisfies the following expression

AJA† = J (1.22)

6If this move bothers you, then let ∂H∂Pi

= ∂H∂pi

∂pi∂Pi

and use Eq. (1.16) for pi and discard higherorder terms.


where J is defined as in Eq. (1.3) and † denotes the transpose. Eq. (1.22) is knownas the symplectic condition. In order to make the connection between Eq. (1.22) andcanonical transformations, let us define the Jacobian matrix DM belonging to thetransformation from η to ξ as

DM =∂ξ

∂η, (1.23)

with entries defined as DMij = ∂ξi∂ηj

. It can be proven7 that the Jacobian matrix DM

of a canonical transformation satisfies8 the symplectic condition, that is,

DM J DM† = J. (1.24)

The symplectic condition is a necessary and sufficient condition for the transformationto be canonical. In other words, a canonical transformation maps a set of canonicalvariables onto a new set of variables that is guaranteed to be canonical; the form ofHamilton’s equations does not change. Notice that so far in this section we have notonce used the Hamiltonian corresponding to either η or ξ. This is because canonicaltransformations are independent of any Hamiltonian.

We can use the symplectic condition for a canonical transformation to see whathappens to a volume element in phase space under a canonical transformation. Itis well known from multivariable calculus that the relationship between two volumeelements is given by

N∏i=1

dQidPi = | det (DM)|N∏i=1

dqidpi, (1.25)

where the matrix DM is defined as in Eq. (1.23). To prove that the volume element inphase space is conserved, we need to show that the absolute value of the determinantof the Jacobian is equal to one. Remembering that the Jacobian of the transformationsatisfies the symplectic condition (cf. Eq. (1.24)), we take the determinant of Eq.(1.24) and, using the fact that det (J) = 1 and det (DM†) = det (DM) [7], we havethat

det (DM) = ±1 (1.26)

and its absolute value is one. Thus we conclude that the absolute value of the deter-minant of a symplectic matrix is always unity. From this it follows that the volumeelement in phase space is preserved under canonical transformations (cf. Eq. (1.25)).

Example Here is a neat application. Eqs. (1.20) and (1.21) indicate that the vari-ables η(t) and η(t+dt) are related by a canonical transformation. Since we justproved that phase space volume under a canonical transformation is conserved,

7The proof outlined in [3] uses Eq. (1.1) as a starting point. Thus the symplectic condition for acanonical transformation is a direct consequence of the symplectic structure of Hamilton’s equationsof motion.

8As a fun connection between the symplectic condition and generating functions, the fact thatthe Jacobian of a canonical transformation satisfies the symplectic condition implies the existenceof a generating function that relates the old variables (q,p) and the new variables (Q,P) [3].


let us consider a volume at time t that encloses a set of points in phase spaceinside a boundary Ω

V (t) =

∫Ω

dη. (1.27)

Let us denote the variables η evaluated at t = t + dt as η = η(t + dt). Thevolume of phase space points at t+ dt is [8]

V (t+ dt) =

∫Ω

dη =

∫Ω

∣∣∣∣ det

(∂η

∂η

)∣∣∣∣dη =

∫Ω

dη = V (t). (1.28)

That is, the volume of phase space points at time t+ dt is equal to the volumeat time t. This incompressibility of the Hamiltonian flow in phase space isknown as Liouville’s theorem and we just proved it. I want to emphasize thatthe key ingredient of our proof was the fact that the change of variables fromη(t) to η(t + dt) is canonical (used to set the determinant of the Jacobianequal to one, the third equality in Eq. (1.28)). This in turn follows fromthe symplectic structure of the canonical transformation. In fact, without thesymplectic structure of the canonical change of variables, phase space volumewould not be preserved. So we can think of the incompressibility of phase spacevolume as a consequence of the symplectic structure of Hamilton’s equations ofmotion.

1.4 Hamilton-Jacobi Theory and Action-Angle Vari-

ables

The existence of generating functions to obtain a new pair of canonical variables givesus infinite choices for our new variables. We could (in theory) generate a new set ofcanonical variables from any generating function. Taking into account that we areinterested only in time-independent Hamiltonians H = H(q,p), there is one particu-lar choice for our new variables that turns out to simplify the problem considerably:we want the new momenta to be constant. We relabel the new variables (Q,P) as(Q,P) ≡ (β,α). Remembering the generating function F2 that we worked with beforein Eqs. (1.12), (1.13), and (1.14), we are going to relabel F2 as F2(q,P) ≡ W (q,α),where W is called Hamilton’s characteristic function. Using Eqs. (1.12), (1.13), and(1.14) we get

pi =∂

∂qiW (q,α) (1.29)

βi =∂

∂αiW (q,α) (1.30)

K(α) = H

(q,∂W

∂q

). (1.31)

1.4. Hamilton-Jacobi Theory and Action-Angle Variables 11

Eq. (1.31) is known as the Hamilton-Jacobi equation, albeit for time-independentHamiltonians9. Since the new Hamiltonian K given by Eq. (1.31) is a function ofconstants, then the Hamiltonian itself is a constant. From the equality we can seethat the old Hamiltonian is equal to this constant. The value of this constant is thevalue of the Hamiltonian, and we interpret it as the energy of the system.

Let us get more specific and consider a system with one degree of freedom H =H(q, p). In this case there is only one new momentum α. Since the new HamiltonianK is a function of the new momentum we can set the new Hamiltonian to be the newmomentum K(α) = α. It follows that α is also the energy of the system. In this casethe energy will be the constant of motion. Now I want us to look at the equations ofmotion for the new variables. These are

α = −∂K∂β

= 0 =⇒ α = constant (1.32)

β =∂K

∂α= 1 =⇒ β = t (1.33)

where we set the (trivial) integration constant when solving for β equal to zero.We can see how easy a one-degree-of-freedom problem becomes if we decided to useHamilton-Jacobi theory: the new momentum is the constant energy and the newcoordinate is time, how ideal!

Let us get even more specific and consider systems whose possible solutions areperiodic. Periodic motion occurs when either: both q and p are periodic in time withthe same period (oscillation), or when p is a periodic function of q (rotation) [9]. Anexample of an oscillation is a simple harmonic oscillator, and an example of a rotationis the motion of the Earth around the Sun. Our goal is to find coordinate variablesthat increase by 2π after each period (hence the name “angle” variables) along withconstant momentum. We assume this can be done since there is already some kindof periodicity built into the system. Let us denote this kind of coordinate variable asθ and its conjugate momentum as J . We only require that it satisfies

H(q, p) = α = K ′(J) (1.34)

where K ′(J) is a function of J10. Now, what is the conjugate momentum J to be?To find this out, let us take the derivative of Eq. (1.13) (and renaming the differentvariables correspondingly) with respect to the old coordinate q

∂θ

∂q=

∂

∂J

(∂W (q, J)

∂q

)=⇒ dθ =

∂

∂J

(∂W (q, J)

∂q

)dq (1.35)

The coordinate variable θ is defined as increasing by 2π after one complete period Ω:

2π =

∮Ω

dθ =∂

∂J

∮Ω

∂W (q, J)

∂qdq =

∂

∂J

∮Ω

p dq. (1.36)

9The more general Hamilton-Jacobi equation is given by H

(∂S∂q ,q, t

)+ ∂S

∂t = 0 where S =

S(q,α, t) has an explicit time dependence. To get the Hamilton-Jacobi equation for time indepen-dent Hamiltonians, write S as S(q,α, t) = W (q,α)− Et where E is the energy of the system.

10What Eq. (1.34) says is that the new constant momenta (let us go back to an N -dimensionalsystem for a second) can be taken as any combination of the α’s.


To satisfy Eq. (1.36), J has to be defined as [10]:

J =1

2π

∮Ω

p dq. (1.37)

We call the variable J the “action” variable, due to its resemblance to the definitionof the Action S =

∫Ldt =

∫p dq. Thus we have found the definition of the action

variable. But where have we made use of the Hamilton-Jacobi equation? Well, fromEq. (1.34) we learn that p is determined by α, that is, p = p(q, α) and we obtainedthis equation with Hamilton-Jacobi theory. Thus, the p in Eq. (1.37) is the p thatwe obtain with the Hamilton-Jacobi equation.

Now, what is the form of the new Hamiltonian? Since the starting Hamiltonian istaken to be time-independent, we have that the old and the new Hamiltonians havethe same value (cf. Eq. (1.31))

H

(q,∂W

∂q

)= E = K(J) (1.38)

and that the new Hamiltonian depends only on the new constant momentum (i.e.,the action variable) J . Now that we have the form of the new Hamiltonian, we useHamilton’s equations of motion to get

J = − ∂

∂θK(J) = 0 (1.39)

θ =∂

∂JK(J) = ω(J) (1.40)

which yield:J(t) = constant θ(t) = ω(J) t+ θ0. (1.41)

We can see that θ evolves linearly in time with frequency ω(J) (with initial phaseθ0). This method allows us to obtain the frequencies of the system without eversolving the dynamical problem (that is, obtaining (q, p)). More importantly, action-angle variables are the most natural choice to look into perturbations to an integrablesystem, since, as we will see, the action variables (by their property of being constant)can be taken to be the constants of motion, such as the angular momentum in arotating system or the energy in a simple harmonic oscillator. Such systems withconstants of motion are called integrable systems, and we will look into them in thenext section.

1.5 Integrable Systems

To introduce the idea of integrable systems, let us look at an example. Let us assumewe have a one-dimensional simple harmonic oscillator,

H(q, p) =p2

2m+

1

2kq2, (1.42)

1.5. Integrable Systems 13

where k is the spring constant, m is the mass of the body undergoing oscillation, andA is the maximum amplitude of the oscillation. We can obtain the total energy ofthe system by setting the momentum equal to zero. This will give us that the totalenergy of the system is E = 1

2kA2. The energy does not depend on time and is thus

a constant.Now, let us tie this result to the Hamilton-Jacobi equation. Using Eq. (1.38) we

obtain1

2m

(∂W

∂q

)2

+1

2kq2 = α1. (1.43)

Thus we have that the left hand side of Eq. (1.43) is equal to the constant α1. Wepreviously learned that the constant quantity (I use the instead of a since this is aone-dimensional system) of the harmonic oscillator is the energy. Thus the quantityα1 is the energy of the system E. Additionally, we learned that we can consider theconstants αi’s as the constants of integration for the Hamilton-Jacobi equation. Thus,the constant of integration for the one-dimensional harmonic oscillator is the energy.

The one-dimensional harmonic oscillator is an easy-to-grasp example of an in-tegrable system. Now, let us consider the more general case of an N dimensionalHamiltonian system. An integrable system is considered such if the Hamilton-Jacobiequation can be separated into N independent equations [9]. In our example above,the Hamilton-Jacobi equation was separable by virtue of the time-independence ofthe system. When the system is integrable, we can write Hamilton’s characteristicfunction as

W (q,α) =N∑k=1

Wk(qk,α), (1.44)

where α = (α1, α2, ..., αN). Once we separate the Hamilton-Jacobi equation, we needseparation constants. These constants are the αi’s. In our example above, the energyα1 = E was the separation constant. Let us remember that for time-independentsystems, Hamilton’s characteristic function W can be seen as a generator. Thismeans that, if W can be written as in Eq. (1.44), we have that the old momenta canbe written as

pk =∂Wk

∂qk. (1.45)

Each of the old momenta is then a function exclusively of one of the old coordinates:pk = pk(qk,α). Assuming that the motion of the system is periodic, we can constructa set of action variables as

Jk =1

2π

∮Ωk

pk(qk,α) dqk for k = 1, ..., N. (1.46)

Now that we have the new action variables in terms of the constant set of αi’s, wecan define the angle variables as

θk =N∑l=1

∂

∂JkWl(ql,α(J)) (1.47)


and again, solving Hamilton’s equations, we get the previously stated result

Jk(t) = constant θk(t) = ωk(J) t+ λk. (1.48)

We just obtained a set of N constant quantities. These constant quantities (or com-binations of them) constitute the set of integrals of motion of the system. To check ifa set of constants Ji are effectively integrals of motion, then the set of constants hasto satisfy the following:

[H, Ji] = 0. (1.49)

where [ , ] are the Poisson brackets11. Additionally, this set of constants have to bein involution [10], that is, they all have to satisfy

[Ji, Jk] = 0 for all i, k = 1, ..., N. (1.50)

Keep in mind that to obtain the integrals of motion of a system as action variableswe had to assume that Hamilton’s principal function W is separable (cf. Eq. (1.44)).The set of 2N variables given by Eq. (1.48) fully describes the evolution of the systemin phase space.

The motion in phase space of an integrable system is very unique, so let us men-tion some of its properties. By having N of the 2N variables constant, the motion isrestricted to an N -dimensional manifold. Additionally, we must remember that theangle variables have such a name because they describe periodic motion, and have theproperty that they increase by 2π after a complete rotation Ωk in their coordinate:2πδij =

∮Ωidθj, where Ωk is the path along which the motion of the kth degree of

freedom is periodic. This resembles the motion along an N -dimensional torus, calledN torus. To make this idea more clear, let us assume that we have an integrable sys-tem of 2 degrees of freedom. Since it is integrable, the system will have two integralsof motion, J1, J2. This means that the motion along phase space will be restricted toa 2-dimensional manifold. We can construct a 3-dimensional (i.e., regular) torus bytaking the two action variables (integrals of motion) as the radii of the torus, and theangle variables as the two angles of the torus. To picture it better refer to Fig. 1.2.

Now let us go back to a N torus. We have attached a meaning to the actionvariables Jk and to the angle variables θk, but it is still unclear what the ωk represent.Unsurprisingly, each ωk is the frequency at which each angle variable rotates as timeincreases. Now, as time increases, all of the angle variables increase by their respectiveωk. It should then be interesting to look into the different relationships that thefrequencies ωk can have among each other.

Let us consider again a system of 2 degrees of freedom. This system will clearlyhave two frequencies, ω1, ω2 corresponding to θ1 and θ2 respectively (see Fig. 1.2).We relate the two frequencies by obtaining their ratio:

ω1

ω2

= r, (1.51)

where r = r(J1, J2). The ratio r is called the rotation number, and can be eitherrational or irrational. If the rotation number is rational, then the trajectory along

11See Appendix A.


Figure 1.2: An integrable system with two degrees of freedom represented as a torus.

(a) (b)

Figure 1.3: Motion along the torus for rational r. The trajectory closes on itself andthus it does not densely cover the torus. In (a) an integrable system with r = 4 isshown. In (b) an integrable system with r = 1/4 is shown.

the torus will be periodic. In this case, we are able to write r as r = ab, where

a, b ∈ Z. The trajectory along the torus will close on itself after a revolutions ofθ1 and b revolutions of θ2. If, on the other hand, the rotation number is irrational,then the trajectory along the torus will be quasiperiodic [10]. This means that thetrajectory will never close on itself, and as time approaches infinity the trajectorywill come arbitrarily close to every point of the torus. To better undestand the newlyintroduced ideas, in Fig. 1.3a and 1.3b we have periodic systems with r equal to 4and 1

4respectively. In Fig. 1.4a, 1.4b, and 1.4c we have a quasiperiodic trajectory.

We can see that, as the number of revolutions around the torus (both in the θ1 andθ2 directions) increases, the trajectory does not close in itself.

Now, let us consider the general case where we have N degrees of freedom. Inthis case, if, for a given J, there exists an N -dimensional vector n whose entries arenonzero integers that satisfies the following:

n · ω(J) = 0, (1.52)


(a) (b) (c)

Figure 1.4: An integrable system with r = 2π. The motion is quasiperiodic andthus densely covers the torus. The system has done: (a) three revolutions, (b) tenrevolutions, (c) sixty revolutions in θ1.

then the orbit will close on itself. When the frequencies of the system satisfy Eq.(1.52) then we say that the frequencies are commensurable, or linearly dependent. Ifthe only n that satisfies Eq. (1.52) is the zero vector, then the orbit will never returnto itself and thus will densely fill the surface of the N torus, getting arbitrarily closeto every point. The ith degree of freedom will have done a complete revolution in atime t equal to its period, which is given by:

Ti =2π

ωi. (1.53)

1.5.1 Example: The Pendulum

The study of the pendulum is necessary because later on we will encounter systemsthat closely resemble the pendulum. The pendulum Hamiltonian is given by:

H =1

2Mp2 − U0 cos q (1.54)

where M is the moment of inertia M = ml2 where m is the mass and l is the lengthof the rope, U0 = mgl, and q is the angular displacement of the pendulum, as shownin Fig. 1.5. We start by obtaining the equilibrium points of the system. To get theequilibrium points, we use Hamilton’s equations of motion on Eq. (1.54) and set thetime derivatives of q, p equal to zero:

∂H

∂p

∣∣∣∣p=p0

=p0

M= 0 =⇒ p0 = 0 (1.55)

−∂H∂q

∣∣∣∣q=q0

= −U0 sin q0 = 0 =⇒ q0 = nπ for n ∈ Z (1.56)

To see if they are stable or not, we resort to the second derivative test for a functionof two variables; in our case, the function to be evaluated is the Hamiltonian. Thistest proceeds as follows [11]. We obtain the determinant of the Hessian matrix of the


Hamiltonian in Eq. (1.54):

D(q, p) = det

(∂2H∂q2

∂H∂q

∂H∂p

∂H∂p

∂H∂q

∂2H∂p2

)

= det

(U0 cos q 0

0 1M

)=U0

Mcos q (1.57)

The classification of stationary points is given by [11]:

If D(q0, p0) > 0 then the fixed point is an elliptic point. (1.58)

If D(q0, p0) < 0 then the fixed point is a hyperbolic point. (1.59)

As we will prove in section 3.2, elliptic points are stable and hyperbolic points areunstable. Since, by virtue of our system being periodic, the angle q0 corresponding ton even are all the same as are the angles corresponding to n odd, we separate thesetwo cases. Plugging in Eqs. (1.55) and (1.56) into Eq. (1.57) we get

D(2πn, 0) =U0

M(1.60)

D((2n+ 1)π, 0) = −U0

M(1.61)

Thus we get that points corresponding to q = 2πn with n even are stable and thepoints corresponding to n odd are unstable.

Now we want to obtain the natural frequency of the system ω0, which is thefrequency for small oscillations. Small oscillations correspond to oscillations in whichthe displacement q from stable equilibrium is small. This means that q = q0 +δq, withq0 = 2πn and δq is small. Plugging in this q into Eq. (1.54) and Taylor expandingwe have

H(2πn+ δq, p) =1

2Mp2 − U0 cos (2πn+ δq)

≈ 1

2Mp2 − U0

[cos (2πn)− δq sin (2πn)− δq2

2cos (2πn)

]=

1

2Mp2 − U0

[1− δq2

2

], (1.62)

and ignoring the constant term −U0 since it does not affect the motion of the system,we get

H(δq, p) =1

2Mp2 +

U0

2δq2, (1.63)

which is the Hamiltonian for a simple harmonic oscillator. The Hamiltonian is con-served, since it is time independent. We call the value of the Hamiltonian the energy,and we relabel it as H = E. This follows directly from Eqs. (1.38) and (1.43). To


obtain the frequency we do a change of variables to action-angle variables. Rememberfrom Eq. (1.37) that we need the momentum12. Solving for the momentum in Eq.(1.63) we get

p = ±

√2M

(E − U0

2δq2

)(1.64)

The integral∮pd(δq) has to be done over a complete cycle. The extreme values that

δq can have happen when the momentum is zero

δqmax =

√2E

U0

δqmin = −√

2E

U0

. (1.65)

Then, the action variable for the harmonic oscillator is

J =1

2π

(∫ δqmax

δqmin

√2M

(E − U0

2δq2

)d(δq) +

∫ δqmin

δqmax

−

√2M

(E − U0

2δq2

)d(δq)

)

=1

2π

(πE

√M

U0

+ πE

√M

U0

)(1.66)

= E

√M

U0

(1.67)

where, in the first line, we used the positive momentum in the first integral and thenegative momentum in the second integral. Solving for E in Eq. (1.67) we get

E = J

√U0

M(1.68)

and by Eq. (1.38) we have that

K(J) = J

√U0

M. (1.69)

Eq. (1.69) is the Hamiltonian for the harmonic oscillator in action-angle variables.To obtain the frequency we use Hamilton’s equations of motion

ω0 = θ

=∂

∂JK(J)

=

√U0

M. (1.70)

Thus the frequency for small oscillations of the pendulum is given by Eq. (1.70).The value given to the energy E determines qualitatively the motion of the pendu-

lum. There are three cases of interest to us: E < U0, E = U0, and E > U0. Fig. 1.6

12We also relabel q → δq in Eq. (1.37).


Figure 1.5: The pendulum system. The coordinate variable q can be clearly seen.

Figure 1.6: Phase space of the pendulum Hamiltonian for various values of E. Thearrows on the curves indicate the direction of the flow.

shows a plot of the phase space trajectory of the pendulum Hamiltonian for differentvalues of E. For E < U0 the motion is bounded to a region of phase space and is anoscillation. Notice that in this case the momentum p is equal to zero for some timet. For E > U0 the pendulum never stops moving (i.e., the momentum is never zero).This kind of motion is a rotation. Notice that both of these motions are periodic inq. Finally, for E = U0 the motion is such that the period becomes infinite [9]. Inphase space, this motion corresponds to the separatrix. The separatrix is the frontierbetween oscillatory and rotational trajectories. The motion near the separatrix isunstable, in the sense that a small perturbation can change the system from rotatingto oscillating and vice versa. The motion in phase space along the separatrix can beobtained by setting E = U0 in Eq. (1.54) and solving for p:

psx = ±√

2MU0(1 + cos q), (1.71)

where the subscript sx is used to refer to the motion along the separatrix. Using the


trigonometric identity cos q2

=√

1+cos q2

, we rewrite Eq. (1.71) as

psx(q) = ±√

4MU0 cos

(q

2

)= ±2Mω0 cos

(q

2

)(1.72)

= ±∆psx cos

(q

2

), (1.73)

where we used Eq. (1.70) in the second line. We define

∆psx = 2√MU0 (1.74)

as the half-width of the separatrix. The positive and negative signs correspond to theupper and lower branches of the separatrix respectively.

Additionally, we notice that the momentum will be zero at the same point atwhich the upper and lower branches of the separatrix intersect. We can get the valueof q, which we denote by q0, for which this happens by setting psx(q0) = 0:

±2ω0 cos

(q0

2

)= 0 =⇒ q0 = (2n+ 1)π for n ∈ Z. (1.75)

This point in phase space corresponds to the hyperbolic fixed point we found in Eq.(1.61). Thus the points of intersection of the branches of the separatix are hyperbolicfixed points.

We are now going to focus on the interval θ ∈ [−π, π], that is, between the fixedpoints corresponding to n = −1 and n = 0 in Eq. (1.75). The time evolution ofthe angle q(t) gives us additional information about the separatrix. Using Hamilton’sequations of motion with H(q, p) given by Eq. (1.54), we get

d

dtqsx =

psxM

(1.76)

which, after plugging in Eq. (1.72) and integrating [12], yields

qsx(t) = 4 arctan (eω0t)− π. (1.77)

On the separatrix the angle approaches ±π as t→ ±∞. In phase space, this meansthat the motion along the separatrix is confined between −π and π, that is, q ∈(−π, π). We can generalize this result and say that q ∈ [(2n+ 1)π, (2n+ 3)π] for anyinteger n.

Using qsx(t) we can obtain psx as a function of time. Replacing Eq. (1.77) in Eq.(1.73) we get

psx(t) = ±∆psx cos

(qsx(t)

2

)= ±∆psx cos

(4 arctan (eω0t)− π

2

)= ±∆psx sech (ω0t) (1.78)


So far, we have pointed out different properties of the separatrix of the pendulumHamiltonian: the intersection of its branches correspond to hyperbolic fixed points,the motion along it is confined to q ∈ [(2n+ 1)π, (2n+ 3)π], its period is infinite, andthe motion near the separatrix is unstable. As a consequence of the third propertylisted we can deduce that the period of a trajectory near the separatrix approachesinfinity as the initial conditions of the trajectory approach the separatrix.

The pendulum Hamiltonian is of utmost importance in the study of integrablesystems with many degrees of freedom that are slightly perturbed. In particular, themotion nearby the separatrix will be important to understand the first manifestationsof chaos. The solution to the pendulum Hamiltonian is easier to obtain using action-angle variables, and we proceed to that next.

1.5.2 The Solution to the Pendulum in terms of Action-Angle Variables

Since the system is periodic for all values of H, switching to action-angle variablesseems to be the natural choice to get the solutions of the system. Solving for p in Eq.(1.54) and plugging it into Eq. (1.37) we get

J(H) =1

2π

∮pdq

=1

2π

∮ √2M(H + U0 cos q)dq

= 4

(1

2π

∫ qmax

0

√2M(H + U0 cos q)

)dq

=2

π

∫ qmax

0

√2M(H + U0 cos q)dq (1.79)

where in the third line we used the fact that the integral over a whole cycle is equalto four times the integral from 0 to qmax, as can be seen in Fig. 1.6. With the helpof elliptic integrals, Eq. (1.79) can be written as [9]

J(H) =√MU0

8

π

E(κ)− (1− κ2)K(κ) if κ ≤ 112κE( 1

κ) if κ ≥ 1,

(1.80)

where K and E are the complete elliptic integrals of the first and second kind respec-tively, defined as

K(κ) =

∫ π2

0

dθ√1− κ2 sin2 θ

, (1.81)

E(κ) =

∫ π2

0

√1− κ sin2 θdθ, (1.82)

and the quantity κ is a dimensionless parameter that relates the energy of the systemH and the maximum potential energy U0 in the following way

κ2 =H + U0

2U0

(1.83)


and satisfies

κ

> 1 for rotations

= 1 for motion along the separatrix

< 1 for oscillations

(1.84)

Now that we have the action variable, we can obtain the frequency for any value ofH by using Eq. (1.40) and writing ω = 1

dJ/dH[9]:

ω(κ) =π

2ω0

1

K(κ)if κ ≤ 1

2 κK(1/κ)

if κ ≥ 1(1.85)

where ω0 is the frequency for small oscillations defined in Eq. (1.70). We know thatclose to the separatrix, the frequency approaches zero. We can take the limit of ω(κ)as κ→ 1 to obtain the frequency in the vicinity of the separatrix [9]

limκ→1

ω(κ) = ωsx(κ) =π

2

ω0

ln

[4√

1− κ2

] if κ ≤ 1

ω0

ln

[4√

κ2 − 1

] if κ ≥ 1(1.86)

Note that ωsx is not the frequency at the separatrix, but rather in its vicinity. Tosimplify Eq. (1.86) we solve for the terms inside the logarithms

√1− κ2 =

1√2

√U0 −HU0

(1.87)

and similarly

√κ2 − 1 =

1√2

√H − U0

U0

(1.88)

Notice that both Eqs. (1.87) and (1.88) depend on the difference between total energyand potential energy |H−U0|. In order to have only one expression for the frequencyso that we do not have to take the oscillation and the rotation cases separately, wedefine a quantity w as

|w| =∣∣∣∣H − U0

U0

∣∣∣∣ =

∣∣∣∣U0 −HU0

∣∣∣∣ (1.89)

where |w| goes to zero as we approach the separatrix. Using |w| we rewrite Eqs.(1.87) and (1.88) as

√1− κ2 =

1√2

√|w| (1.90)

√κ2 − 1 =

1√2

√|w| (1.91)


Since the two terms in Eq. (1.86) differ only by the term in the denominator and wejust found that they can be set to be equal, we write the frequency of oscillations inthe vicinity of the separatrix, ωsx(κ) in Eq. (1.86), as

ωsx(κ) =πω0

ln

(32

|w|

) (1.92)

and the period is given by

Tsx =2π

ωsx=

2

ω0

ln

(32

|w|

). (1.93)

Notice that the period goes to infinity as |w| approaches zero, or, equivalently, as weapproach the separatrix.

Chapter 2

Mappings

The study of a dynamical system can be made considerably easier by introducingmappings.

2.1 Surface of Section

The time evolution of a Hamiltonian system is given by Hamilton’s equations ofmotion. If we had a system with one degree of freedom (N = 1), we can easily visualizeits evolution in phase space, since such a system would only have one momentumvariable and one coordinate variable. However, this visualization process becomescomplicated for any system with N > 1, that is, more than one degree of freedom.Even for a system with two degrees of freedom, whose evolution in coordinate spacerequires no more than three dimensions (q1, q2, t) and thus can be easily visualizedusing a computer and even a flat surface like a board, the visualization of phase spaceevolution of such a system happens in 4 dimensions. This is, for obvious reasons,problematic. To solve this problem, we introduce the surface of section. Imagine wehave a Hamiltonian system with 2 degrees of freedom

H(q1, q2, p1, p2) =1

2mp2

1 +1

2mp2

2 + V (q1, q2). (2.1)

If the energy is constant, then the flow in phase space will lie in a 2N − 1 = 3dimensional manifold. In this 3 dimensional manifold we can fix one of the variables,say, q2. So by fixing this coordinate we have a 2 dimensional manifold defined bysetting q2 ≡ constant = qc. This manifold is the surface of section. The “normal”Hamiltonian flow takes place in the 3 dimensional manifold. This flow, in turn,crosses the surface of section. By recording all the crossings (in one direction) theHamiltonian flow has with the surface of section, we construct a map that still hasinformation from the original system but is much easier to work with. Figs. 2.1 and2.2 show how to construct a surface of section.

26 Chapter 2. Mappings

Figure 2.1: By setting our variable q2

as a constant we define the surface ofsection in the (p1, q1) plane.

Figure 2.2: The surface of section forthe Hamiltonian flow in Fig. 2.1.

2.2 Mappings

The surface of section is a powerful tool to simplify the study of the dynamics of asystem. In order to witness its usefulness, let us recall our discussion of integrablesystems1. Consider the trajectory on a torus shown in Fig. 1.4c. It should be clearthat it is hard to deduce anything from the system, except that the ratio of itsfrequencies r (cf. Eq. (1.51)) is irrational. In order to better see what is happeningwith the dynamics of the system, we make use of the surface of section by letting θ1

be constant (cf. Fig. 1.2). This allows us to only see the crossing of the trajectorywith a plane defined by θ1. Figs. 2.3a through 2.4c correspond to the surfaces ofsection of Figs. 1.3a through 1.4c. It is clear that visualization of the dynamics ofthe system is much easier this way.

For a periodic system (e.g. Figs. 2.3a and 2.3b), the surface of section consistsonly of a discrete number of points. This is because the trajectory on the torus closeson itself and thus it crosses the surface of section only a discrete number of times.For a quasiperiodic system (e.g. Figs. 2.4a, 2.4b and 2.4c), the surface of section isfilled up by an infinite number of points since the trajectory on the torus never closeson itself.

The trajectory that intersects the surface of section is given by Hamilton’s equa-tions of motion (θ1 = ω1(J1, J2) = ∂H

∂J1and θ2 = ω2(J1, J2) = ∂H

∂J2), so whatever

information is on the surface of section originated from these equations. It seemsconvenient to think of the intersections on the surface of section as a mapping thatobeys Hamilton’s equations of motion. Let us define a two-dimensional mapping Mas

M(Jn, θn) = (Jn+1, θn+1). (2.2)

The way a mapping works in practice is as follows: the initial conditions (J0, θ0) arepicked and plugged into Eq. (2.2) to obtain the values of (J1, θ1).Then these new

1The action angle variables used in this section are defined on a torus as in Fig. 1.2 and the ratioof the frequencies is defined as r = r(J1, J2) = ω1/ω2 (cf. Eq. (1.51)).

2.2. Mappings 27

(a) (b)

Figure 2.3: Surface of section with θ1 constant for an integrable system with (a) r = 4for only one initial condition, and (b) r = 1/4 for two initial conditions denoted by •and N.

(a) (b) (c)

Figure 2.4: Surface of section with θ1 constant for an integrable system with r = 2π.The system has done (a) three revolutions, (b) ten revolutions, and (c) six hundredrevolutions in θ1.


values are plugged into the mapping equations to obtain the values of (J2, θ2). Thisis repeated until desired.

Since we are concerned exclusively with Hamiltonian systems, we expect a map-ping that describes such systems to have some property analogous to the invariance ofphase space volume. We are talking about a preservation of area, and mappings thatdescribe Hamiltonian systems are area-preserving mappings. To prove this, recallthat a Hamiltonian flow is nothing more than a canonical change of variables thattranslates the system in time with the Hamiltonian as the generating function of thistranslation. So we can think of the mapping (Jn, θn) → (Jn+1, θn+1) as a canonicaltransformation2. From Eq. (1.26) it follows that a mapping M(Jn, θn) = (Jn+1, θn+1)is area preserving if and only if∣∣∣∣ det

(∂(Jn+1, θn+1)

∂(Jn, θn)

)∣∣∣∣ = 1. (2.3)

Hence all surface of section maps derived from a Hamiltonian system satisfy Eq. (2.3).

2.2.1 The twist map, resonant tori, and nonresonant tori

Let us obtain the equations that describe the mapping corresponding to the figuresabove. We know that both J1 and J2 are constants and that the energy of the systemis E = E(J1, J2). Now set the surface of section at some θ1. This makes θ1 a constantand makes the mapping describe the dynamics of the variables (J2, θ2). By Eq. (1.53)the trajectory intersects the surface of section every ∆t = 2π

ω1. In this time θ2 has

increased by ∆tω2 = 2πr where r is the rotation number (cf. Eq. (1.51)). Usingthis result and the fact that J1 = J1(E, J2) to make r a function of J2 only, theequations that describe the mapping corresponding to Figs. 2.3a through 2.4c (fortheir respective values of r) are3

Jn+1 = Jn (2.4)

θn+1 = θn + 2πr(Jn+1) mod 2π. (2.5)

If we assume that r increases with Jn+1, then the map just derived is called the twistmap. The twist map is a very simple mapping and is helpful to understand some ofthe basic properties of a mapping. We define the twist map T as

T (Jn, θn) =

(Jn+1

θn+1

)=

(Jn

θn + 2πr(Jn+1)

). (2.6)

2Remember that if a canonical transformation exists, then there is a generating function thatgenerates this canonical transformation [3]. It follows that the equations that describe a mappingcan also be obtained with a generating function alone.

3Do not confuse the action angle variables of the original system (J1, θ1, J2, θ2) with the mappingvariables (Jn, θn). Both have numerical subindices; however, the former correspond to the variablesthat live in full phase space, whereas the latter correspond to the mapping of the original variables(J2, θ2) into the surface of section at some constant θ1. If you decide to refer to, say, J2, you needto specify if you are talking about the original action variable that lives in full phase space, or if youare talking about the value of the Jn for the second iteration of the map.

2.2. Mappings 29

Figure 2.5: The twist map for different values of r. For r = 1/4 two trajectories areplotted, each one with different initial conditions from the other. For r = 1/6 onlyone trajectory is plotted. For r irrational, the trajectories correspond to only oneinitial condition. The green arrows indicate the location of the invariant resonanttori and the blue arrows indicate the location of the nonresonant tori. As indicated,the nonresonant tori are densely covered by a single trajectory.

First of all, note that the action variable is constant throughout iterations of the map.Similarly, r is a parameter determined by Jn+1, but since the action is constant wehave that r(Jn+1) = r(J0), with J0 the initial action. Since r determines if the motionis periodic or quasiperiodic by virtue of Eq. (1.51), and r is determined by J0, thenit follows that J0 determines if the motion is periodic or not. A picture with variousiterations of the twist map for different values of r(J0) is given in Fig. 2.5.

Notice how for a given J0 the trajectory lies on a curve. We call these curvesinvariant curves. In the case of the twist map the invariant curves are circles ofradius J0. For r irrational, a single trajectory densely covers the invariant curve.For r rational, a single trajectory covers only a discrete set of points that lie on theinvariant curve4. If we write r as r = a/b, then the trajectory will consist of b discretepoints. These discrete points are said to be fixed points of period b of the map T . Fig.2.5 shows the cases for r both rational and irrational.

Let us recount what we have done so far. Recall that the twist map was con-structed by using a surface of section placed at a constant θ1 in phase space5. Wethen defined the invariant curves as the cross section of the tori in phase space.The tori corresponding to invariant curves with r rational are called resonant tori.Similarly, the tori corresponding to invariant curves with r irrational are called non-resonant tori. Invariant curves that live on the surface of section give us informationabout the tori that live in phase space. Usually invariant curves are called resonantand nonresonant tori, and similarly invariant curves are sometimes called invariant

4It is the set of points that define the invariant curve, not the other way around. From thisperspective, let us pick an initial J0 such that r is rational, and let us relabel it as J0 ≡ Jp. The setof all points (Jp, θ0) where θ0 ∈ [0, 2π] defines the invariant curve.

5θ1 is shown in Fig 1.2.


tori.So far the treatment of resonant and nonresonant tori has been specific to the

twist map. Now let us translate these ideas to a system with N degrees of freedom.Let ω(J) be the vector whose entries are the frequencies of the system. The resonancecondition6 for such a system is7

n · ω(J) = 0, (2.7)

where n is some vector whose entries are integers. The J for which Eq. (2.7) holdsfor nonzero vector n define the resonant tori of the unperturbed system [13]. Whenresonance happens there is a big exchange of energy between degrees of freedom8.

The nonresonant tori are defined by the J for which Eq. (2.7) holds only for thevector n whose entries are all zero, ni = 0 for i = 1, ..., N . In the language of linearalgebra, this means that the entries of the vector ω(J) are linearly independent.We will encounter once we get to the KAM theorem that the condition for linearindependence of the frequencies can also be written as

det

∣∣∣∣∂ω∂J

∣∣∣∣ 6= 0. (2.11)

2.3 Nonintegrable systems

A system of N degrees of freedom is nonintegrable if it has less that N integralsof motion. Although integrable systems are nice since we know that we can (in

6This condition was already introduced when treating integrable systems (cf. Eq. (1.52)).7Eq. (2.7) is exactly the same condition we encountered for the twist map. If this is not clear,

then let ω = ω(J1, J2) =

(ω1

ω2

)=

(θ1

θ2

)where the angles are shown in Fig. 1.2. Then n · ω =

n1ω1 +n2ω2 = 0→ ω1

ω2= −n2

n1= r = r(J2). We can clearly see that there is a J2 such that r is made

up of a division of two integers, and thus it is rational. The different J2 for which r is rational definethe resonant tori. Additionally, r depends only on J2 because we know that the energy is constant,E = E(J1, J2). This allows us to write J1 as J1(E, J2) and then r(J1, J2) = r(J1(E, J2), J2) =r(E, J2) = r(J2).

8This is easily seen by considering a simple example. Let us consider a driven simple harmonicoscillator

x+ ω2x = A cos (Ωt), (2.8)

whose solution is

x(t) = a sin (ωt+ φ) +A

ω2 − Ω2cos (Ωt). (2.9)

If we have that ω−Ω = 0, or, equivalently n ·ω ≡ (1,−1) ·(ω,Ω) = 0 then the system is in resonance,and the second term of the solution blows up. However, for the limit where ω → Ω, we can rewritethe solution so that it has the form [10]

x(t) = a sin (ωt+ φ) +A

2ωt sinω. (2.10)

Notice that in Eq. (2.10) the second term is linear in t, which means that the amplitude x(t) growslinearly in time without bound. We call such terms secular terms. This growth represents the bigexchange of energy between degrees of freedom that occurs when the system is in resonance.

2.3. Nonintegrable systems 31

theory) solve them or at least write the solution as integrals, chaos only manifests innonintegrable systems. Integrable systems provide us with a basis to study systemsthat are nonintegrable. For example, in an elementary classical mechanics course, themotion of the Earth around the Sun is studied as a two-body system, with the Sunas the only source of the potential that the Earth is subject to. However, in realitywe need to consider the effects of other celestial bodies, such as the Moon. Since thepotential due to the Moon is way smaller than the potential due to the Sun, we canconsider the potential due to the Moon as a perturbation on the original two-bodysystem. In what follows we consider only periodic systems because once we want tostudy the manifestations of chaos, we will be only interested in such systems.

Formally, we write the Hamiltonian of a nonintegrable system as the sum of anintegrable Hamiltonian H0 and a nonintegrable small perturbation H1 periodic in θ:

H(J,θ) = H0(J) +H1(J,θ) (2.12)

where H1(J,θ + 2π) = H1(J,θ). Since the perturbation H1 is periodic in the anglevariable, we can write it in terms of its Fourier series

H1(J,θ) =∑n

H1,n exp(in · θ) (2.13)

where the H1,n are the Fourier coefficients, the components of the vector n are allintegers, and the term n = (0, 0, ..., 0) is excluded from the sum9. The study ofnonintegrable systems is the subject of perturbation theory, whose goal is to findsolutions to these systems. This task, as easy as it sounds, is not simple. Let us takea look at canonical perturbation theory and it’s main drawback.

Canonical perturbation theory’s main objective is to find a generating functionW (I,θ) such that, using Eq. (1.14) we obtain a new Hamiltonian K(I) that is in-tegrable. I will not lay out the details of how this is done. The interested readeris referred to Chapter 3 of [10] for a thorough treatment of how to obtain thegenerating function that gives the new integrable Hamiltonian. The basic gist isthat we assume the generating function can be written as a power series W (I,θ) =I · θ + εW1 + ε2W2 + .... Then, one can plug this into the Hamilton-Jacobi equation,and equating terms with equal powers of ε, one obtains (in theory) the generatingfunction. A problem arises when solving for W1, since when one solves for W1, onegets

W1(I,θ) = i∑n

H1,n

n · ω0(I)exp(in · θ), (2.14)

where ω0 is the vector whose components are the frequencies of all the degrees offreedom. The problem with Eq. (2.14) is that if the system is at a resonance then ω0

satisfies Eq. (2.7) and the series for W diverges. Also, if the frequencies are close toresonance, Eq. (2.14) becomes very large. If the frequencies do not obey Eq. (2.7),

9The term n = (0, 0, ..., 0) is excluded so that there is not a constant term in H1. If there was aconstant term in H1, the assumption that the perturbation is purely periodic would not be satisfied.


we can always find an n big enough to make the denominator arbitrarily small10.This problem is known as the small divisors problem, and it was a fundamental

problem of classical mechanics. It was believed that there were no solutions thatdescribed the long-term behavior of perturbed systems [14]. There is, however, a the-orem that tells us that under certain conditions, solutions do exist in certain regionsof phase space. This theorem is known as the KAM theorem, and the solutions cor-respond to surviving nonresonant tori. The method used to find solutions accordingto the KAM theorem is not that of canonical perturbation theory, but rather that ofsuperconvergent perturbation methods [9].

Remember that an integrable system can have two kinds of motion: periodic andquasiperiodic. The motion happens on resonant or nonresonant tori respectively.Under perturbation, we can see from Eq. (2.14) that what happens to the resonanttori is unclear. The Poincare-Birkhoff theorem deals with the resonant tori, while theKAM theorem tells us what happens to nonresonant tori.

As an example of a perturbed system, if we perturb the twist map we obtain

Jn+1 = Jn + εf(Jn+1, θn) (2.15)

θn+1 = θn + 2πr(Jn+1) + εg(Jn+1, θn) mod 2π. (2.16)

where one clearly sees the perturbation terms, f, g.

2.3.1 Mappings for time-periodic systems

Let us consider a time-dependent Hamiltonian with only one pair of conjugate vari-ables that is periodic in time

H(J, θ, t) = H(J, θ, t+ T ) (2.17)

where T is the period of the system. For such a system we can construct a map byrecording the values of (J, θ) at times nT where n ∈ Z. In practice, if we let the initialtime be t0 = 0, our map will consist of the points (J(t = nT ), θ(t = nT )) = (Jn, θn).It should go without saying that Eq. (2.3) also holds for these mappings, since theystill originate from a Hamiltonian. The generating function F2 that gives us themapping as a canonical transformation is now going to be time-dependent (cf. Eqs.1.12 and 1.13).

The Hamiltonian for a time periodic system is written as composed of two parts:an integrable part whose solutions we knowH0(J), and a (small) perturbationH1(J, θ, t)that is periodic in t

H(J, θ, t) = H0(J) +H1(J, θ, t). (2.18)

A system that only has (J, θ, t) as its variables is said to be a one-and-a-half degreeof freedom system, N = 3/2.

10To see how this is true, let us work with a system with two degrees of freedom. Let m1ω1 +m2ω2 = r. Solving for ω1

ω2we have ω1

ω2= r

m1ω2− m2

m1. If we let r

ω2 m2 then the second term will

dominate and ratio of the frequencies will be approximately rational ω1

ω2≈ −m2

m1.

2.4. Introducing the Standard Map 33

Figure 2.6: The kicked rotor. A delta function kicks the rotating mass every timet = n in the direction of the arrows, regardless of where the mass is.

2.4 Introducing the Standard Map

The standard map is a map that describes the dynamics of certain time-periodicsystems. One such system is the delta-kicked rotor, and a derivation of the standardmap will be given using this system for the reader to get an intuitive sense of theorigin of the map. This particular rotor consists of a mass attached to the end of arod, while the other end is bolted to a frictionless pivot. Gravity plays no role in thissystem. So far the system is free to rotate with no loss of energy11. To make thingsmore interesting, let us introduce a force that acts on the rotor. Let this force be aseries of delta kicks that act only in one direction, F = F0

∑∞n=−∞ δ(t/T − n)(−y).

This is pictured in Fig. 2.6. We know from Newton’s Second Law for rotations that

I θ = |R× F| = RF0T sin θ∞∑

n=−∞

δ(t− nT ). (2.19)

11Think of a free particle (H(J) = J2/2m) whose momentum will not change until some potential(and hence a force) perturbs it. For our case, the rotor-mass body will rotate forever with the sameangular momentum until we perturb it somehow.


Writing θ as θ = dω(t)dt

and defining the constants in front of the sine term as A ≡ RF0TI ,

we can write the equations of motion for the delta-kicked rotor:

dω(t)

dt= A sin θ

∞∑n=−∞

δ(t− nT ) (2.20)

dθ(t)

dt= ω(t) (2.21)

where Eq. (2.21) comes from the definition of angular velocity. In order to make thefinal result look clean, let us make the equations of motion dimensionless. For thetime variable, let the new variable a be defined as t = aT . Plugging this into bothequations of motion we get

dω(a)

da= A sin θ

∞∑n=−∞

δ(a− n) (2.22)

dθ(a)

da= ω(a)T. (2.23)

Similarly, for the angular velocity we define the new variable ν as ω = ν/T . Pluggingthis into the equations above and defining the dimensionless constant12 K ≡ AT weget

dν(a)

da= K sin θ

∞∑n=−∞

δ(a− n) (2.24)

dθ(a)

da= ν(a). (2.25)

Now that we have the equations of motion, we need to get a mapping out of them.We are going to do this by using the special properties of the Dirac delta functions inEq. (2.24)13. For notational convenience, we are going to denote a function at “time”a = n as f(a = n) ≡ fn. Let the nth kick happen immediately after n. Then thedelta kick happens in the time interval a ∈ [n, n], where n = (n + ε) and ε 1. Letus go ahead and integrate Eq. (2.20) in this interval:∫ νn

νn

dν =

∫ n

n

K sin θ∑n

δ(a− n) da

νn − νn = K sin θn. (2.27)

12A has dimensions of 1[t] as the reader can easily prove.

13We are particularly interested in the following property of the Dirac delta function:∫ b

a

f(x)δ(x− x0)dx =

f(x0) if a ≤ x0 ≤ b0 if x0 > b or x0 < a

(2.26)


This is starting to look like a map, although a map goes from a set of variables at nto the same set of variables at n + 1. To get νn+1 we need to find its relationship toνn. For this purpose we integrate Eq. (2.20) in the interval a ∈ [n, n+ 1]∫ νn+1

νn

dν =

∫ n+1

n

K sin θ∑n

δ(a− n) da

νn+1 − νn = 0 =⇒ νn+1 = νn. (2.28)

Plugging in Eq. (2.28) into Eq. (2.27), we have

νn+1 = νn +K sin θn. (2.29)

Eq. (2.29) tells us that the angular velocity ν changes by a value of K sin θn afterthe nth kick, and Eq. (2.28) tells us that it remains constant between kicks. We nowhave half of our mapping. What about the other half? Well, for the angle variable θwe know that it has to be continuous everywhere. This amounts to the requirementthat we cannot have a particle teletransporting around. Then, when the delta spikeacts on the system in the interval a ∈ [n, n] we require that

θn = θn. (2.30)

Now we know what happens to θ during the delta kick: nothing. Since nothingexciting happens to θ during the only exciting part of a cycle, θ will spend the timeuntil the next delta kick rotating with constant angular velocity. We can see howthis is true by solving Eq. (2.25) with ν(a) ≡ ν constant: θ(a) = ν a + θ0. If wefocus our attention on the interval a ∈ [n, n + 1] then this equation is valid betweenθn and θn+1, and we require θ0 = θn and ν = νn+1. Also, since the amount of timebetween θn and θn+1 is a = 1 (or, in t, t = T ) we have that for θ(a = n + 1) ≡ θn+1

the equation describing it is

θn+1 = θn + νn+1. (2.31)

Eqs. (2.29) and (2.31) form the mapping denominated standard map. The importanceof this map lies both in its ubiquity and its simplicity. It is clearly nonlinear thanksto the sine term, it is two-dimensional which makes it easy to visualize, and both ofthe coordinates can be restricted to live between 0 and T , the period of the kickingterm. Since θ is an angle, I let the period of the kicking be T = 2π, so that we cantake the both coordinates as mod 2π.

2.4.1 Hamiltonian Formulation of the Standard Map

Now that we have an intuitive sense of how the standard map can be derived, letus take a more formal approach. Let us take a free particle and add a periodicperturbation to it

H(J, θ, t) =J2

2+H1(J, θ, t) (2.32)


where H1(J, θ, t) = H1(J, θ, t + T ), the frequency of the perturbation is Ω = 2π/T ,and H1 is small. Let our perturbation have the form

H1(J, θ, t) =A

Tcos θ δT (t) (2.33)

where δT (t) = T∞∑

n=−∞

δ(t− nT ). Then the Hamiltonian of the standard map can be

written as

H(J, θ, t) =J2

2+A

Tcos θ δT (t). (2.34)

That Eq. (2.34) is in fact the Hamiltonian for the standard map becomes evidentonce we get the equations of motion, which are

J = A sin θ∞∑

n=−∞

δ(t− nT ) (2.35)

θ = J. (2.36)

By comparing Eqs. (2.35) and (2.36) with Eqs. (2.20) and (2.21), we can see thatEq. (2.33) is in fact the Hamiltonian corresponding to the standard map. Rememberthat we introduced dimensionless variables when deriving the standard map. In thiscase, we keep the transformation for time t = aT and we add the one correspondingto the action variable J = I/T . The corresponding equations are

In+1 = In +K sin θn mod 2π (2.37)

θn+1 = θn + In+1 mod 2π, (2.38)

which are, again, the pair of equations corresponding to the standard map. Forthe rest of the thesis, we are going to use Eqs. (2.37) and (2.38). The standardmap is plotted in Figs. 2.7-2.9. The plots of the standard map are not usuallygiven for a single pair of initial conditions. Instead, one picks a bunch of initialconditions and iterates the map as many times as one desires. In the plots includedwe iterate the standard map for various numbers of initial conditions, each withdifferent perturbation parameter K.

The standard map can be obtained in a more straightforward manner by using agenerating function F2 and Eqs. (1.12) and (1.13). The F2 that gives us the standardmap is

F2(In+1, θn) = In+1θn +1

2I2n+1 +K cos θn. (2.39)

Since the standard map can be obtained from a generating function, it means thatthe map is canonical and thus area preserving.

Notice the similarity between Eqs (2.37), (2.38) and Eqs. (2.15), (2.16). Bycomparing them we can think of the standard map as the perturbed twist map withεf = K sin θn, g = 0, and 2πr = In+1. From this comparison we obtain that therotation number for the standard map when the system is unperturbed is equal to


(a) (b) (c)

Figure 2.7: 1000 iterations of the standard map for only one pair of initial conditions,(I0 = π, θ0 = π + .3). The value of K was: for (a) K = 0, for (b) K = 0.5, and for(c) K = 2.5.

(a) (b) (c)

Figure 2.8: 10000 iterations of the standard map for four pairs of initial conditionsall with θ0 = π + .3 and each individual pair with I0 = 2π/5, 4π/5, 6π/5, 8π/5. Eachinitial condition is marked in the plot. The value of K was: for (a) K = 0, for (b)K = 0.5, and for (c) K = 2.5.


(a) (b) (c)

Figure 2.9: 1000 iterations of the standard map for thirty pairs of initial conditionsall with θ0 = π + .3 and K = 2.5 and each individual pair with I0 = 2πn

31where

n ∈ [1, 30]. Each initial condition is marked in the plot. The value of K was: for (a)K = 0, for (b) K = 0.5, and for (c) K = 2.5.

rSM = In+1/2π. For us “unperturbed” means that K = 0. We can then writeIn+1 = In = ... = I0. Thus the rotation number for the standard map is14

rSM(I0) =I0

2π. (2.40)

The rotation number then only depends on I0. The motion is periodic if I0 = 2π ab,

where a, b are integers with no common factors. In that case we can write the rotationnumber as rSM = a/b and the number of fixed points is given by b. Notice how inFig. 2.7a the initial condition is J0 = π, which, plugging into Eq. (3.18) we get thatb = 2, and, effectively, two fixed points are shown. Similarly for the case in Fig. 2.8awe expect 5 fixed points given that I0 = n2π/5 with n integer, and we do get the 5fixed points.

By convention we usually map the twist map with the variable Jn as a radiusand θn as an angle. Even though the standard map is the twist map with an addedperturbation, it is customary to plot the standard map with the variable In as they-axis and the θn variable as the x-axis. To show how these two options of plottingaffect our visualization of the map, we show both in Fig. 2.10.

We can obtain the fixed points of the standard map. We are interested only inthose fixed points that are valid for K = 0 and K & 0. I will not prove it, but onlythe fixed points of period 1 and period 2 satisfy this requirement [15].

To obtain the period 1 fixed points, we set In+1 = In and θn+1 = θn + 2πm with

14The denominator of Eq. (3.18) is determined by the period of the variable θn. Some authorsprefer to write θn as mod 1 in Eq. (2.38), and in that case the denominator of the rotation numberwould be 1.


(a) In, θn as Cartesian variables (b) In, θn as polar variables

Figure 2.10: Two different options for plotting the standard map. The preference fora visualization as in (a) is evident by noticing that the islands and invariant curvesare easier to see in (a).

m an integer15. This gives us

In = In +K sin θn

arcsin 0 = θn

θn = πm (2.41)

and

θn+1 = θn + 2πm = θn + In+1 =⇒ In+1 = In = 2πm (2.42)

So, taking into account that the standard map is mod 2π, the period 1 fixed pointsare

(I0, θ0) = (0, 0) (2.43)

(I0, θ0) = (0, π) (2.44)

We have finally introduced the main object of study of this thesis. There is a richcontent in the pair of equations that describe the standard map that we will soonsee and as it should be evident from how complex the plots in this section are. It isimportant to not forget about the Hamiltonian, since our study of the standard mapcannot go far without it. Note that the Hamiltonian for the standard map has theform of an integrable part plus a perturbation. Since this kind of Hamiltonian is anonintegrable Hamiltonian we need to study some general properties of this kind ofHamiltonian and we will see how these properties manifest in the standard map aswe move on.

15Since θn is mod 2π, it has the same value after 2πn, hence any value of θn will be same afterone iteration if we add any integer multiple of 2π.

Chapter 3

Phase Space Topology ofNonintegrable Systems

As we saw in section 2.3, the solutions for long term behavior of nonintegrable systemscannot be obtained by usual canonical perturbation methods. This problem wasattributed to the fact that the series that would give us a solution diverges due tointernal resonances of the system. It was not until recently that, thanks to the workof Kolmogorov, Arnold, and Moser, solutions for integrable systems are known toexist far from resonance. This is known as the KAM theorem. As for what happensnear resonance, Poincare discovered that the motion near hyperbolic fixed points is ofa very wild nature, and Birkhoff demonstrated that a resonant torus does not surviveunder perturbation but in its place elliptic and hyperbolic fixed points appear. Thisis known as the Poincare-Birkhoff theorem. Both theorems just mentioned will befleshed out in this chapter, as they are essential to our understanding of the originsof chaos. We also use the standard map as an example of a system where we canappreciate what is being described.

3.1 KAM theorem

For perturbed systems that are integrable in the absence of the perturbation, theKAM theorem guarantees the existence of invariant tori. If these tori exist, thenall the integrals of motion of the unperturbed system still exist. Conversely, in thevicinity of resonances, the integrals of motion of the system are destroyed [14]. Thefollowing discussion of the KAM theorem follows very closely the one given in section3.2 of [9].

We assume that the Hamiltonian consists of an integrable part H0 and a smallperturbation H1

H(J,θ) = H0(J) +H1(J,θ), (3.1)

where H1 satisfies H1(J, θ + 2π) = H1(J,θ), that is, H1 is periodic in the anglevariables. Let us start by assuming that invariant tori exist under perturbation. We

42 Chapter 3. Phase Space Topology of Nonintegrable Systems

call one such torus a KAM torus. A KAM torus can be parameterized by ξ as

J = J0 + v(ξ, ε) (3.2)

θ = ξ + u(ξ, ε) (3.3)

with v, u periodic in ξ and equal to zero for ε = 0. The parameter is the vectorwhose components are the frequencies of the unperturbed torus, ξ ≡ ω = ∂H0/∂J.The conditions for the KAM torus to exist are listed below.

1. The frequencies of the unperturbed system, given by ω(J) = ∂H0/∂J, have tobe linearly independent,

n · ω(J) 6= 0. (3.4)

for some values of J1, where n is a vector whose entries are all integers, ni ∈ Z.

2. The perturbation H1 has to be differentiable at least 2N − 2 times, where N isthe number of degrees of freedom. This condition is said to be the sufficientlysmooth condition2.

3. The initial conditions have to be sufficiently far away from resonance so thatthey satisfy

|n · ω(J)| ≥ γ|n|−τ (3.5)

for all n. τ depends on how smooth H1 is and on the number of degrees offreedom of the system, and γ is proportional to the magnitude of the pertur-bation H1, the value of ε and the inverse of the nonlinearity parameter, wherethe nonlinearity parameter is ∂2H0/∂J2. For systems of two degrees of freedom,this condition is [10] ∣∣∣∣r − a

b

∣∣∣∣ > γm,nb2.5

for a, b ∈ Z, (3.6)

where [12]

γm,n ∝

√ε|Hm,n|

∂2H0

∂J12 . (3.7)

As we will see later, Hm,n is the coefficient of the perturbation written as aFourier series.

Right off the bat we identify some limitations the conditions impose on the exis-tence of the KAM tori. If γ gets too big, then Eq. (3.6) cannot be satisfied. Since γis proportional to |Hm,n|, we require a sufficiently small perturbation if we want theKAM tori to exist. We now look at the first and third conditions in detail, the secondone being too technical.

1We say some values of J because the frequencies ω are a function of J, and there exist some Jfor which the left hand side of Eq. (3.4) is zero.

2This allows perturbations that are not analytic, that is, have an infinite number of continuousderivatives.

3.1. KAM theorem 43

3.1.1 Linear Independence Condition

Let us consider a system with two degrees of freedom. Let the two frequencies of thesystem ω1(J1, J2), ω2(J1, J2) be related by

f(ω1, ω2) = 0, (3.8)

where we do not know the form of f . If we take the total derivative of f , we get

df =∂f

∂ω1

(∂ω1

∂J1

dJ1 +∂ω1

∂J2

dJ2

)+

∂f

∂ω2

(∂ω2

∂J1

dJ1 +∂ω2

∂J2

dJ2

)= 0

df = dJ1

(∂ω1

∂J1

∂f

∂ω1

+∂ω2

∂J1

∂f

∂ω2

)+ dJ2

(∂ω1

∂J2

∂f

∂ω1

+∂ω2

∂J2

∂f

∂ω2

)= 0 (3.9)

Since Eq. (3.9) holds for any dJ1 and dJ2, we require the terms in parenthesis tovanish. We can write this requirement in matrix form as(

∂ω1

∂J1

∂ω1

∂J2∂ω2

∂J1

∂ω2

∂J2

)(∂f∂ω1

∂f∂ω2

)= 0 =⇒ ∂ω

∂J

∂f

∂ω= 0. (3.10)

If the matrix ∂ω∂J

is invertible, that is, if det (∂ω∂J

) 6= 0, then we require that ∂f∂ω

= 0.This rules out the possibility of having an f of the form

f = n1ω1 + n2ω2 = 0, (3.11)

with n1, n2 nonzero integers. In other words, we have that in Eq. (3.11), the fre-quencies are linearly dependent. Since f cannot take this form when we assume thatdet (∂ω

∂J) 6= 0, then the frequencies are necessarily linearly independent. Thus, the

condition

det

(∂ω

∂J

)6= 0 (3.12)

requires the frequencies to be linearly independent. This requirement essentiallymeans that the frequencies depend on the action variables, that is, ω = ω(J). Thatthe frequencies depend on the action variables is a particular characteristic of non-linear systems. If the system was linear, then the frequencies would be constant. Aswe will see later, the dependence of the frequencies on the action variables stabilizesthe motion.

3.1.2 Sufficiently Far From Resonance

We know that the resonant tori do not survive under perturbation, and also the regionnear them. We ask ourselves then, what is the width of this region? Or, equivalently,how far away from resonance does the nonresonant torus have to be in order tosurvive? Since this question is a question of rational versus irrational numbers, let uslook at how far away are the irrational numbers from the rational numbers.


Let k be an irrational number. We can approximate any irrational number witha rational number a/b. For example, let k =

√2 = 1.414213.... We can approximate√

2 with different rational numbers, such as

√2 ≈

1410141100141410001414210000...

(3.13)

Although this method works, it is very rudimentary. A better approximation isachieved with continued fractions. We start by taking the integer part of k andcalling it s0. Then, we compute 1/(k − s0) and call its integer part s1. Then, wecompute 1/(s0 − s1) and call its integer part s2. This is done as many times aswanted:

k = s0 +1

s1 +1

s2 +1

s3 +1

s4 + . . .

(3.14)

and we can write Eq. (3.14) compactly as

k = [s0, s1, s3, s4, . . . ]. (3.15)

The convergence of the sum to the number k is slower the smaller the numberss0, s1, . . . are [16]. From this assertion, it follows that the hardest number to approx-imate is the one given by having all the coefficients si = 1. This number is called thegolden mean

ϕ =

√5− 1

2= [1, 1, 1, . . . ]. (3.16)

Later we will see that the golden mean plays an important role in determining whenglobal chaos makes its appearance. For now let us continue our discussion of continuedfractions.

For practical purposes we truncate the sequence in Eq. (3.14) so that we onlyiterate n+ 1 times. This gives us

k ≈ kn =anbn

= s0 +1

s1 +1

s2 +1

s3+.. .

sn−1

1

sn

, (3.17)

3.2. Fixed Points and their Linear Stability 45

(a) (b)

Figure 3.1: The shaded regions are those for which a KAM torus cannot exist. Theshaded regions correspond to those r that satisfy

∣∣r− ab

∣∣ < γb2.5

, where a/b correspondsto each rational number shown on the axis. The values of γ used were: for (a) γ = 0.12and for (b) γ = 2. Notice that as γ increases, the region where the KAM tori canexist decreases.

Figure 3.2: Same as Fig. 3.1 but with γ = 4. For this value of γ the regions whereKAM tori can live is very small compared to the region of Fig. 3.1a.

where an/bn satisfies ∣∣∣∣k − anbn

∣∣∣∣ < 1

bnbn−1

, (3.18)

where bn−1 is given by the denominator in Eq. (3.17) if we only iterate n times. Eq.(3.18) is a measurement of how close the irrational number k is to the rational numberan/bn.

Now that we have an idea of the relation between rational and irrational numbers,we go back to the nonresonant KAM tori problem. We focus on a two-degree-of-freedom system for simplicity. For the “sufficiently irrational” condition of the KAMtheorem, the condition for the rotation number r = ω1/ω2 is [10]∣∣∣∣r − a

b

∣∣∣∣ > γ

b2.5for a, b ∈ Z (3.19)

where

γ ∝

√ε|H1|

∂2H0

∂J12 . (3.20)

A plot of the “allowed zones” where r can be is given in Figs. 3.1 and Figs. 3.2.

3.2 Fixed Points and their Linear Stability

From our study of the twist map T , defined in Eq. (2.6), we know that there arevalues of Jn+1 such that the rotation number r is rational. For such a Jn+1 let uswrite r as

r =a

b, (3.21)


with a, b ∈ Z and a and b have no common factors. Then the iteration of the map willconsist of only b points. Let these points define the set3 (Jb,n, θb,n) whose elementsare (Jb,0, θb,0), (Jb,1, θb,1), ..., (Jb,(b−1), θb,(b−1)) and (Jb,(b+i), θb,(b+i)) = (Jb,i, θb,i). If weapply the map to any of these points, then the map will return to that point after biterations

T b(Jb,n, θb,n) = (Jb,n, θb,n), (3.22)

where T b indicates that the map T was applied b times. We call such a point a fixedpoint of period b. Then for r = a/b, there are b fixed points of period b. Note thatsuch fixed points are not fixed points of the twist map T , but rather they are fixedpoints of the twist map iterated b times: T b. Only if b = 1 can we talk about a fixedpoint of the twist map.

The existence of fixed points does not limit itself to the twist map. Let us havean N -dimensional map M that acts on the variables x where x = x1, x2, ..., xN. Ifthe map M outputs a periodic orbit that consists of the points x0,x1, ...,xb−1 withxb+i = xi then a fixed point of period b of the map M is

M b(xi) = xi for i = 0, 1, ..., b− 1. (3.23)

Fixed points are somewhat analogous to equilibrium points: a system in an equi-librium point remains there for an infinite amount of time. Similarly, a system ina fixed point remains there for an infinite amount of map iterations. Just as equi-librium points can be either stable or unstable, and this gives us information abouthow the system behaves in their vicinity4, fixed points can be elliptic, hyperbolic, orparabolic, and depending on what type of fixed point we have, we immediately haveinformation about the stability of the system in the vicinity of the fixed points. Letus work with a two-dimensional mapping U where

U

(Jnθn

)=

(Jn+1

θn+1

)=

(f(Jn, θn)g(Jn, θn)

). (3.24)

Since we are only interested in the behavior of the map in the vincinity of a singlepoint, we Taylor expand about one such point. For δJn+1:

Jn+1 + δJn+1 = f(Jn + δJn, θn + δθn)

Jn+1 + δJn+1 ≈ f(Jn, θn) + δJn∂f

∂J

∣∣∣∣J=Jn,θ=θn

+ δθn∂f

∂θ

∣∣∣∣J=Jn,θ=θn

δJn+1 ≈ δJn∂Jn+1

∂Jn+ δθn

∂Jn+1

∂θn. (3.25)

Similarly, for θn+1:

δθn+1 ≈ δJn∂θn+1

∂Jn+ δθn

∂θn+1

∂θn. (3.26)

3The first subscript of the variables indicates of what period they are a fixed point of, and thesecond subscript indicates which fixed point they are.

4A system near a stable equilibrium stays in the vicinity whereas a system near an unstableequilibrium diverges.


If we let

(Jn = Jfθn = θf

)be a fixed point, then the Eqs. (3.25) and (3.26) describe the

behavior of a point in the vicinity of a fixed point. By making the following definitions

U11 ≡∂Jn+1

∂Jn

∣∣∣∣Jn=Jf ,θn=θf

(3.27)

U12 ≡∂Jn+1

∂θn


(3.28)

U21 ≡∂θn+1

∂Jn


(3.29)

U22 ≡∂θn+1

∂θn


(3.30)

and defining the Jacobian matrix U of the map U as

U ≡(U11 U12

U21 U22

)(3.31)

we can write Eqs. (3.25) and (3.26) as(δJn+1

δθn+1

)= U

(δJnδθn

). (3.32)

We now have an equation that describes the behavior close to a given fixed point.The only piece of information that we need to know what happens near a fixed pointis the form of the eigenvalues. We start by solving the characteristic equation:

det

(U11 − λ U12

U21 U22 − λ

)= λ2 − λTr U + 1 = 0, (3.33)

where we set the determinant of the matrix equal to one due to the area-preservingnature of the map (cf. Eq. (2.3))5. Solving for λ in Eq. (3.33), we get

λ± =Tr U

2±

√(Tr U

2

)2

− 1 (3.34)

= t±√t2 − 1, (3.35)

where t is defined as t ≡ TrU2

. Eq. (3.35) puts a restriction on the eigenvalues: theyneed to satisfy

λ+ + λ− ∈ R, (3.36)

and the fact that the determinant of the matrix U has to be one imposes the condition

λ+λ− = 1. (3.37)

5Eq. (2.3) says that the magnitude of the determinant is one, not that the determinant itself isone. However, it can be proven, albeit rather laboriously, that if the entries of a symplectic matrixsuch as U are real, then the determinant is 1. A proof of this can be found in [17].


We know from linear algebra that, once we find the eigenvalues, we can find theeigenvectors of the matrix U and use them transform to a basis that diagonalizes U.If we denote the transformation matrix as T and the diagonal matrix as V we havethat

V = TUT−1 =

(λ+ 00 λ−

), (3.38)

T

(δJnδθn

)=

(δσnδνn

)(3.39)

and (δσn+1

δνn+1

)=

(λ+ 00 λ−

)(δσnδνn

)(3.40)

where the columns of T−1 are the eigenvectors of U. Note that the new point

(δσnδνn

)is still the old point

(δJnδθn

), it is only written in the new basis.

If we look at the square root in Eq. (3.34) the nature of both eigenvalues λ+ andλ− depend on the trace of U. We study each case in detail.

Case 1. For |Tr U| < 2 we have that the square root in Eq. (3.34) gives acomplex number. Then we can write Eq. (3.35) as6

λ± = t± i√

1− t2

λ± =

√t2 + (

√1− t2)2 e±i arctan (

√1−t2/t)

λ± = e±iα (3.42)

where we defined α ≡ arctan (√

1− t2/t) and α ∈ R. Note that α cannot be zero,since for that to happen the trace of U would have to be equal to 2, and that violatesour original assumption. Now that we have the eigenvalues we can see what happensin the vincinity of the fixed point:(

δσn+1

δνn+1

)=

(eiα 00 e−iα

)(δσnδνn

)=⇒ δσn+1 = δσne

iα

δνn+1 = δνne−iα (3.43)

This is nothing more than a rotation about the fixed point.7 Then this fixed pointis stable and we call it an elliptic fixed point. Since the orbits in the vincinity of anelliptic fixed point are stable, we should expect invariant curves around it.

6We use the identity

x+ iy =√x2 + y2ei arctan (y/x). (3.41)

7This can be easily pictured if we do the following. Since the trace is independent of the basiswe are working with, the trace for this case is

TrU = λ+ + λ− = 2 cosα. (3.44)


Case 2. For |Tr U| > 2 we have that the square root in Eq. (3.34) gives a realnumber. Then we can write Eq. (3.35) as

λ± = t±√t2 − 1. (3.47)

Thus, the eigenvalues for this case are real and satisfy λ+ = 1/λ−, that is, they arereciprocals of each other. With these eigenvalues, the behavior in the vicinity of thefixed point can be described by(

δσn+1

δνn+1

)=

(λ 00 1/λ

)(δσnδνn

)=⇒ δσn+1 = δσn λ

δνn+1 = δνn 1/λ(3.48)

The trajectory given by Eq. (3.48) lies on a hyperbola. However, the trajectory isdifferent for positive and negative λ. For positive λ, the trajectory lies on a singlebranch of the hyperbola. We call such a fixed point a hyperbolic fixed point. Fornegative λ, the trajectory alternates between the two branches of the parabola. This isbecause every iteration of the map changes the sign of the variables without changingits magnitude. Such a fixed point is called a hyperbolic-with-reflection fixed point.In Figs. 3.3a and 3.3b we can see the successive iterations of the map given by Eq.(3.48) for both positive and negative λ respectively. Both kinds of hyperbolic fixedpoints are unstable, since trajectories retreat from the fixed point as n→∞.

Case 3. For |Tr U| = 2 we have that the square root in Eq. (3.34) vanishes.This means that both eigenvalues have the same value. Additionally, since we onlyrequire that the magnitude of the trace is equal to 2 the trace can have two values8,Tr U = ±2. Then, the eigenvalues are

λ± = 1 (3.49)

or

λ± = −1. (3.50)

Given the value of the eigenvalues, we could naively conclude that they do nothingmore than keep or change the sign of the magnitude of points close to the fixed point.This conclusion only applies to the eigenvectors of U. So far they have been useful,

The rotation matrix is defined as (cosα − sinαsinα cosα

). (3.45)

It can be easily verified that its characteristic equation is, when set equal to zero:

λ2 − 2 cosα+ 1 = 0 (3.46)

which, when using Eq. (3.44), is nothing more than Eq. (3.33). Then the eigenvalues of the rotationmatrix are the same as the eigenvalues of U when |TrU| < 2. Thus the rotation matrix is nothingmore than the matrix we have in Eq. (3.43) written in a different basis.

8We can only consider real numbers due to the requirement that the sum of the eigenvalues hasto be real (cf. Eq. (3.36)).


(a) (b)

Figure 3.3: The mapping given in Eq. (3.48) with δσ0 = 2, δν0 = 2 and: for (a)λ = 1.3 and for (b) λ = −1.3.

but for this case we are better off working with our original basis,

(δJnδθn

). Then, for

this case the behavior in the vincinity of the fixed point can be described by(δJn+1

δθn+1

)=

(λ c0 λ

)(δJnδθn

)=⇒ δJn+1 = δJn λ+ δθnc

δνn+1 = δθn λ(3.51)

or (δJn+1

δθn+1

)=

(λ 0c λ

)(δJnδθn

)=⇒ δJn+1 = δJn λ

δνn+1 = δJnc+ δθn λ(3.52)

Eq. (3.51) describes a displacement only in the J direction, whereas Eq. (3.52)describes a displacement only in the θ direction. Such mapping is know as a shearmapping. As in the hyperbolic fixed point case, the sign of λ affects the trajectory.For λ = 1, the trajectory only increases the value of the iterated points, and so thetrajectory lies on only one line. For λ = −1, the trajectory increases the magnitudeof the iterated points but it changes their sign at every iteration. Then the trajectoryalternates between two lines. Such fixed points are called parabolic fixed points.

We have already encountered parabolic fixed points without being aware of them.If in Eq. (3.51) we set δθn = 0, we have that, for λ = 1, the Jn nearby the elliptic fixedpoint are fixed points of period 1. If these adjacent fixed points are also parabolicfixed points, then there are more fixed points further away from the original parabolicfixed point. If these new fixed points happen to be parabolic, then we find more fixedpoints even further away from the original fixed point, in a region where our originalapproximation of the map is probably not valid anymore. Additionally, the set ofthese adjacent fixed points lie on a curve. This is very reminiscent of the resonant

3.3. Poincare-Birkhoff Theorem 51

tori in unperturbed systems. In fact, the points that lie on the resonant tori areparabolic fixed points.

The condition for fixed points to be parabolic is very specific. Remember thatthe trace of U depends on the form of the map, and if the map we are working withhas a parameter that varies, such as the strength of a perturbation, then by slightlychanging this parameter we change the trace of U, and this makes having parabolicfixed points very rare. This is another aspect that makes resonant tori special.

3.2.1 Application: The Standard Map

We want to determine the stability of the fixed points of the standard map. Recallthat the period 1 fixed points are (0, 0) and (0, π). The Jacobian matrix U of thestandard map is

U =

(1 K cos θf1 1 +K cos θf

). (3.53)

The trace of U is

Tr U = 2 +K cos θf . (3.54)

Now we study in detail each fixed point of period 1.

• For the fixed point (0, 0) with θf = 0, we have that the absolute value of thetrace is

|Tr U| = |2 +K| (3.55)

and it will be positive for any nonzero perturbation, that is, for all K > 0. Thenthis fixed point is a hyperbolic fixed point for K positive.

• For the fixed point (0, π) with θf = π, we have that the absolute value of thetrace is

|Tr U| = |2−K|. (3.56)

From examining the trace we can conclude that for K < 4, the fixed point is anelliptic fixed point. For K > 4, the fixed point becomes hyperbolic, and thusunstable.

3.3 Poincare-Birkhoff Theorem

The KAM theorem guarantees that when the system is under a small perturbation,some tori survive. One of the conditions for such tori to exist is that the unperturbedsystem has to be sufficiently far away from resonance. This obviously rules out torithat are resonant, and less obviously the region in their proximity. Thus, underperturbation, the KAM theorem tells us nothing about the resonant tori. This iswhere the Poincare-Birkhoff theorem comes in to help us understand what happensto the resonant tori. Once again, we make use of the twist map T (cf. Eqs. (3.33)and (3.34)) to explain this theorem. For the twist map, the rotation number r(Jn+1)


(a) (b)

Figure 3.4: Behavior of the invariant curves C, C+, C− under the map T b. In (b) wecan see one iteration of the twist map for five pairs of initial values (J0, θ0) all withθ0 = π/2. The stationary point (green point) corresponds to J0 = 3

increases with Jn+1 (cf. Eq. (2.6)). To be on a resonant tori, we need the rotationnumber to be a rational number. For this purpose let us write r as

r(Jn+1) =a

b. (3.57)

Let us call C the invariant curve defined by Eq. (3.57). All the points that lie on thisinvariant curve are period b fixed points. This, in turn, implies that all the pointsthat lie on C are period one fixed points of the map T b: T b(C) = C. Thus for themap T b all points that lie on C map onto themselves, and hence do not move. Thesepoints are then stationary under the map T b. Now let us consider two curves, onedenoted by C+ defined by r+ ≡ r > a/b with r irrational and the other denoted byC− defined by r− ≡ r < a/b with r also irrational. Under the map T b the points lyingon the curve C+ rotate counterclockwise; similarly, under the map T b the points lyingon the curve C− rotate clockwise. This can be seen in Fig. 3.4.

Now that we are acquainted with the behavior of the unperturbed system near aresonant invariant curve, let us perturb the system. The perturbed twist map, whichwe denote by Tp, can be written as:

Jn+1 = Jn + εf(Jn+1, θn) (3.58)

θn+1 = θn + 2πr(Jn+1) + εg(Jn+1, θn) mod 2π. (3.59)

with f and g periodic in θ and picked such that the map Tp is area-preserving [9].Since the rational numbers are everywhere dense [13] we can pick r+ and r− that areclose to r = a/b such that C+ and C− are KAM curves. The KAM theorem guaranteesus that under a small perturbation, the curves C+ and C− survive. They, however,


take a new form: Cp+ and Cp− where the new curves are slightly deformed versions9

of C+ and C−, as sketched in Fig. 3.5a. Points on such curves are mapped onto thesame curves: Tp(Cp+) = Cp+ and Tp(Cp−) = Cp−. Additionally, the rotation of the curvesCp+ and Cp− remain counterclockwise and clockwise respectively when acted on by themap T bp .

In the unperturbed system we have two curves that rotate in different directionsunder the map T b. We expect that between these two curves there is a set of pointsthat marks the transition of direction, i.e., a set of points that does not rotate underthe map T b. From such a set of points we expect to find a subset of points that donot change their radius under the map T b. This subset of points was given by thepoints that lie on C. It seems reasonable to expect to also find such a set of stationarypoints between the perturbed tori Cp+ and Cp−, since the form of r is independent of theperturbation and so it has the same form for both the perturbed and the unperturbedsystem. Let the set of points that do not rotate under T bp lie on the curve Cp, whereCp differs slightly from the unperturbed curve C as pictured in Fig. 3.5a.

When T bp acts on Cp most of the points that lie on this curve change their radiusJn. I say “most” because there are some points that do not change their angle northeir radius when T bp acts on them. Then, even though Cp is not an invariant curveof the map T bp , the fixed points of this map lie on this curve. To find them, we recallthat the area enclosed by the curve Cp and C ′p ≡ T bp (Cp) has to be the same sincethe mapping must be area-preserving. This can only be so if the curves Cp and C ′pintersect an even number of times [9]. Such intersections define the fixed points ofthe map T bp . Since we require an even number of fixed points, we say that thereare 2kb fixed points, where k ∈ Z+. The fixed points of T bp lie on the unperturbedinvariant curve C, and so they are also fixed points of T b, as can be seen in Fig. 3.5b.Now we can state the Poincare-Birkhoff theorem [10]: For an unperturbed resonantcurve defined by r = a/b, only an even number of fixed points 2kb remain underperturbation.

By examining the flow near the fixed points as in Fig. 3.6 we can determinetheir type: they alternate between elliptic and hyperbolic. This means that there isan equal number of elliptic and hyperbolic points. Next we study the motion nearelliptic points and hyperbolic points.

3.3.1 Motion in the vicinity of an elliptic point

Both elliptic and hyperbolic points are what remains of the resonant invariant curve.Now we examine the motion in the region close to elliptic points. So far we have beenworking with a two-degree-of-freedom Hamiltonian, but in this section we focus onsystems that have one and a half degrees of freedom, that is, where the Hamiltonianhas the form H = H(J, θ, t). The results obtained in this section can then easily beapplied to the standard map.

Let us assume that we have a one-degree-of-freedom Hamiltonian that is perturbedby a periodic external force. The potential corresponding to this force can be written

9See Eqs. (3.2) and (3.3) to see the form that the surviving tori take under perturbation.


(a) (b)

Figure 3.5: The new shape of the invariant curves of the unperturbed system underperturbation. (a) shows that the (small) perturbation barely affects the nonresonantcurve, whereas the resonant curve changes its shape drastically. In (b) we can see thatby iterating the map T b on the points lying on Cp we obtain C ′p. The intersections ofCp and C ′p are the fixed points of the system.

Figure 3.6: The flow around the fixed points depends both on the direction themapping moves the points on the curve Cp and on the direction the mapping movesthe points on the curves Cp+, Cp−. This flow indicates to us what kind the fixed pointsare, if elliptic or hyperbolic.


as a Fourier series

V (φ) =∞∑

n=∞

Vneinφ, (3.60)

whereφ = φ(t) = Ωt+ φ0, (3.61)

and Ω is the driving frequency of the external perturbation. When formulating theHamiltonian corresponding to the system with this perturbation, the phase φ is takenas a variable. We write the Hamiltonian as

H(J, θ, t) = H0(J) + εH1(J, θ, t) (3.62)

where H1 is periodic and thus can also be written as a Fourier series. Doing this wehave

H(J, θ, t) = H0(J) + ε∑m,n

Hm,n(J)ei(mθ+nφ), (3.63)

where Hm,n are the Fourier coefficients. The frequency corresponding to the unper-turbed system H0 is

ω(J) =∂H0

∂J. (3.64)

For the resonance condition to be satisfied, we need to find a value of the actionvariable J such that Eq. (2.7) is satisfied. Let such action variable be J = Jr. Wethen have

bω(Jr)− aΩ = 0 =⇒ ω(Jr)

Ω=a

b(3.65)

where a, b are integers.Since the system we are working with is nonlinear, i.e., the frequency depends on

the action variable10 (which, in turn, depends on the energy), then, when the systementers resonance, it does not remain in resonance. This is because, when a systemis in resonance, there is an exchange of energy between the degrees of freedom11.This exchange of energy changes the frequency, and this brings the system out ofresonance12.

In general, there are many Jr’s that define resonances since resonances themselvesmake up an everywhere dense set13, that is, they are found easily and very close to eachother. This means that there are some m 6= a, n 6= −b such that resonance happens.However, since we are interested in the motion in the vicinity of a single resonancewe assume that other resonances are far away and negligible, and we assume that

10See Eq. (3.12).11For the system we are working with (cf. Eq (3.63)), the system absorbs energy from the external

perturbation.12Then we can say that the nonlinearity of the system stabilizes the motion by virtue of the

dependence of the frequencies on the action variables. If the system would not get out of resonance,then the exchange of energy would not stop and the amplitude of oscillation of the system would goto infinity as time increases. See Eq. (2.10) for an example of this.

13The rotation number r has to be rational for a resonance to exist. The subset of rational numbersis an everywhere dense subset of the set of real numbers.


only one resonance exists, the one given by Eq. (3.65). This assumption simplifiesthe Hamiltonian, which now we write as

H(J, θ, t) = H0(J) + εHb,−a(J)ei(aθ−bφ) (3.66)

Now that we have the Hamiltonian, let us look into the motion of the system closeto resonance. First, let us examine the action variable. Using Hamilton’s equationsof motion we get

J = −εibHb,−a(J)ei(aθ−bφ) (3.67)

As we recently noted, since the system is nonlinear, when it enters resonance it getsout of it. Thus the actual motion of the system takes place in the vicinity of resonance.This means that the quantity

|J − Jr| (3.68)

where J is given by Eq. (3.67), is nonzero. It is, however, small. To examine whathappens to the phase, we construct a new angle variable as

γ = bθ − aφ. (3.69)

Let us assume for a second that the frequency ω did not depend on the action variable,such that θ = ω. Then, if we take the time derivative of Eq.(3.69) we get

γ = bθ − aφ = bω − bΩ = 0, (3.70)

where we used Eqs. (3.61) and (3.65). Thus for a system that can remain in resonanceforever, the phase that we just cooked up is constant in time. This is what being atresonance means: resonances correspond to fixed points.

Eq. (3.70) does not hold for a nonlinear system. However, the result we justobtained helps us make assumptions of the behavior of γ in the vicinity of resonance.assuming that γ is well-behaved, since it is stationary at resonance, we infer that closeto resonance it (and its harmonics) oscillates slowly, i.e. slowly compared to termswith m 6= a, n 6= −b in Eq. (3.63):

γ mθ + nφ for n 6= −a and m 6= b (3.71)

In general, γ measures “the slow deviation from resonance” [10].Let us perform a canonical change of variables so that we can use the newly

introduced angle variable γ and an action variable similar to the term in Eq. (3.68).We do this with the help of the generating function

F2 = (bθ − aφ)I + θJr. (3.72)

Using Eqs. (1.12), (1.13), and (1.14) we have that the new Hamiltonian and therelations between the new and old coordinates are

J =∂

∂θF2 = bI + Jr (3.73)

γ =∂

∂IF2 = bθ − aφ (3.74)

K = H(J, θ, t) +∂

∂tF2 = K0(I) + εK1(I, γ, φ, t)− aΩI, (3.75)


where the integrable part of the Hamiltonian is

K0(I) = H0(J(I)) (3.76)

and the perturbation term is

K1 =∑m,n

Kmn(I) exp

[i

(maφ+ γ

b+ nφ

)]=∑m,n

Kmn(I) exp

[i

b

(mγ + (ma+ nb)φ

)], (3.77)

withKmn(I) = Hmn(J(I)). (3.78)

Notice the similarities between Eqs.(3.68), (3.69) and (3.73), (3.74). The Hamiltonianin Eq. (3.75) then gives the motion in the vicinity of resonance. We now make anotherassumption and say that close to resonance

γ φ = Ω. (3.79)

That is, the perturbation term oscillates much faster than γ. We then proceed toaverage the Hamiltonian over one revolution of the fast variable φ since its averagecontribution to the system is zero:

K =1

2π

∫ 2π

0

K(I, γ, φ, t)dφ

=1

2π

∫ 2π

0

K0(I)dφ+1

2π

∫ 2π

0

εK1(I, γ, φ, t)dφ− 1

2π

∫ 2π

0

aΩIdφ

= K0 + ε∑m,n

Km,n(I)exp

[i

bmγ

]1

2π

∫ 2π

0

exp

[i

b(ma+ nb)φ

]dφ− aΩI.

Recall that Eq. (3.79) is only valid if we are close to resonance, and we also assumedthat, for simplicity, there is only one resonance in the system, the one given by Eq.(3.65). Then14, we keep only the terms corresponding to resonance and its harmonics,that is, m = pb, n = −pa with p ∈ N

K = K0 + ε∞∑

p=−∞

Kpb,−pa(I) exp[ipγ]1

2π

∫ 2π

0

exp

[i

b(pba− apb)φ

]dφ− aΩI

= K0 + ε∞∑

p=−∞

Kpb,−pa(I) exp[ipγ]1

2π

∫ 2π

0

exp[0]dφ− aΩI

= K0 + ε

∞∑p=−∞

Kpb,−pa(I) exp[ipγ]− aΩI. (3.80)

= K(I, γ) (3.81)

14The following assumption is essentially the same as the one that was used to bring Eq. (3.63)to Eq. (3.66), except that here we also consider higher harmonics of the resonance.


and, since we got rid of the time dependence, we end up with an integrable Hamil-tonian. We simplify it further by assuming that the higher-order harmonics of γ aresmall compared to the lowest harmonic, that is

Kpb,−pa Kb,−a for |p| = 2, 3, ... (3.82)

Thus we only keep the terms for p = −1, 0, 1. Additionally, we assume that Kb,−a =K−b,a [9]. Using these two assumptions, Eq. (3.80) becomes

K = K0(I) + εK0,0(I) + εKb,−a(I) exp[iγ] + εK−b,a(I) exp[−iγ]− aΩI

= K0(I) + εK0,0(I) + εKb,−a(I)(

exp[iγ] + exp[−iγ])− aΩI

= K0(I) + εK0,0(I) + ε2Kb,−a(I) cos γ − aΩI. (3.83)

So far we have been using the fact that we are close to resonance only to makeassumptions about γ. Now we use this fact to assume that the action I is small.Then we Taylor expand Eq. (3.83).

The first term in Eq. (3.83) gives us

K0(I) = K0(I(Jr)) + I∂K0

∂I

∣∣∣∣I=I(Jr)

+I2

2

∂2K0

∂I2

∣∣∣∣I=I(Jr)

+O(I3)

= H0(Jr) + bI∂H0

∂J

∣∣∣∣J=Jr

+b2I2

2

∂2H0

∂J2

∣∣∣∣J=Jr

+O(I3)

= H0(Jr) + bIω(Jr) +b2I2

2

∂2H0

∂J2

∣∣∣∣J=Jr

+O(I3), (3.84)

where in the second line we used Eq. (3.76). The second term in Eq. (3.83) gives us

εK0,0(I) = εK0,0(I(Jr)) + εI∂K0,0

∂I

∣∣∣∣I=I(Jr)

+O(εI2). (3.85)

The third term in Eq. (3.83) gives us

ε2Kb,−a(I) cos γ = ε2 cos γ

(Kb,−a(I(Jr)) + I

∂Kb,−a

∂I

∣∣∣∣I=I(Jr)

+O(I2)

). (3.86)

Plugging these expansions into Eq. (3.83) and neglecting the terms of order O(εI),we get

K = K0(I(Jr))+bIω(Jr)+b2I2

2

∂2H0

∂J2

∣∣∣∣J=Jr

+εK0,0(I(Jr))+ε2Kb,−a(I(Jr)) cos γ−aΩI

(3.87)By using Eq. (3.65), the second and last terms cancel. Dropping the first andfourth terms because they are constant and do not affect the dynamics (they do notcontribute when we use Hamilton’s equations of motion) we get

K =b2I2

2

∂2H0

∂J2

∣∣∣∣J=Jr

+ ε2Kb,−a(I(Jr)) cos γ, (3.88)


which, if we define

M−1 = b2∂2H0

∂J2

∣∣∣∣J=Jr

V = −ε2Kb,−a(I(Jr)), (3.89)

can be written as

K =I2

2M− V cos γ. (3.90)

This is the Hamiltonian for a pendulum (cf. Eq. (1.54)). Thus, the motion inthe vicinity of any resonance is qualitatively similar to that of a pendulum. Theword “qualitatively” is key here. The motion in the vicinity of a resonance does notfully resemble that of a pendulum. For example, in following sections we will learnthat for a nonintegrable system the region where we would expect the separatrixto be is in fact home to chaotic trajectories. Moreover, in the region where weexpect oscillatory motion there are second-order resonances, which will be explainedshortly. It is important to keep in mind that we obtained Eq. (3.90) by assuming thatthe perturbation is small enough for our approximations to hold. Since Eq. (3.90)describes the motion in the vicinity of all resonances, it is denominated the standardHamiltonian [9].

The fixed points of the pendulum correspond to elliptic and hyperbolic points (cf.Eqs. (1.60) and (1.61)). We already stated that the elliptic points correspond toresonance (this whole section was devoted to the motion in their vicinity, after all).Now we want to show that the hyperbolic points of the new system correspond tothe hyperbolic points of the original system. For the fixed points of the new system,we have that γ = 0. In the original variables this means that bθ − aφ = 0. We makeone final approximation: θ ≈ ω, which is equivalent to ∂H/∂J ≈ ∂H0/∂J . Thenγ = 0 translates to bω − aΩ = 0, which is the original resonance (cf. Eq. (3.65)).This allows us to make the connection between the fixed points of the newly foundstandard Hamiltonian and the fixed points of the original Hamiltonian.

Now we proceed to find the equation of the separatrix in the original variables. Aswe will see shortly, the region close to the separatrix is the first region where chaosappears. The equation of the separatrix is given by Eq. (1.73)

(I)sx = 2√MV cos

(γ

2

)(3.91)

and the frequency of the motion near resonance is given by Eq. (1.70)

ωJr =

√V

M. (3.92)

Rewriting the separatrix equation in terms of the old variables we have

Jsx = Jr ±∆Jr cos

(bθ − aφ

2

), (3.93)

where∆Jr = 2b

√MV . (3.94)


is the resonance half-width. We can express ∆Jr in terms of the frequency as

∆ωr =∂ω

∂J∆Jr

=1

b2M2b√MV

=2

b

√V

M. (3.95)

The assumptions that we made are: neglecting the nonresonant terms by assumingthey oscillated fast, neglecting the harmonics of the resonance Kpb,−pa Kb,−a, andneglecting terms of order O(ε∆I1). These assumptions are valid if the moderatenonlinearity condition [12] is satisfied:

ε α 1

ε, (3.96)

where

α =J

ω

∣∣∣∣∂ω∂J∣∣∣∣ (3.97)

measures the degree of nonlinearity of oscillations [18].The amplitude term of the separatrix gives us an idea of the dependence on the

perturbation strength of the appearance of a pendulum-like phase space around anelliptic point. We have that

2√MV = 2

√M(−ε2Kb,−a(I(Jr))) ∼ ε1/2 (3.98)

and so the fact that the phase space close to resonance resembles that of a pendulumdepends on the perturbation as ε1/2.

So far our discussion has involved resonances between the internal frequency of thesystem ω and the frequency of the perturbation Ω. We call such resonances first orderresonances. Notice that the motion close to resonance has its own frequency given byEq. (3.92). Then resonance can happen between the frequency of the system closeto resonance ωJr and the frequency of the perturbation Ω. We call such resonancessecond order resonances. These second order resonances form “islands” around theelliptic point, and we call these island chains. The process to obtain the motionnear the first order resonance can also be applied to the second order resonances,and resonance between the perturbation and phase oscillations of the second orderresonances would produce third order resonances. This can be done ad infinitum.It can be shown [9] that the dependence on the perturbation strength of the effectof a secondary resonance is of order 1/(ε−1/2)!. Thus, for second order resonancesto have a considerable effect on the motion of the system, the perturbation has tobe significantly strong. This does not mean that secondary resonances do not existfor small ε, it just means that they are negligible. In fact, as long as ε is nonzero,secondary (and higher) resonances exist. Thus we have that, around elliptic points,there is a rich and intricate formation of resonances.


Application: The Standard Map

From the analysis of fixed points of the standard map done back in section 3.2.1, welearned that the fixed points of period 1 are (0, 0) and (0, π)15. So we already knowthat resonance happens for I ≡ Ir = 016. However, we obtained this result by lookingat the mapping formulation of the standard map, Eqs. (2.37) and (2.38). In ourdiscussion above we obtained the values of J and θ for which resonance happens bylooking at the Hamiltonian rather than at the map, and that is what we will do inthis section for the standard map .

Let us start by considering the Hamiltonian for the standard map given in Eq. (2.34),where the frequency of the perturbation is Ω = 2π/T . We rewrite the standard mapHamiltonian using some properties of the Dirac delta function given in Appendix B,letting the period be T = 2π, and defining the perturbation parameter K = AT :

H(J, θ, t) =J2

2+

K

(2π)2

∞∑n=−∞

cos (θ − nt). (3.99)

It is clear from the perturbation term that what we previously thought of as periodicdelta spikes can be represented as the sum of all the harmonics of the perturbationfrequency Ω (which in this case is equal to 1). Let us obtain the equation of motionfor θ:

θ =∂H

∂J= J. (3.100)

We are lucky that for the standard map the perturbation parameter does not depend

on the action J . Thus, the unperturbed frequency of the unperturbed system ∂(J2/2)∂J

=

J = ω(J) is the same as the frequency of the perturbed system, θ = J . FromEq. (3.100) we have that θ = Jt = ω(J)t.

To obtain the values of J for which resonance happens, we let the argument ofthe cosine terms in the perturbation be equal to zero. Thus we have that:

θr − nt = 0

θr = nt

ω(Jr)t = nt

Jr = n. (3.101)

Hence the resonances for the standard map happen when J = n ≡ Jr. If we definethe slowly varying phase as γn = θ − nt, we can rewrite the Hamiltonian Eq. (3.99).If we consider only one resonance

H(J, θ, t) =J2

2+

K

(2π)2cos γn, (3.102)

where each γn corresponds to a different resonance Jr = n. Thus the motion arounda resonance resembles that of a pendulum, as expected.

15Or, equivalently, (2πn, 2πm) and (2πn, (2m+ 1)π) for m,n integer.16These resonances are the ones corresponding to the lowest harmonic of the perturbation.


(a) (b)

Figure 3.7: Iterations of initial conditions close to resonance. The phase space is verysimilar to that of the pendulum, which is shown in Fig. 1.6. The map was iteratedfor perturbation strength (a) K = 0.4 and (b) K = 0.95. A zoom of the region insidethe rectangle is shown in Fig. 3.8.

(a) (b)

Figure 3.8: Blow up of the rectangle in Fig. 3.7a. This is a secondary resonance, withits own pendulum-like motion in its vicinity. (a) was iterated with the same initialconditions as those in Fig. 3.7a, whereas for (b) we picked initial conditions so thatthe structure of phase space in the vicinity of this point is clearer.

We now know that the value of J at which resonance takes place is Jr = n. But theresult we obtained from our previous analysis indicates that the resonances happenat values of 2πn. That is because, when deriving the standard map, we performed achange of variables: J = I/T . Since we have taken the period to be T = 2π (as wealso did when deriving the standard map) we write Ir in terms of Jr as

Ir = TJr = 2πn (3.103)

which is the result we obtained back when we obtained the fixed points of period 1 ofthe standard map (cf. Eq. (2.42)). In Fig. 3.7a we can see the motion in the regionwhere In ≈ 0, 2π. We can see that, comparing to Fig. 1.6, the motion in the vicinityof resonance resembles that of a pendulum.

In Fig. 3.8 we show one of the second order islands surrounding the elliptic point(0, 0). Notice that in its vicinity, phase space resembles that of a pendulum. If we


Figure 3.9: The curves H+ and H− are the same. The intersection is at the hyperbolicpoint H.

carefully observe Fig 3.8b, we can see little dots next to the solid, almost vertical line,about three quarters to the right of the plot. These little dots are are also resonances,corresponding to fixed points of very high period.

3.3.2 Homoclinic and Heteroclinic Points

Now that we know what happens to the motion around elliptic points we want toknow what happens around the hyperbolic points. We are already slightly familiarwith hyperbolic points. In the pendulum phase-space, the intersection of the separa-trices correspond to hyperbolic points. Fig. 1.6 shows two curves H+ moving towardsthe hyperbolic point H and two other curves H− moving away from it. This is char-acteristic of all hyperbolic points, that is, having two stable manifolds approachingit and two unstable manifolds moving away from it. From Eq. (1.77) we know thata point lying on H+ approaches the hyperbolic point H as t→∞ and a point lyingon H− approaches the hyperbolic point H as t→ −∞.

In terms of mappings, assume we have a point q lying on one of these curves.Then

limn→∞

T nq → H for q ∈ H+ (3.104)

limn→−∞

T nq → H for q ∈ H− (3.105)

The curves H+ and H− can interact in different ways. If the curves H− and H+

intersect and they are outgoing and incoming trajectories respectively from the samehyperbolic point, then the intersection is referred to as a homoclinic point. An ex-ample of this is the pendulum, whose unstable fixed points θ = −π, π are the same.If an intersection happens between one curve corresponding to T s and another curvecorresponding to T s+r then the intersection is not a homoclinic point, but rather aheteroclinic point.

Integrable systems exhibit a simple structure in the construction of homoclinicpoints because the curves H+ and H− are allowed to intersect each other, as shownin Fig. 3.9. For nonintegrable systems the curves H+ and H− are not allowed tointersect each other, as shown in Fig 3.10a. What happens in this case is neatlyexplained by Tabor in [10] and the following discussion follows his. In Fig. 3.10a we


(a) (b)

Figure 3.10: A homoclinic point and the confusion regarding its position after appyingthe map T to it. In (a) the curves H+ and H− originate from the same hyperbolicpoint and intersect each other at the homoclinic point q. Notice how the curve H+

does not intersect itself, and neither does H−. In (b) The nearby points q, q′ aremapped to Tq, Tq′ respectively. It is unclear where we should put Tq.

can clearly see the hyperbolic point lying at the intersection on the left. On the right,the intersection of H+ and H− corresponds to the homoclinic point q. Now picturetwo other points that live close to q, one that lies on H+ which we call q′ and anotherthat lies on H− which we call q′′. If we map these two points, then by the flow of thecurves H+ and H− it is unambiguous where they have to be mapped. This is shownin Fig. 3.10b. However, it is unclear where to put Tq in this picture. Since it lies onboth H+ and H−, it is supposed to follow the flow of both curves. So we can eitherput it on H− to the right of Tq′′ or on H+ to the left of Tq′. Since both of theseoptions have to be satisfied, the curve H− folds such that the point Tq satisfies bothrequirements. This folding can be seen in Fig. 3.11. Notice that there is a secondintersection between H+ and H−. This corresponds to another homoclinic point.

The solution just given was for the case of only one iteration of the map T .What about T 2? We run into the same problem as before. However, we now knowthat folding the curve H− solves it. This is shown in Fig. 3.12. Since the point q isapproaching the hyperbolic point, and we know that the hyperbolic point is reached asn→∞ where n in the number of iterations (cf. Eq. (1.79) and following discussion),then the distance between Tq and T 2q is smaller than the distance between q andTq. Additionally, since the map has to be area-preserving, the gray areas in Fig. 3.12have to be equal. Thus H− is longer and is thinner in the second folding than in thefirst folding. Moreover, another homoclinic point appears.

This folding process process continues ad infinitum17, with Eq. (3.104) satisfied,an infinite number of homoclinic points appearing, and increasingly thinner and longerfoldings of H−. Then we say that a single intersection of a stable curve H+ and anunstable curve H− implies an infinite number of intersections [13]. But that is notthe end of the story. This complex folding of curves happens as we move backwardsin time, that is, as limn→−∞ T

nq. In this case, the curve H+ does the folding processas q approaches the separatrix along the curve H−. This is shown in Fig. 3.13.

17This complex behavior was first discussed by Poincare when was working on the stability of thethree body problem.


Figure 3.11: The problem of where to put Tq is solved by folding the curve H− sothat q follows both flows after one iteration of the map T . The new crossing of H+

and H− corresponds to a new homoclinic point.

(a) (b) .

Figure 3.12: A second iteration of the map T on q. In (a) we see that for a seconditeration of the map T on q, the curve H− has to fold a second time. The distancebetween Tq and T 2q is smaller than the distance between q and Tq and thus, sincethe mapping is area-preserving, the folding this time is thinner and longer. (b) showsan easier visualization of Fig. 3.12a.

As a general remark, the behavior of a system close to homoclinic and heteroclinicpoints resembles that of the horseshoe map [4], a map that exhibits chaotic behavior.Thus we can infer that the behavior in the vicinity of the separatrix is chaotic18. Forsmall perturbations, the region where chaotic motion takes place in phase space isbounded by KAM curves as shown in Fig. 3.14.

For a single hyperbolic point the complete picture is given in Fig. 3.15.In the presence of a single resonance, the only region where chaotic motion is

present is around the separatrix. If the perturbation is strong enough, then second-order resonances stop being negligible and start having an effect on the system. Thesenew resonances have their own elliptic and hyperbolic points. This means that theH+ and H− curves of the secondary resonances can (and do) intersect the H+ and H−

curves of the first-order resonances, leading to the appearance of heteroclinic points.In this section we have seen that the motion close to the hyperbolic points takes on

a very wild behavior19. Moreover, we have gained an additional piece of information:

18The presence of chaos in the vicinity of the separatrix can also be said to exist by virtue ofthe embedding of Bernoulli shifts close to homoclinic and heteroclinic points [19]. Bernoulli shiftsare, just as positive Lyapunov exponents a signature of chaos. Lyapunov exponents are treated inAppendix C.

19In [20] the Lyapunov exponents near hyperbolic points of the standard map are shown to be


Figure 3.13: As we iterate the map backwards in time, the curve H+ also undergoesthe same folding that we described for H−.

Figure 3.14: The problem of where to put Tq is solved by folding the curve H− sothat q follows both flows after one iteration of the map T . The new crossing of H+

and H− corresponds to a new homoclinic point.

Figure 3.15: Both stable and unstable curves are shown. They both behave in a wildmanner as they approach the hyperbolic point.


Figure 3.16: Visualization of the motion close to the hyperbolic point (2π, 0) forK = 0.5. We picked (J0 = 0.01, θ0 = 0.01) as the initial point from which weobtained the line segment, and from the line segment a set of 1000 initial conditions.Each initial condition was iterated 100 times.

the chaotic motion is somehow related to the manifolds that, for integrable systems,form a smooth separatrix joining two hyperbolic points. So the next step is to considerthe motion in the region close to the separatrix, and we will see that it is in this regionthat chaotic motion first occurs. We call such regions stochastic layers.

Application: The Standard Map

From our result in section 3.2.1 we know that the hyperbolic fixed point of period 1of the standard map is at (0, 0) (or, equivalently, (2π, 0)). To visualize the motion inthe vicinity of the hyperbolic point we pick as our initial conditions a set of pointsthat lie close to the separatrix. The way this set of points is picked is by choosing oneinitial condition close to a hyperbolic point, iterating it once, tracing a line betweenthe initial condition and the first iteration, and finally picking the points lying onthis line as your initial conditions. We also get rid of the limitation that Jn and θnhave to live in [0, 2π] so that the whole trajectory can be seen around the hyperbolicpoint. The result of this procedure can be seen in Fig. 3.16.

positive.


3.4 Motion in the Vicinity of the Separatrix

From our discussion of homoclinic and heteroclinic points, it is clear that chaos makesits first appearance in perturbed systems in the region close to the separatrices evenfor arbitrarily small perturbations. Separatrices, in turn, show up when we examinethe motion close to a resonance. The system close to a resonance is given by theHamiltonian in Eq. (3.90), that is, by the Hamiltonian of a pendulum. For a onedimensional, integrable pendulum, the motion close to the separatrix is well behaved:the energy of the system experiences no change at all. For the separatrix of a res-onance, chaos appears in its vicinity. This means that nearby trajectories diverge,and thus the energy of a trajectory is not constant. With this in mind, we proceedto construct a map that describes the change in the energy when the system is nearthe separatrix.

Before we construct the map, there are two things I want to mention. First, themaps that we have considered so far have consisted of a pair of canonical variables(Jn, θn). In a similar vein we consider the conjugate variable to H, which is time t,in order to have a map consisting of a pair of canonical variables. Second, we willnot consider the actual energy H or the time t, but rather functions of both of thesequantities which we will call (w(H), φ(t)), where w(H) is defined as in Eq. (1.89) andφ(t) = Ωt. These two functions will still make a pair of canonical variables. Withthese clarifications in mind let us proceed to the construction of the mapping.

Let us consider the pendulum system that is constructed in the vicinity of res-onance (cf. Eq. (3.90)) under a perturbation that is periodic in time with periodT = 2π/Ω. Let us write the Hamiltonian for such a system as

H(q, p, t) =1

2Mp2 − U0 cos q + εU0 cos q cos (Ωt+ φ0) (3.106)

where φ0 = Ωt0 and t0 is the initial time. We want to consider only the changes inenergy of the original, unperturbed system. We write the change in energy of theoriginal system as20

dH0

dt=∂H0

∂t+ [H0, H]

=∂H0

∂t+

(∂H0

∂q

∂H

∂p− ∂H0

∂p

∂H

∂q

)= U0 sin q

p

M− p

M(U0 sin q − εU0 sin q cos (Ωt+ φ0))

= εU0

Mp sin q cos (Ωt+ φ0) (3.107)

20Chirikov [12] proves that the change in energy is the same for both the unperturbed HamiltonianH0 and the perturbed Hamiltonian H, so it suffices to consider the change in energy of only H0.This result is equivalent to saying that the contribution of the high frequency term to the fluctuationin energy averages to zero.

3.4. Motion in the Vicinity of the Separatrix 69

and the change of energy during one period of the pendulum is

∆H0 =

∫ T/2

−T/2

dH0

dtdt (3.108)

= εU0

M

∫ T/2

−T/2p sin q cos (Ωt+ φ0)dt. (3.109)

Since the period of the pendulum goes to infinity at the separatrix, we adjust thelimits of Eq. (3.109) accordingly. We also use psx as defined in Eq. (1.78) and qsx asdefined in Eq. (1.77). Then Eq. (3.109) becomes

∆H0 = εU0

M

∫ ∞−∞

∆psx sech (ω0t) sin (4 arctan (eω0t)− π) cos (Ωt+ φ0)dt

= εU0

M∆psx

πΩ2

ω30

(− csch

(πΩ

2ω0

))sinφ0

= 2MεπΩ2

(− csch

(πΩ

2ω0

))sinφ0, (3.110)

where in the third line we used Eq. (1.70). To simplify Eq. (3.110) we assume21 thatthe frequency of the perturbation is much bigger than the natural frequency of thesystem, Ω ω0. This gives us that

− csch

(πΩ

2ω0

)= − 2

eπΩ/2ω0 − e−πΩ/2ω0

≈ − 2

eπΩ/2ω0

= −2e−πΩ/2ω0 (3.111)

and we write Eq. (3.110) as

∆H0 ≈ −4MεπΩ2exp

(− πΩ

2ω0

)sinφ0. (3.112)

This is the change of energy after one period near the separatrix. Notice that ∆H0

depends on the initial phase of the perturbation φ0. This phase also changes afteronly half of the period Tsx (cf. Eq. (1.93)) as follows22:

∆φ = ΩTsx2

=Ω

ω0

ln

(32

|w|

). (3.113)

21This is equivalent to the assumption given in Eq. (3.79).22If you are wondering why only half of the period and not a full period, let me explain. If a

trajectory at t = t0 is close to a hyperbolic point (q = −π, p = 0) and H < U0, then it takes a fullperiod Tsx for it to come back to (q = −π, p = 0). However, we are interested in what happens to thetrajectory when it reaches another hyperbolic point, in this case the one located at (q = π, p = 0).The trajectory takes only half the period Tsx to get close to this other hyperbolic point. This is notrestricted to H < U0, as can be easily seen in Fig. 1.6.


Here w, defined in Eq. (1.89), is used. Since the change in phase φ depends on wrather than the energy H0, we plug ∆H0 into w to obtain ∆w:

∆w =(H0 + ∆H0)− U0

U0

− H0 − U0

U0

=∆H0

U0

= −4M

U0

επΩ2exp

(− πΩ

2ω0

)sinφ0

= −4Ω2

ω20

επexp

(− πΩ

2ω0

)sinφ0. (3.114)

Now that we have the changes in phase φ and the relative energy w after one period

near the separatrix, we can write a map that takes the values of

(wnφn

)at time t = nT

and maps them to a new set of values

(wn+1 = wn + ∆wφn+1 = φn + ∆φ

)at time t = (n+1)T . This

map takes the form

wn+1 = wn − 4Ω2

ω20

επexp

(− πΩ

2ω0

)sinφn (3.115)

φn+1 = φn +Ω

ω0

ln

(32

|wn+1|

). (3.116)

We can make the map prettier by defining a pair of parameters as

λ =Ω

ω0

(3.117)

W = −4λ2επexp

(− λπ

2

), (3.118)

where λ measures the ratio of the frequencies. With these parameters in place, themapping now takes the form

wn+1 = wn +W sinφn (3.119)

φn+1 = φn + λ ln

(32

|wn+1|

). (3.120)

The mapping given by Eqs. (3.119) and (3.120) describes the fluctuations in energyand phase along the separatrix, and it is referred to as the whisker map, where the“whiskers” are the stable and unstable curves H+ and H− arriving to and leavingfrom a hyperbolic point.

Similar to the standard map, the whisker map is a perturbed twist map. Bycomparing Eqs. (3.119), (3.120) with (2.15), (2.16) we have that εf = W sinφn,

g = 0, and λ ln(

32|wn+1|

)= 2πr. The perturbation term is then proportional to W ,

3.4. Motion in the Vicinity of the Separatrix 71

which is linear in ε from Eq. (3.118). So, no matter how small the perturbation, therewill be fluctuations in the energy and stochasticity close to the separatrix.

Why do the fluctuations in the energy given by Eq. (3.119), which are seem-ingly small, give way to chaos? This is because the frequency of oscillations dependsstrongly on the energy when the system is close to the separatrix (when |H−U0| 1),as opposed to when the system is close to the elliptic fixed point, where the frequencycan even be approximated to be independent of the energy. The strong dependenceof the frequency on the energy leads to considerable changes in the phase φ. Thisleads to chaos in the vicinity of the separatrix [18].

If we set W = 0, we have no perturbation and the whisker map becomes unper-turbed. The resonant values of wn can be obtained by setting wn+1 ≡ wr, where wrcan be defined in terms of the rotation number r by solving:

λ ln

(32

|wr|

)= 2πr(

32

|wr|

)= e2πr/λ

|wr| = 32e−2πr/λ (3.121)

Whether the unperturbed trajectory is periodic or quasiperiodic depends on the ro-tation number r. For r rational, wr defines a resonant trajectory.

The fixed points of period one, which we denote with the subscript 1f , are givenif we set wn+1 = wn ≡ w1f and φn+1 = φn + 2πm = φn ≡ φ1f with m an integer23.Applying the first of these conditions we have

w1f = w1f +W sinφ1f

0 = W sinφ1f

arcsin 0 = φ1f

φ1f = 0, π (3.122)

and the second condition gives us

|w1f | = 32e−2πm/λ, (3.123)

which is equivalent to setting r = m/1 in Eq. (3.121), as we should have expected.If λ is big, which is equivalent to saying that the driving frequency is much faster

than the pendulum’s frequency Ω ω0, we can linearize the mapping in w expandingaround a fixed point of period 1 by letting wn+1 → wn+1 = w1f + wn+1, the secondterm in Eq. (3.120) becomes:

λ ln

(32

|wn+1|

)= λ ln

(32

|w1f + wn+1|

)= λ ln

(32

|w1f |

)− λwn+1 − w1f

w1f

+O(wn+12) (3.124)

23The second condition has that form because the phase φn is an angle and returns to its positionafter 2πm.


Then, dropping second order terms and defining a new variable as In = −λwn−w1f

w1f,

Eq. (3.120) becomes

φn+1 = φn + λ ln

(32

|w1f |

)− λwn+1 − w1f

w1f

= φn + λ ln

(32

|w1f |

)+ In+1

= φn+1(w1f ) + In+1

= φn + In+1, (3.125)

where going from the third to the last line we used the fact that w1f is a fixed pointof period 1. As for Eq. (3.119), we plug in the new variable In:

wn+1 = wn +W sinφn

−In+1w1f

λ+ w1f = −Inw1f

λ+ w1f +W sinφn

−In+1w1f

λ= −Inw1f

λ+W sinφn

In+1 = In −λW

w1f

sinφn

In+1 = In +K sinφn, (3.126)

where we define the parameter K = −λWw1f

. The new pair of equations (3.125) and

(3.126) have the same form as the standard map (cf. Eqs. (2.37) and (2.38)). We nowhave arrived at a surprising result: the motion close to the separatrix of a perturbedsystem can be approximated by the standard map. Recall that the standard map hasits own separatrices. Thus the motion of the standard map close to these separatricescan be modeled by itself. Step by step, it goes like this24: first, we consider thesystem given in Eq. (2.34), whose perturbation has a frequency Ω. The system canbe written as the standard map given in Eqs. (2.37) and (2.38). The system haselliptic fixed points or resonances, in whose vicinity the motion resembles that of apendulum. There is a frequency ωJr associated with the motion close to resonance.Then resonance happens between Ω and ωJr, which we call second order resonances.The motion close to these resonances can be described by the standard map givenin Eqs. (3.125) and (3.126). They both have the same form, but they describephenomena at different scales. We already knew about higher order resonances, butwhat we did not know is that the motion close to resonance can be described bythe standard map at all higher levels. This gives a quantitative explanation for theself-similarity present in the standard map.

3.4.1 Application: The Standard Map

We can visually appreciate the stochastic layers that take place in the vicinity of theseparatrix in Fig. 3.17.

24The following discussion is essentially the same that was given after Eq. (3.98).

3.5. Self-Similarity 73

(a) (b)

Figure 3.17: The standard map iterated with I0 = 10−8, θ0 = 0, initial conditions thatare very close to the hyperbolic point and are very likely to be within the chaotic layerclose to the separatrix. For (a) K = 0.6, and for (b) K = 0.9.

3.5 Self-Similarity

According to the KAM theorem, one of the conditions for the KAM tori to exist isthat the perturbation should be small enough. If KAM tori exist, they span acrossphase space, acting as barriers that separate regions of regular and chaotic motion.If we increase the strength of the perturbation past a certain value, KAM tori do notexist anymore. Once KAM tori stop existing, then chaotic regions are not boundedanymore and chaotic trajectories are free to take on any value of In. We concernourselves with this so called global chaos in the following chapter.

If the perturbation is small enough, then KAM tori do exist, and they bound thechaotic motion that happens in the vicinity of the separatrix. But there is one thingwe have not noted yet: since the period close to the separatrix approaches infinity,the rotation number r goes to zero [9]. This means that there are primary resonancesclose to the separatrix for arbitrarily small perturbations, and they correspond toelliptic fixed points of very high periods.

To see this, we use the standard map as an example of a map where these phenom-ena manifest. Let us consider Fig. 3.18. We know that the chaotic motion is boundedby KAM curves to a layer along the separatrix, and this can be easily visualized inFig. 3.18b. We can also see empty “islands” where, allegedly, the motion is stable.Many of such empty islands are in the highlighted region of the plot.

Let us focus on the empty islands for a bit. Fig. 3.19a is the highlighted regionfrom Fig. 3.18b, and Fig. 3.19b is the same as Fig. 3.19a just with more iterations ofthe map. From these two plots, it is clear that the chaotic trajectories do not enter theislands. To see what the motion inside the islands look like, we pick initial conditionsthat fall inside these islands. Fig. 3.20a is plotted in the same range as Fig. 3.19a,but this time we strategically pick initial conditions that fall inside the islands. Oncewe iterate the map for such initial conditions we can see that, amazingly, the motioninside these islands is regular! On top of that, the phase space resembles that of apendulum. We know that such motion happens around elliptic points, and indeedthese are second-order resonances with very high rotation numbers.


(a) (b)

Figure 3.18: K = 0.1 and with 106 number of map iterations. In (a) the range of theplot does not let us appreciate the chaotic layer, making the trajectory seem regular.In (b), however, we zoom in, and we find that the region of chaotic motion is clearlybounded. Islands of stable motion, the regions inside the chaotic sea that are blank,are unexplored by the chaotic trajectories.

(a) (b)

Figure 3.19: Amplification of the highlighted region in Fig. 3.18b. In (a) we cansee the islands of stability corresponding to primary resonances of very high rotationnumber. (b) is the same as (a) but iterated 107 times.

3.5. Self-Similarity 75

(a) (b)

Figure 3.20: Same conditions as Fig. 3.19 but with initial conditions that fall insidethe islands. (a) The motion can be seen to be stable inside these islands. (b) Weamplify the highlighted island in (a).

(a) (b)

Figure 3.21: Blow up of the highlighted region in Fig. 3.20b. (a) was plotted withthe same initial conditions as Fig. 3.20b, whereas for (b) we picked different initialconditions so that the islands of stability could be visible.

However, in our simulation we picked the initial conditions to be inside only oneisland, yet in Fig.3.20a we see that more than one island was plotted. This is because,since the period close to the separatrix goes to infinity, these islands of stabilitycorrespond to a family of elliptic fixed points of a very high period. In terms of thePoincare-Birkoff theorem, the value of b of the rotation number r = a/b is very high,b 1, and for such a rotation number there are kb elliptic fixed points, with k aninteger.

Fig. 3.20b shows the highlighted region of Fig. 3.20a, and it makes it easier toappreciate the structure of phase space in this region.

Chapter 4

Global Chaos and Diffusion in theStandard Map

The description given to us by the Poincare-Birkhoff theorem gave us a better senseof the dynamics in the region close to the hyperbolic points. Based on the qualitativedescription of homoclinic and heteroclinic intersections, it was clear that the chaoticmotion extended itself along the region where a separatrix joining two hyperbolicpoints should have been present. This motivated us to derive a map that describedthe motion in the vicinity of the separatrix, which is called the whisker map [12]. Thewhisker map, in turn, could be linearized around a fixed point, and this linearizationmade the whisker map take the form of the standard map. We already knew thatchaos took place in the standard map, along with regions of regular motion. This wasyet another confirmation that the motion along the separatrix was chaotic.

This means that for the standard map, for low enough K the chaotic motionis local, that is, it manifests itself in the region close to the separatrices delimitedby KAM tori. In this chapter we explore the transition of chaos from being local(happening only in bounded regions of phase space) to being global (happening forall values of In) as a function of the perturbation strength. Once chaos is global,studying individual trajectories does not give us any information and for this reasonwe need to use to statistical mechanics to effectively describe the motion when chaosovertakes phase space.

This chapter deals exclusively with the standard map, as opposed to previouschapters where we kept things more general.

4.1 Overlapping Resonances

By visual inspection of the simulations of the standard map for different initial con-ditions, we note that, for low enough values of K, the chaotic motion is bounded toa layer around the multiple separatrices by KAM tori, as shown in Fig. 4.1. Theselayers have finite widths. As the value of K increases, the layers of chaotic motionstart becoming wider and coming closer together. Additionally, the width of the sep-aratrix layers around resonances corresponding to fixed points of higher periods start

78 Chapter 4. Global Chaos and Diffusion in the Standard Map

Figure 4.1: The standard map iterated with K = 0.9. Chaotic regions have finitewidths and are bounded by KAM curves.

growing, and since chaos also takes place in these layers, we cannot consider suchresonances negligible anymore. Thus we can deduce that, as K increases, the layerswhere chaotic motion happens will eventually touch and overlap. This will allowtrajectories from one layer to have access to a second layer, and if the latter layer istouching a third layer, then the trajectory that was originally confined to the firstlayer can now explore all three layers. Thus, as K increases, chaotic trajectories startovertaking phase space and can now take on values of In that previously they couldnot. In fact, for a critical value of K, which we call Kc, all chaotic layers touch andoverlap each other, and chaotic trajectories are then allowed to take on all possiblevalues of In. This is what we refer to when we talk about global chaos. Since theKAM tori bound chaotic motion, another method to predict global chaos is by findingthe value of Kc for which the last KAM torus disappears.

Chirikov introduced a method that roughly predicts the onset of chaos [12]. Thismethod consists of obtaining the value of K for which two primary resonances corre-sponding to fixed points of period 1 overlap; by overlap we mean that their separatricestouch each other. For the standard map, it goes as follows.

Consider the Hamiltonian for the standard map for only one resonance (cf. Eq.3.102))1:

H(J, θ, t) =J2

2+

K

(2π)2cos γn (4.1)

where γn is the varying phase corresponding to the motion around the resonance givenby Jr = n. The resonance half-width2 for a resonance of the standard map is

∆Jr = 2

√K

(2π)2=

1

π

√K. (4.2)

1Remember that the period of the kicks is T = 2π.2We defined this in Eq. (3.94), and for this case b is equal to one since the resonance we are

working with corresponds to an elliptic fixed point of period one.

4.1. Overlapping Resonances 79

Figure 4.2: The separatrices of the resonances corresponding to In = 0 and In = 2πare seen.

Notice how, as we expected, the width3 of the resonance is a function of K. Toknow when they overlap, we need to know the distance between two resonances. Thisdistance in equal to

δr = Jn+1 − Jn= (n+ 1)− n= 1. (4.3)

By inspection of Fig. 4.2 we can see that, for the separatrices of the two primaryresonances to touch, we need to require their resonance half-width ∆Jr to be equalto half the distance between the primary resonances. Using this condition we have

∆Jr =1

2δr

1

π

√K =

1

2

K =π2

4≈ 2.467 (4.4)

So we have that, according to the overlap criterion, chaos becomes global for K ≈2.467.

As the reader can probably tell, this method makes gross approximations. Weassumed that the separatrices of the two resonances are smooth lines, when they arein fact layers. Also, we actually have more than two primary resonances (in fact, we

3If you are having trouble interpreting what this quantity means, see Fig. 1.6. There we called∆psx the half-width of the separatrix. The resonance half-width is equivalent to the half-width ofthe separatrix. The difference is that the half-width of the separatrix tells us where the separatrix is,whereas the resonance half-width tells us where the separatrix would be if the system was integrable.Remember that in nonintegrable systems, instead of a separatrix there is a chaotic layer.


Figure 4.3: The standard map is iterated for various initial conditions with K ≈0.971635.

have as many as there are rational numbers). These other primary resonances havetheir own chaotic layers that interact with the period one chaotic layers. Ideally, wewould consider all primary resonances along with their widths.

The overlap criterion would work if our system had only two resonances. Thestandard map has more than two resonances, as we can see in Fig. 4.1. Besides theresonances at In = 0 and In = 2π there is another primary resonance at In = πcorresponding to elliptic points of period two.

4.2 Global Chaos

The (accepted) critical value of K is [15]

Kc ≈ 0.971635. (4.5)

found by Green. We know from the KAM theorem that KAM tori survive if theirrotation number is sufficiently irrational (see Figs. 3.1 and 3.2). Green’s methodto determine the critical value Kc assumes that those tori with the most irrationalrotation numbers are the last ones to be destroyed as the perturbation K increases. Aswe pointed out already, the “most irrational” number is the golden mean ϕ =

√5−12

.Thus the last KAM torus to disappear corresponds to that with rotation numberr = ϕ, and this disappearance happens at K = Kc. A plot of the standard map withK = Kc is shown in Fig. 4.3.

For K slightly bigger than Kc, there are still big regions in phase space wheremotion is regular. These regions correspond to elliptic points, the biggest of these

4.3. Diffusion in the Standard Map 81

Figure 4.4: The standard map is iterated for various initial conditions with K = 1.Even though the last KAM torus has been destroyed, there are still regions where themotion is regular.

regions being around the elliptic point of period one. Such a picture of phase spaceis shown in Fig. 4.4.

We have so far assumed that the variable In is mod 2π. Since phase space wasdivided by KAM tori and thus the excursion of chaotic trajectories in the variableIn was bounded, this assumption caused no trouble. However, once phase space isnot divided by KAM tori anymore, chaotic trajectories are allowed to take on anyvalue of In. If we still assume that In ∈ [0, 2π], our description of the dynamics ofthe system would not be correct. So, once K > Kc, we need to allow In ∈ [−∞,∞] .Once chaotic trajectories are free to have any value of In we say that there is diffusionin the variable In.

4.3 Diffusion in the Standard Map

As we have stated many times already, once K > Kc, trajectories in the chaoticsea diffuse in In. The “degree” of diffusion, that is, the actual randomness of thetrajectory depends on the value of K. For low K there still are islands of stability inphase space, and they are of considerable size. A trajectory that starts in a chaoticregion can get trapped by one of the islands of stability for an arbitrary amountof time. As we increase K, these islands of stability start becoming smaller andnegligible. For this reason, we consider values of K for which K Kc, that is, whena chaotic trajectory can explore almost all4 of phase space without getting trappedinto any particular region for arbitrary amounts of time. We can then approximate the

4Small islands where regular motion takes place exist even as K →∞ [12].


system as ergodic5 [12], that is, as a purely random process. We may then treat themap statistically and study ensemble averages, as opposed to individual trajectories.

5Ergodicity (in terms of the standard map) is defined as follows [9]. Let there be a functiong(Jn, θn) where the evolution of the variables is given by the standard map M . Let us define thetime average of g as

g = limN→∞

1

N

N−1∑n=0

g(Mn(Jn, θn)) (4.6)

and the average over all of phase space as

〈g〉 =

∫Ω

g(Jn, θn)dµ. (4.7)

The system is ergodic if

〈g〉 = g (4.8)

where, for the standard map we can write∫

Ωdµ =

∫ 2π

0

∫∞−∞ dJn dθn.

A simple example of ergodicity is the quasiperiodic motion of a trajectory with irrational rotationnumber on a torus [10]. It should be clear that in Fig. 3.4, as time t → ∞, the trajectory getsarbitrarily close to all points on the torus. Similarly, it is uniformly distributed on the torus for allt.

Conclusion

A treatment of the origins of chaos in nonlinear oscillatory Hamiltonian systems wasgiven that would suit the skill level of an undergraduate physics student. With thehelp of the standard map, most of the concepts treated herein were explained. Allof the material in this thesis can be found in the literature. However, most of theexisting literature is difficult to understand with only a background of undergraduateclassical mechanics. I have, to my best effort, tried to provide a thorough expositionof the origins of chaos, making explicit some of the assumptions and mathematicalsteps that are skipped in the literature. I hope that undergraduate students can usethis thesis as a first step in their understanding of chaos.

For projects following this thesis, the interested student could provide a compre-hensive introduction to quantum chaos, and present an exposition of the diffusionpresent in the standard map in the quantum regime.

Appendix A

Poisson Brackets

Let us define the Poisson bracket of two variables A and B with respect to thecanonical variables (q, p) as

[A,B]q,p =N∑i=1

[∂A

∂qi

∂B

∂pi− ∂A

∂pi

∂B

∂qi

](A.1)

or, making use of the symplectic notation, we can write Eq. (A.1) as

[A,B]η =∂A

∂η

†J∂B

∂η. (A.2)

We can immediately derive some of the properties of the Poisson bracket. By pickingA and B as qj and pk, we have the following relations:

[qj, qk]q,p = 0 [pj, pk]q,p = 0 [qj, pk]q,p = δjk − [pj, qk]q,p = δjk. (A.3)

The relations given in Eq. (A.3) can be written as one single statement:

[η,η]η = J (A.4)

where ([η,η]η)ij = [ηi, ηj]η. Now that we have the Poisson bracket as a tool, wecan use it to relate the new canonical variables ξ that we obtained from a canonicaltransformation from the old variables η. Let us evaluate the Poisson bracket of thenew variables with respect to the old variables:

[ξ, ξ]η =∂ξ

∂η

†J∂ξ

∂η. (A.5)

Luckily for us, the Jacobian of the transformation shows up. We have defined thisquantity as in Eq. (1.23). Using this, Eq. (A.5) can be written as

[ξ, ξ]η = DM† J DM. (A.6)

Since the transformation from η to ξ is canonical, the right hand side of Eq. (A.6)satisfies the symplectic condition (cf. Eq. (1.24)). Then we can rewrite Eq. (A.6) as

[ξ, ξ]η = J. (A.7)

86 Appendix A. Poisson Brackets

If Eq. (A.7) holds, then the symplectic condition for the Jacobian matrix of thetransformation holds and so the transformation is canonical. Eq. (A.7) is surprisinglysimilar to Eq. (A.4), [ξ, ξ]ξ = J. From this similarity we can conclude that thePoisson brackets of a pair of canonical variables (i.e., by taking the variables A andB in Eq. (A.2) to be a pair of canonical variables) are invariant under a canonicaltransformation [3]. Expressly, the pair of canonical variables with respect to whichwe take the Poisson brakets is irrelevant.

Why is this relevant for us? Because from the invariance of Poisson bracketswe can state a very important property of canonical transformations: a canonicaltransformation is independent of any particular Hamiltonian [8].

For the sake of completeness, we assert that it can easily be proven that theinvariance of Poisson brackets applies when we pick a pair of any (i.e., arbitrary Aand B) functions.

Appendix B

Dirac Delta Properties

B.1 Fourier Series

In the Hamiltonian of the standard map, Eq. (2.34), the perturbation term consistsof a sum of Dirac Delta functions

∞∑n=−∞

δ(t− nT ). (B.1)

We now obtain its Fourier series expansion starting from

∞∑n=−∞

δ(t− nT ) =∞∑

n=−∞

cnei2πnt/T . (B.2)

To get the Fourier coefficients cn we multiply both sides by e−i2πmt/T and then averageover a period T . The right hand side of Eq. (B.2) gives us

1

T

∫ T/2

−T/2

∞∑n=−∞

cnei2πnt/T e−i2πmt/Tdt =

∞∑n=−∞

cn1

T

∫ T/2

−T/2ei2π(n−m)t/Tdt

=∞∑

n=−∞

cnδmn

= cm. (B.3)

The left hand side of Eq. (B.2) is nonzero only for n = 0 because the only Deltaspike within the range on the integral [−T/2, T/2] corresponds to n = 0. With thisin mind, Eq. (B.2) gives us

1

T

∫ T/2

−T/2

∞∑n=−∞

δ(t− nT )e−i2πmt/Tdt =1

T

∫ T/2

−T/2δ(t)e−i2πmt/Tdt

=1

T(B.4)

88 Appendix B. Dirac Delta Properties

Combining our two recently obtained results, we have that the Fourier coefficientsare

cm =1

T. (B.5)

Thus we have that all the Fourier coefficients have the same value. Then, the sum ofDirac Delta functions can be written in terms of its Fourier series as

∞∑n=−∞

δ(t− nT ) =1

T

∞∑n=−∞

ei2πnt/T . (B.6)

B.2 Rewriting the Perturbation Term of the Stan-

dard Map Hamiltonian

We prove that the Hamiltonian for the standard map can be written as in Eq. (3.99).We start with the perturbation of Eq. (2.33) and make use of the Fourier series ofthe sum of Dirac Delta functions

cos θ∞∑

n=−∞

δ(t− nT ) =eiθ + e−iθ

2

1

T

∞∑n=−∞

ei2πnt/T

=1

2T(eiθ + e−iθ)

(0∑

n=−∞

e2πint/T +∞∑n=1

e2πint/T

)

=1

2T

[0∑

n=−∞

ei(2πnt/T+θ) +0∑

n=−∞

ei(2πnt/T−θ) +∞∑n=1

ei(2πnt/T+θ) +∞∑n=1

ei(2πnt/T−θ)

]

=1

2T

[∞∑n=0

ei(−2πnt/T+θ +0∑

n=−∞

ei(2πnt/T−θ) +−1∑

n=−∞

ei(−2πnt/T+θ) +∞∑n=1

ei(2πnt/T−θ)

]

=1

2T

[∞∑

n=−∞

ei(−2πnt/T+θ) +∞∑

n=−∞

e−i(−2πnt/T+θ)

]

=1

T

∞∑n=−∞

cos (θ − 2πnt/T ) (B.7)

where, in going from the fourth line to the fifth, the terms corresponding to n = 0from the first two terms are “donated” to the last two terms.

Appendix C

Defining Chaos

In this chapter we want to define chaos. Chaos is a rather newly discovered phe-nomenon that occurs in classical systems. In an elementary classical mechanics course,all the systems that are subject to study are integrable and can be solved analytically.Such systems exhibit regular motion. Regular motion can be identified by noticingthat nearby trajectories diverge at most linearly with time. This allows a predictabil-ity that is absent when the motion is chaotic. Chaotic motion, on the other hand,can be identified by noticing that nearby trajectories diverge exponentially. Thisphenomenon is usually referred to as the high sensitivity to initial conditions.

There are different criteria to determine if the trajectories of a region in phasespace are regular or chaotic. In what follows we investigate the Lyapunov exponents,which quantify the rate at which nearby trajectories diverge.

C.1 Lyapunov Exponents

Lyapunov exponents give us a quantitative measure of the degree of chaos in a givenregion [9]; they provide us with the rate of divergence of nearby trajectories. We firstdiscuss continuous systems and then mappings.

Let us have a set of differential equations that describe the motion of an N -dimensional system

dx(t)

dt= F(x). (C.1)

Let us pick two initial conditions that are nearby to each other: x(0) and x(0) +δx(0) where δx(0) is small. We are interested in how the distance between the twotrajectories δx(t) evolves in time. To do this, let us plug in the second initial conditioninto Eq. (C.1):

d

dt(x + δx) = F(x + δx)

dx

dt+dδx

dt= F(x) + δx · dF(x)

dxdδx

dt= δx · dF(x)

dx(C.2)

90 Appendix C. Defining Chaos

Keep in mind that Eq. (C.2) is actually N differential equations, each describing theevolution of the distance between two initial trajectories along their correspondingdirection. However, what we want is the evolution of the norm of the distance betweenthe two trajectories. This is given by

d(t) =√δx(t) · δx(t). (C.3)

If we assume that nearby trajectories diverge exponentially, we can write the equationfor d(t) as

d(t) = d(0)eσt, (C.4)

where σ measures the rate at which nearby trajectories diverge. Solving for σ wehave

σ =1

tlnd(t)

d(0). (C.5)

and we let the time go to infinity and the initial separation to zero:

σ = limd(0)→0

limt→∞

1

tlnd(t)

d(0)(C.6)

If σ is positive, then trajectories that start near each other will diverge as t increases.This is what we mean when we say that the system is sensitive to initial conditions.

Now that we have an intuitive sense of where σ comes from, we look into theanalog of σ in mappings. Let us have a map U

U(xn) = xn+1. (C.7)

Following the procedure above we pick a set of initial conditions xn and xn + δxn.We plug the second initial condition into Eq. (C.7) to get

xn+1 + δxn+1 = U(xn + δxn) (C.8)

δxn+1 = δxn∂xn+1

∂xn(C.9)

We define the Jacobian matrix U of the mapping U as

U =∂xn+1

∂xn. (C.10)

Eq. (C.9) describes the motion after only one iteration. This is not enough informa-tion to determine if nearby trajectories diverge. First, we need the trajectory that apoint takes. The trajectory of an initial condition x1 after n iterations can be writtenas x1,x2, . . . ,xn. We introduce the matrix An

An = [U(xn) ·U(xn−1) · · · · ·U(x1)]1/n (C.11)

that satisfies the following equation

δxn = Annδx1 (C.12)

C.1. Lyapunov Exponents 91

(a) (b)

Figure C.1: In (a) we can see chaotic trajectories only as layers, whereas in (b) chaotictrajectories can take any value of I.

as the reader can easily verify by substituting U by Un in Eq. (C.8). Now we get theeigenvalues of An which we denote λi(n). We can then define the quantity analogousto σ above, which we denote σi [10]:

σi = limn→∞

ln |λi(n)| (C.13)

We call the σi’s Lyapunov exponents. For regular motion, all the Lyapunov exponentsare zero, whereas for chaotic motion, at least one Lyapunov exponent is positive. Thuswe define a chaotic trajectory as that which has positive Lyapunov exponents. Thiscan be summarized as follows: If there exists at least one Lyapunov exponent that ispositive, then the motion is chaotic.

Since the quantity xn is N -dimensional, there will be N Lyapunov exponents. ForHamiltonian maps, since they are area-preserving, the Lyapunov exponents have tosatisfy the following condition

N∑i=1

σi = 0 (C.14)

otherwise phase space volume would not be conserved [9].The chaotic trajectory does not necessarily have to cover all phase space; a tra-

jectory can be confined to a region of phase space and still be chaotic. If chaos onlyhappens in a region of phase space, we call it local chaos, whereas if it spans all phasespace, we call it global chaos. To picture how this can happen, we show in Fig. C.1athe standard map with some regions of phase space having chaotic trajectories andother regions having regular trajectories. In Fig. C.1b we can see global chaos, which,in the standard map, means that the momentum of a chaotic trajectory will comearbitrarily close to any possible value I ∈ [−∞,∞] as the number of iterations n goesto infinity.

For an exposition of the Lyapunov exponents of the standard map, see [20].

References

[1] H. Poincare, Science and Method (Thomas Nelson, 1914).

[2] D. Goroff and H. Poincare, New Methods of Celestial Mechanics, History ofModern Physics and Astronomy (American Inst. of Physics, 1992).

[3] H. Goldstein, C. Poole, and J. Safko, Classical Mechanics (Addison Wesley,2002).

[4] G. L. Baker and J. P. Gollub, Chaotic Dynamics: An Introduction (CambridgeUniversity Press, 1990).

[5] J. Franklin, Advanced Mechanics and General Relativity (Cambridge UniversityPress, 2010).

[6] L. D. Landau and E. M. Lifshitz, Mechanics (Butterworth-Heinemann, 1976).

[7] R. Larson and D. Falvo, Elementary Linear Algebra, Enhanced Edition (CengageLearning, 2009).

[8] H. Iro, A Modern Approach to Classical Mechanics (World Scientific, 2002).

[9] A. J. Lichtenberg and M. A. Lieberman, Regular and stochastic motion, vol. 38(Springer Science & Business Media, 1983).

[10] M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction,Wiley-Interscience publication (Wiley, 1989).

[11] J. Stewart, Multivariable Calculus, Hybrid (Cengage Learning, 2011).

[12] B. V. Chirikov, “A universal instability of many-dimensional oscillator systems,”Physics reports 52, 263 (1979).

[13] E. Ott, Chaos in Dynamical Systems (Cambridge University Press, 2002).

[14] L. Reichl, The Transition to Chaos: Conservative Classical Systems and Quan-tum Manifestations, Institute for Nonlinear Science (Springer, 2004).

[15] J. M. Greene, “A method for determining a stochastic transition,” Journal ofMathematical Physics 20 (1979).

94 References

[16] M. Gutzwiller, Chaos in Classical and Quantum Mechanics, InterdisciplinaryApplied Mathematics (Springer New York, 1991).

[17] D. Rim, “An Elementary Proof That Symplectic Matrices Have DeterminantOne,” arXiv preprint arXiv:1505.04240 (2015).

[18] G. M. Zaslavski, R. Z. Sagdeev, D. A. Usikov, and A. A. Chernikov, Weak Chaosand Quasi-Regular Patterns (Cambridge University Press, 1991).

[19] J. Moser, Stable and Random Motions in Dynamical Systems: With Special Em-phasis on Celestial Mechanics, Hermann Weyl lectures (Princeton UniversityPress, 2001).

[20] U. Tirnakli and E. P. Borges, “The standard map: From Boltzmann-Gibbs statis-tics to Tsallis statistics,” arXiv preprint arXiv:1501.02459 (2015).

[21] D. Giancoli, Physics for Scientists and Engineers with Modern Physics, Physicsfor Scientists & Engineers with Modern Physics (Pearson Prentice Hall, 2008).

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