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COMPLEX HAMILTONIAN DYNAMICS

Tassos BOUNTIS

Department of Mathematics and Center for Research and Applications of Nonlinear

Systems (CRANS),

University of Patras, GR–26500, Rion, Patras, Greece

Lecture 1:

Introduction to Hamiltonian Dynamics: Stability and Chaos

Workshop and Latin–American School on “Foundations of Comp lexity”

Rio de Janeiro, Brasil, October 4–30, 2015

COMPLEX HAMILTONIAN DYNAMICS – p. 1/37

Contents

1. Dynamical Systems and Stability of Equilibrium Points

2. Hamiltonian Systems of N=1 and 2 Degrees of Freedom

3. Stability Analysis of Periodic Orbits

4. Local Dynamics of N–Degree– of–Freedom Hamiltonians

5. An Analytical Criterion of “Weak” Chaos

6. The Spectrum of Lyapunov Exponents and “Strong” Chaos

7. Conclusions

Acknowledgement:This research has been co-financed by the European Union (European Social Fund–ESF) and Greek national funds through the Operational Program "Education andLifelong Learning" of the National Strategic Reference Framework (NSRF)–ResearchFunding Program: Thales. Investing in knowledge through the European Social Fund.


Dynamical Systems and Stability of Equilibrium Points

Dynamical systems in continuous time are described by systems of Ordinary DifferentialEquations (ODEs), in n real dependent variables (xk(t), k = 1, 2, . . . , n, which constitute astate ~x(t) = (x1(t), . . . , xn(t)) in the phase space of the system D ⊆ R

n and are functionsof the single independent variable of the problem: the time t ∈ R. Their dynamics isdescribed by the system of first order ODEs

dxk

dt= fk(x1, x2, . . . , xn), k = 1, 2, . . . , n (1)

Since the fk do not explicitly depend on t the system is called autonomous. The functions fkare defined everywhere in D and are assumed analytic in all their variables, meaning thatthey can be expressed as convergent series expansions in the xk (with non-zero radius ofconvergence) near one of their equilibrium (or fixed) points, located at the origin of phasespace ~0 = (0, . . . , 0) ∈ D, where

fk(~0) = 0, k = 1, 2, . . . , n. (2)


What can we say now about the solutions of these equations in a small neighborhood of theequilibrium point (2), where the series expansions of the fk converge? Is the motion “regular”or “predictable” there? This is the question of stability of motion first studied systematicallyby the great Russian mathematician A. M. Lyapunov, more than 110 years ago.

The first and simplest notion of stability is called asymptotic stability and refers to the casewhere all solutions xk(t) of (1), starting near the origin, tend to 0 as t → ∞. A lessrestrictive situation arises when we can prove that for every 0 < ε < ε0, no matter how small,all solutions starting at t = t0 within a neighborhood of the origin K(ε) j B(ε), where B(ε)

is a “ball” of radius ε around the origin, remain inside B(ε) for all t ≥ t0.

This so-called neutral or conditional stability will be of great importance to us, as it frequentlyoccurs in conservative dynamical systems, among which are the Hamiltonian systems.These conserve phase space volume and hence cannot come to a complete rest at anyvalue of t, finite or infinite. As we shall discuss later, conditional stability characterizesprecisely the systems for which Lyapunov could prove the existence of families of periodicsolutions around the origin.


To discuss the question of stability of the motion near an equilibrium point, we need to knowsomething about the behavior of the solutions of the linearized equations about that point.Thus one might try to compare these solutions to an exponential function of time, with thepurpose of identifying a particular exponent, which we call today the Lyapunov characteristicexponent (LCE).

Let us identify the meaning of these exponents for our problem. Indeed, they are directlyrelated to the eigenvalues of the n× n matrix J = (pjk), j, k = 1, 2, . . . , n, obtained as theroots of the characteristic equation

det(J − λIn) = 0, (3)

λ1, λ2, . . . , λn, In being the n× n identity matrix.

In modern terminology, therefore, consider a dynamical system (1), with an equilibrium pointat (0, 0, . . . , 0) and a constant Jacobian matrix

Jk,j = pkj =∂fk

∂xj

(0, . . . , 0)/ j, k = 1, 2, . . . , n, (4)


The above analysis translates to the following well-known result:

Theorem (see p. 181 Hirsch and Smale, 1974) If all eigenvalues of the matrix J havenegative real part less than −c, c > 0, there is a compact neighborhood U of the origin,such that, for all (x1(0), x2(0), . . . , xn(0)) ∈ U , all solutions xk(t) → 0, as t → ∞.Furthermore, one can show that this approach to the fixed point is exponential: Indeed, if wedenote by | · | the Euclidean norm in R

n and define ~x(t) = (x1(t), x2(t), . . . , xn(t)), it canbe proved that for all ~x(0) in U , |~x(t)| ≤ |~x(0)|e−ct , and |~x(t)| is in U for all t ≥ 0.

Lyapunov paid particular attention to the case where one (or more) of the eigenvalues of thelinearized equations have zero real part. This was the beginning of what we now callbifurcation theory, as it constitutes the turning point between stability of the fixed point (alleigenvalues have negative real part) and instability, where at least one eigenvalue haspositive real part.

This theory can be found, not only in Lyapunov’s treatise “Stability of Motion” but also in amany textbooks on the qualitative theory of ODEs (Hirsch, Smale and Devaney, 2004, Perko,1995 and Wiggins, 1990). One more result of Lyapunov’s theory, concerning simple periodicsolutions of Hamiltonian systems, will be described in the next section.


Hamiltonian Systems of N=1 and 2 Degrees of Freedom

Let us now apply the above theory to the case of Hamiltonian dynamical systems of Ndegrees of freedom (dof), where n = 2N and the equations of motion are

dqkdt

=∂H

∂pk,

dpkdt

= − ∂H

∂qk, k = 1, 2, . . . , N, (5)

where qk(t), pk(t), k = 1, 2, . . . , 2N are the position and momentum coordinates and H isthe Hamiltonian. If H does not explicitly depend on t, it is a first integral (or constant of themotion), whose value equals the total energy of the system E.We now assume that our Hamiltonian can be expanded in power series as a sum ofhomogeneous polynomials Hm of degree m ≥ 2

H = H2(q1, . . . , qN , p1, . . . , pN ) +H3(q1, . . . , qN , p1, . . . , pN ) + . . . = E, (6)

so that the origin is always an equilibrium point of the system. H(qk(t), pk(t)) = E thusdefines the so-called constant energy surface, on which our Hamiltonian dynamics evolves.

If the linear equations resulting from (5) and (6), with Hm = 0 for all m > 2, yield a matrix,

whose eigenvalues all occur in conjugate imaginary pairs, ±iωk, these provide the

frequencies of the normal mode oscillations of the linearized system.


We can then change to normal mode coordinates and write our Hamiltonian in the form of Nuncoupled harmonic oscillators

H(2) =ω1

2(x1

2 + y12) +

ω2

2(x2

2 + y22) + . . .+ωN

2(x2

N + y2N ) = E, (7)

where xk, yk, k = 1, 2, . . . , N are the new position and momentum coordinates and ωk

represent the normal mode frequencies of the system.

Theorem (Lyapunov) If none of the ratios of these eigenvalues, ωj/ωk , is an integer, forany j, k = 1, 2, . . . , N , j 6= k, the linear normal modes continue to exist as periodic solutionsof the nonlinear system (5) when higher order terms H3, H4, . . . etc. are taken into accountin (6).These solutions have frequencies close to those of the linear modes and are examples ofwhat we call simple periodic orbits (SPOs), where all variables oscillate with the samefrequency ωk = 2π/Tk , returning to the same values after a single maximum (andminimum).These are also called nonlinear normal modes, or NNMs. As we vary the total energy E in(6) their stability under small perturbations of their initial conditions changes and although alocal property, is often relevant for the more global stability properties of the system!


The case ofN = 1 degree of freedom

One of the first physical systems that we encounters in our studies is the harmonic oscillatorshown in Fig. 9 described by Newton’s second order differential equation

md2q

dt2= −kq, (8)

where k > 0 is a constant representing the hardness (or softness) of the spring. Equation (8)can be easily solved to yield the displacement q(t) as an oscillatory function of time of theform

q(t) = A sin(ωt+ α), ω =√

k/m, (9)

where A and α are free constants corresponding to the amplitude and phase of oscillationsand ω is the frequency.


If we now recall that p = mdq/dt represents the momentum of the mass m, we rewrite (8) inthe form of two first order ODEs

dq

dt=

p

m=

∂H

∂p,

dp

dt= −kq = −∂H

∂q, (10)

derived from the N = 1 dof Hamiltonian function

H(q, p) =p2

2m+ k

q2

2= E, (11)

which represents the total (kinetic plus potential) energy. If we plot the solutions as orbits inthe (q, p) phase space we obtain the curves shown below, where q(t) and p(t) oscillateperiodically with the same frequency ω.


A nonlinear system of N=1 degrees of freedom

A more interesting one dof Hamiltonian system representing the motion of a simplependulum shown in Fig. 11. Its equation of motion is

ml2d2θ

dt2= −mlg sin θ, (12)

If we now write this equation as a system of two first order ODEs, we find again that they canbe cast in Hamiltonian form

dq

dt= p =

∂H

∂p,

dp

dt= − g

lsin q = −∂H

∂q, (13)

where q = θ.


In this case, the energy integral is

H(q, p) =p2

2+

g

l(1− cos q) = E. (14)

Plotting this family of curves in the (q, p) phase space for different values of E we now obtaina much more interesting picture than Fig. 10 depicted in Fig. 12 below. Observe that, besidesthe elliptic fixed point at the origin, there are two new equilibria located at the points (±π, 0).These points A and B are called saddle points and are unstable in contrast to the (0, 0) fixedpoint, which is stable characterized by what we called conditional (or neutral) stability.


The case of N=2 degrees of freedom: Integrability and chaos

Let us extend our study to Hamiltonian systems of two dof, joining at first two harmonicoscillators, as shown in Fig. 13. We shall assume that our oscillators have equal massesm1 = m2 = m and spring constants k1 = k2 = k and impose fixed boundary conditions totheir endpoints.

Newton’s equations of motion give in this case:

md2q1

dt2= −kq1 − k(q1 − q2), m

d2q2

dt2= −kq2 + k(q1 − q2), (15)

where qi(t) are the particles’ displacements from their equilibrium positions at qi = 0,i = 1, 2. If we also introduce the momenta pi(t) of the two particles in terms of theirvelocities, we obtain the Hamiltonian function

H(q1, p1, q2, p2) =p212m

+p222m

+ kq212

+ kq222

+ k(q1 − q2)2

2= E (16)


If we now change variables

Q1 =q1 + q2√

2, Q2 =

q1 − q2√2

, P1 =p1 + p2√

2, P2 =

p1 − p2√2

(17)

we see that, adding and subtracting by sides the two equations in (15) (dividing also by m

and introducing ω =√

k/m), splits the problem into two uncoupled harmonic oscillators

dQi

dt=

Pi

m,

dPi

dt= −ω2

i Qi, i = 1, 2, ω1 = ω, ω2 =√3ω, (18)

with frequencies ω1, ω2. The new Hamiltonian of the system

K(Q1, P1, Q2, P2) =P 21

2m+

P 22

2m+ k

Q21

2+ 3k

Q22

2= E. (19)

is expressed as the sum of the Hamiltonians of these oscillators. Thus, changing variableswe have performed a canonical coordinate transformation and realize that our systempossesses two integrals of the motion

Fi(Q1, P1, Q2, P2)) =P 2i

2m+ ki

Q2i

2= Ei, i = 1, 2, (20)


with k1 = k, k2 = 3k, while Ei are two free parameters of the system to be fixed by the initialconditions qi(0), pi(0), i = 1, 2.The solutions of this system are, in general, linear combinations of trigonometric functionswith frequencies ω1 =

√k, ω2 =

√3k. If these were rationally dependent, i.e. if their ratio

were a rational number ω1/ω2 = m1/m2 (m1,m2 all orbits close on 2-dimensional invarianttori and the motion would be periodic. In our example, this could only happen for initialconditions such that E1 or E2 is zero.For E1 and E2 both non-zero the oscillations are quasiperiodic, as they are thesuperposition of trigonometric terms whose frequencies are rationally independent, since theratio ω2/ω1 =

√3 is irrational. Hence, the orbits in the 4-dimensional phase space are never

closed, i.e. they never pass by the same point and eventually cover uniformly a2-dimensional torus specified by the values of E1 and E2.


A nonlinear system of N=2 degrees of freedom

We now turn to a system of two coupled nonlinear oscillators connected with the famousHénon and Heiles Hamiltonian

H =1

2(p21 + p22) +

1

2(ω2

1q21 + ω2

2q22) + q21q2 − C

3q32 . (21)

describing the motion of a star of mass m in the axisymmetric potential of a galaxy.Introducing the more convenient variables q1 = x, q2 = y, p1 = px, p2 = py , we rewrite theabove Hamiltonian in the form

H =1

2(p2x + p2y) + V (x, y) =

1

2(p2x + p2y) +

1

2(Ax2 + By2) + x2y − C

3y3 = E, (22)

where E is the total energy and we have set ω21 = A > 0 and ω2

2 = B > 0. Newton’sequations of motion associated with this system are

d2x

dt2= −∂V

∂x= −Ax− 2xy,

d2y

dt2= −∂V

∂y= −By − x2 + Cy2. (23)


Note that (22) represents a first integral of this system. If we could also find a second one,the problem would be completely integrable and could be integrated by quadratures. This,however, is highly unlikely in general!The surprising result is there are only 3 known cases in which a second integral exists,allowing one to solve the Hénon-Heiles equations completely:

Case 1 : A = B, C = −1,

Case 2 : A,B free, C = −6,

Case 3 : B = 16A, C = −16. (24)

In these cases, most orbits (in phase space domains of bounded motion) would bequasiperiodic and lie on 2-dimensional tori rendering the dynamics perfectly regular andpredictable. For all other parameter values, one finds (besides periodic and quasiperiodicorbits), a new kind of solution that appears “irregular” and “unpredictable”, which we callchaotic. These solutions tend to occupy densely 3-dimensional regions in the 4-dimensionalphase space and depend very sensitively on initial conditions, in the sense that almost allother orbits in their vicinity deviate exponentially from the chaotic orbit as time increases.


Figure 1: On the Poincare Surface of Section (y, py) for (a) E = 1/24 and (b) E = 1/12, nochaotic orbits are visible. At (c) E = 1/8, we observe islands of order surrounded by chaos.In (d), where E = 1/6, chaotic motion extends over most of phase space.


Stability Analysis of Periodic Orbits

To discuss the stability of periodic orbits, we need two fundamental concepts: The first ismore analytical and is provided by Floquet theory and the second is more numerical andrefers to the so-called Poincaré map and its associated surfaces of section (PSS).In particular, we will assume that our n-dimensional dynamical system, cast in the generalform ~x = ~f(~x) (see (1)) has a periodic solution ~x(t) = ~x(t+ T ) of period T . Let us choosean arbitrary point along this orbit ~x(t0) and define a PSS at that point as follows

Σt0 =

~x(t) / (~x(t)− ~x(t0)) · ~f(~x(t0)) = 0

. (25)

Thus, Σt0 is a (n− 1)-dimensional plane which intersects the given periodic orbit at ~x(t0)and is vertical to the direction of the flow at that point. Clearly now a Poincaré map can bedefined on that plane, by

P : Σt0 → Σt0 , ~xk+1 = P~xk, k = 0, 1, 2, . . . (26)

for which ~x0 = ~x(t0) is a fixed point, since ~x0 = P~x0.


We now examine small deviations about this point,

~xk = ~x0 + ~ηk, ||~ηk|| ≪ ε, (27)

(where ε is of the same magnitude as ||~x0||), substitute (27) in (26) and linearize thePoincaré map to obtain

~ηk+1 = DP (~x0)~ηk, (28)

where we have neglected higher order terms in ~η and DP (~x0) denotes the Jacobian of Pevaluated at ~x0.To determine P we may use the variational equations of the original differential equationsderived by writing ~x(t) = ~x(t) + ~ξ(t), whence linearizing (1) about this periodic orbit leads tothe system

~ξ(t) = A(t)~ξ(t), A(t) = A(t+ T ), (29)

where A(t) is the Jacobian matrix of ~f(~x) evaluated at the periodic orbit ~x(t) = ~x. Thecrucial question, of course, we must face now is: How are the two linear systems (28) and(29) related to each other?


Observe that we have used different notations for the small deviations about the periodicorbit: ~ξ(t) in continuous time (for the differential equations) and ~ηk in discrete time (for thePoincaré map). Furthermore, their dimensionality as vectors in the n-dimensional phasespace R

n (n = 2N for a Hamiltonian system) is different: ~ξ(t) is n-dimensional, while ~ηk is(n− 1)-dimensional! How are we to match these two vectors of small deviations?

The answer comes from what is called Floquet theory (see Perko, 1995,Wiggins, 1990). Firstwe realize that since (29) is a linear system of ODEs it must possess, in general, n linearlyindependent solutions, forming the columns of the n× n fundamental solution matrixM(t, t0) in

~ξ(t) = M(t, t0)~ξ(0), M(t, t0) = M(t+ T, t0) (30)

Now, if we change our basis at the point ~x(t0) so that one of the directions of motion is alongthe direction vertical to the PSS (25), we will observe that the nth column of the matrixM(T, t0) has zero elements except at the last entry which is 1. Thus, if we eliminate fromthis matrix its nth row and nth column, it turns out that its (n− 1)× (n− 1) submatrix isnone other than the Poincaré map (26) itself!


This means that if we could compute the so-called monodromy matrix M(T, t0) numericallywe could evaluate its eigenvalues, µ1, . . . , µn−1 (the last one being µn = 1), which arethose of the Poincaré map and determine the stability of our periodic orbit as follows: If theyare all on the unit circle, i.e. |µi| = 1, i = 1, . . . , n− 1, the periodic orbit is (linearly) stable,while if (at least) one of them satisfies |µj | > 1 the periodic solution is unstable.

But how do we compute the monodromy matrix M(T, t0)? It is not so difficult. Let us first sett0 = 0 for convenience and observe from (30) that M(0, 0) = In. All we have to do isintegrate numerically the variational equations (30) from t = 0 to t = T , n times, each timefor a different initial vector (0, . . . , 0, 1, 0, . . . , 0) with 1 placed in the ith position,i = 1, 2, . . . , n.

Note that since these equations are linear numerical integration can be performed toarbitrary accuracy and is also not too-time consuming for reasonable values of the period T .Once we have calculated M(T, 0), we may proceed to compute its eigenvalues anddetermine the stability of the periodic orbit according to whether at least one of theseeigenvalues has magnitude greater than 1.


Local Dynamics ofN–Degree– of–Freedom Hamiltonians

The Fermi–Pasta–Ulam (FPU)- β model

The FPU -β one-dimensional lattice is described by the Hamiltonian

H =1

2

N∑

j=1

x2j +

N∑

j=0

(

1

2(xj+1 − xj)

2 +1

4β(xj+1 − xj)

4

)

= E (31)

Let us focus on Simple Periodic Orbits (SPOs), where all variables oscillate in or out ofphase and return to their initial state after only one maximum and one minimum in theiroscillation. The SPOs we consider here are:

I. The FPU π-mode under periodic boundary conditions :

xN+k(t) = xk(t), ∀t, k (32)

where the particles execute out-of-phase motion (OPM) with N even

xj(t) = −xj+1(t) ≡ x(t), j = 1, . . . , N. (33)


II. For the FPU model and fixed boundary conditions :

x0(t) = xN+1(t) = 0, ∀t (34)

(a) The SPO1 mode , with N odd,

x2j(t) = 0, x2j−1(t) = −x2j+1(t) ≡ x(t), j = 1, . . . ,N − 1

2. (35)

(b) The SPO2 mode , with N = 5 + 3m, m = 0, 1, 2, . . . particles,

x3j(t) = 0, j = 1, 2, 3 . . . ,N − 2

3, (36)

xj(t) = −xj+1(t) = x(t), j = 1, 4, 7, . . . , N − 1. (37)

Let us see some of these solutions graphically in the figure of the next page:


An application of Floquet stability analysis

Stability analysis of SPOs is performed by studying the eigenvalues of the monodromymatrix. To see how this is done consider the SPO1 mode of FPU, for which the equations ofmotion collapse to a single second order ODE:

¨x(t) = −2x(t)− 2βx3(t) (38)

Its solution is well–known in terms of Jacobi elliptic functions

x(t) = C cn(λt, κ2) (39)

with modulus κ2. Linearizing about this solution xj = xj + ξj , by keeping up to linear termsin ξj , we get the variational equations

ξj = (1 + 3βx2)(ξj−1 − 2ξj + ξj+1), j = 1, . . . , N (40)

which are equivalent to (30). In our case here, these separate into N uncoupled Laméequations

zj(t) + 4(1 + 3βx2)sin2(

πj

2(N + 1)

)

zj(t) = 0, j = 1, . . . , N (41)


where the zj variations are simple linear combinations of the ξj ’s.

Changing variables to u = λt, the above equation takes the form

z′′j (u) + 2(

1 + 4κ2 − 6κ2sn2(u, κ2))

sin2(

πj

2(N + 1)

)

zj(u) = 0, j = 1, . . . , N (42)

where primes denote differentiation with respect to u. According to Floquet theory, itssolution is bounded (or unbounded) depending on whether the eigenvalues of themonodromy matrix are on or off the unit circle .

These periodic solutions all experience a first destabilization at energy densities :

Ec

N∝ N−α, α = 1, or, 2, N → ∞. (43)

More specifically, the first variation zj(u) to become unbounded as κ2 (or, the energy E)increases is j = N−1

2and the energy values Ec/N ∝ 1/N at which this happens are

plotted below.


5 10 100 500

10-2

10-1

100

E c/N

N

Figure 2: Ec

Nof the first destabilization of the SPO1 solution of the FPU system (the

k = (N + 1)/2 mode ) obtained by the monodromy eigenvalues. The dashed line is thefunction ∝ 1

N.

The OPM (or π-mode) on the other hand, becomes unstable, as the eigenvalues of theirmonodromy matrix exit the unit circle on the real axis at -1, giving rise to a sequence ofperiod–doubling bifurcations. It is associated with chaos over a wide region of phase space,since its first destabilization occurs at a relatively high value o f the total energy .


An Analytical Criterion for “Weak” Chaos

As was shown recently by Flach et al. (2005), the linear modes of the FPU β – model can becontinued as SPOs of the corresponding lattice. In fact, the energy threshold for thedestabilization of the low k - modes (k = 1, 2, 3, ..) coincides with the “weak” chaosthreshold shown by de Luca and Lichtenberg (1995) to be associated with the breakup ofthe famous FPU recurrences . By k–modes, we refer here to the linear normal modes of theFPU lattice

Qk =

√

2

N + 1

N∑

i=1

qi sinkiπ

N + 1, Pk = Qk (44)

with energies and frequencies

Ek =1

2

[

P 2k + ωk

2Q2k

]

, ωk = 2 sinkπ

2(N + 1)(45)

Using linear stability analysis, Flach et al. (2005) reported an approximate formula for thedestabilization energy of the low k = 1, 2, 3, .. modes given by

Ec ≈ π2

6β(N + 1). (46)


Evidently, this destabilization occurs at much lower energies than the π-mode, or theSPO1 mode since Ec/N ∝ N−2 for large N and and is thus associated with “thin” chaoticlayers (or “weak chaos”) in phase space . In fact, this energy threshold coincides withthe instability threshold of our SPO2 mode (see Figure 3 below)!

5 6 8 10 20 40 60 80 100 2000.2

0.4

0.60.8

1

2

4

68

10

20

N

E2u

(SPO2) E

c (q=3 breather solution)

Figure 3: The solid curve is the energy Ec/N of the first destabilization of the SPO2 (thek = 2(N + 1)/3 mode ) for β = 0.0315 obtained from the eigenvalues of the monodromymatrix and the dashed line is the approximate formula.


Spectrum of Lyapunov Exponents and “Strong” Chaos

Convergence of Lyapunov Spectra

We now evaluate, in the neighborhood of our SPOs the Lyapunov spectra :

Li, i = 1, . . . , 2N, L1 ≡ Lmax > L2 > . . . > L2N . (47)

If the largest one, L1 ≡ Lmax > 0, the orbit is chaotic. In particular, we compute in the limitt → ∞ the quantities

Kit =

1

tln

‖ ~wi(t) ‖‖ ~wi(0) ‖

, (48)

where ~wi(0) and ~wi(t), i = 1, . . . , 2N are deviation vectors from the given orbit ~x(t). Afterevery Tj , following the Bennetin et al.(1980) algorithm, we ortho–normalize the vectors ~wi(t)

and obtain finally Li by

Li = limt→∞

1

n

n∑

j=1

KiTj

, n → ∞. (49)


Observe that in the figure below we have plotted the Lyapunov spectrum of both the OPMand the SPO1 mode of the FPU Hamiltonian for N = 16 and periodic boundaryconditions at the energy E = 6.82 where both of them are unstable. We see that the twoLyapunov spectra are very close to each other suggesting that the chaotic regions are“connected" .

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160.00

0.01

0.02

0.03

0.04

0.05

0.06E=6.82

(c) FPU OPM FPU OHS

FPU OPM & FPU OHS N=16

L i

i

Figure 4: Lyapunov spectra of the OPM and the SPO1 modes of FPU, for N = 16 and

periodic boundary conditions practically coincide at E = 6.82 where both are unstable.


More systematically, let us plot the Lyapunov spectra near the SPO1 and SPO2 modes, ofthe FPU system, for N = 11, fixed boundary conditions and energy values E1 = 1.94 andE2 = 0.155 respectively, where the SPOs have just destabilized.

1 2 3 4 5 6 7 8 9 10 110.00000

0.00002

0.00004

0.00006

0.00008

0.00010

0.00011(a)

N=11

Li

i

Neighbor orbit of SPO1, Energy=1.94 Neighbor orbit of SPO2, Energy=0.155

1 2 3 4 5 6 7 8 9 10 110.000

0.005

0.010

0.015

0.020(b)

N=11

Li

i



1 2 3 4 5 6 7 8 9 10 110.000

0.004

0.008

0.012

0.016

0.020

0.024

0.028(c)

N=11

Li

i


1 2 3 4 5 6 7 8 9 10 110.0000

0.0075

0.0150

0.0225

0.0300

0.0375

0.0450

0.0525(d)

N=11

Li

i

Neighbor orbit of SPO1, Energy=5 Neighbor orbit of SPO2, Energy=5

Figure 5: Lyapunov spectra near SPO1 and SPO2 for N = 11 at energies E = 1.94 and

E = 0.155, where they have just destabilized. (b) Same as (a) at energy E = 2.1. (c)

Convergence of the Lyapunov spectra near the two SPOs at energy E = 2.62. (d)

Coincidence of Lyapunov spectra continues at higher energy E = 4.


Conclusions

1) We defined dynamical systems of continuous time and concentrated in particular onHamiltonian systems of N degrees of freedom. Focusing on linear and nonlinearHamiltonians of N = 1 and N = 2 degrees of freedom we described the concept ofequilibrium points and discussed their stability analysis.

2) We then examined the important notions of periodic and quasiperiodic orbits andintroduced the topics of integrability (related to regular or predictable behavior) and chaos(connected with unpredictable dynamics). We mentioned Floquet theory as the main tool forstudying the stability of periodic orbits.

3) We studied local and global dynamics of N dof Hamiltonian systems, focusing on simpleperiodic orbits (SPOs) and showed that their first destabilization typically decays as

Ec/N ∝ N−α, α = 1, or, 2. (50)

4) We explained on the FPU β− model how low k normal modes of the linear lattice (butalso relatively higher ones like k = 2(N + 1)/3), are important for the global dynamics, sincetheir destabilization threshold obeys


Ec ≈ π2

6β(N + 1), (51)

which is of the form Ec/N ∝ 1/N2 and is thus connected with the onset of“weak" chaos .

5) On the other hand, there are other linear modes, like the π-mode of the FPU lattice underperiodic boundary condition, which when continued in the nonlinear regime destabilize athigher energies , at thresholds obeying

Ec

N∝ 1

N, N → ∞. (52)

and are thus related to the onset of “strong chaos” in these models.

6) We calculated the Lyapunov spectra in the vicinity of our SPOs solutions and observedthat, as E increases, they attain same functional form,

Li(N) ∝ e−α iN , i = 1, 2, . . . ,K(N) (53)

and eventually converge, implying that the corresponding chaotic regions have “merged” andlarge scale chaos has spread in the system.


References

1. Skokos, Ch. [2001], “Alignment Indices: A New, Simple Method for Determining theOrdered or Chaotic Nature of Orbits", J. Phys. A, 34, pp. 10029 – 10043.2. Skokos, Ch., Antonopoulos, Ch., Bountis, T. C. and Vrahatis, M. N. [2003], “How does theSmaller Alignment Index (SALI) Distinguish Order from Chaos?", Prog. Theor. Phys. Supp., 150,pp. 439 – 4433. Skokos, Ch., Antonopoulos, Ch., Bountis, T. C. and Vrahatis, M. N. [2004], “DetectingOrder and Chaos in Hamiltonian Systems by the SALI Method", J. Phys. A, 37, pp. 6269– 6284.4. Antonopoulos, Ch., Bountis, T. C. and Skokos, Ch., [2006], “Chaotic Dynamics ofN–Degree–of–Freedom Hamiltonian Systems”, International Journal of Bifurcation andChaos, vol.16(6), 1777-1793 , June 2006.5. Antonopoulos, Ch. and Bountis, T., [2006], “Stability of Simple Periodic Orbits and Chaosin an FPU Lattice”, PRE73, 056206, 1-8 (2006).6. T. Bountis, "Stability of Motion: From Lyapunov to N - Degree of Freedom HamiltonianSystems", [2006], "Nonlinear Phenomena and Complex Systems", vol. 9(3) ,209 -239, 2006.7. Skokos, Ch., Bountis, T. and Antonopoulos, Ch. [2007], “Geometrical properties of localdynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method”,Physica D 231, 30.


COMPLEX DYNAMICS OF MULTIDIMENSIONAL HAMILTONIAN SYSTEMS

Tassos BOUNTIS, Haris Skokos, Chris Antonopoulos and Thanos Manos


Systems (CRANS),

University of Patras, GR–26500, Rion, Patras, Greece

Lecture 2:

Order and Chaos in Multi-Dimensional Hamiltonian Systems



COMPLEX DYNAMICS OF MULTIDIMENSIONAL HAMILTONIAN SYSTEMS – p. 1/35

Contents

1. The Method of Lyapunov Exponents

2. GALIk(t): Chaos and Order in Multidimensions

3. Analytical Estimates for the GALIk(t) Indices

4. Numerical Verification and Applications

5. Low - Dimensional Tori and Weak Diffusion

6. Conclusions



Complex Hamiltonian Dynamics

Tassos Bountis and Haris SkokosSpringer Synergetics series, April 2012

Foreword by Sergej Flach

Chapter 1. Fundamental concepts of Lyapunov and Poincare

Chapter 2. Hamiltonian Systems of Few Degrees of Freedom

Chapter 3. Local and Global Stability of Motion

Chapter 4. Normal Modes, Symmetries and Stability

Chapter 5. Efficient Indicators of Stable and Chaotic Motion

Chapter 6. FPU Recurrences and the Transition from Weak to Strong Chaos

Chapter 7. Localization and Diffusion in Nonlinear 1-Dimen sional Lattices

Chapter 8. Complex Statistics of Quasi-Stationary States


Chaotic Dynamics ofN–Degree– of–Freedom Hamiltonian Systems

We consider conservative systems described by the N - degree - of freedom Hamiltonian:

H ≡ H(q1(t), . . . , qN (t), p1(t), . . . , pN (t)) = E (1)

where E is the total (constant) energy and the positions qj(t), j = 1, . . . , N and momentapj(t), j = 1, . . . , N are solutions of Hamilton’s equations:

dqj(t)

dt=

∂H

∂pj(t),

dpj(t)

dt= − ∂H

∂qj(t), j = 1, . . . , N. (2)

which are 2N first order ODEs which describe the orbits of the system in the 2N–dimensionalphase space. One particular example we will be concerned with here is the famous β−FPU1–dimensional lattice described by the Hamiltonian

H =1

2

N∑

j=1

x2j +

N∑

j=0

(

1

2(xj+1 − xj)

2 +1

4β(xj+1 − xj)

4

)

= E (3)

where qj = xj , pj = xj .


Distinguishing Order from Chaos in Many Dimensions

The Method of Lyapunov Exponents

Perhaps the most popular method for identifying solutions of dynamical systems as chaoticis to evaluate, in the neighborhood of a solution the spectrum of Lyapunov exponents :

Li, i = 1, . . . , 2N, L1 ≡ Lmax > L2 > . . . > L2N . (4)

If the largest one, L1 ≡ Lmax > 0, the orbit is chaotic. In particular, we compute for largetime intervals t = Tj the quantities

Kit =

1

tln

‖ ~wi(t) ‖‖ ~wi(0) ‖

, (5)

where ~wi(t), i = 1, . . . , 2N are deviation vectors orthogonal to the vector tangent to thegiven orbit ~x(t). Since orbits in chaotic regions rapidly deviate from each other, we need torepeat the calculation over many time intervals Tj , ortho–normalizing the vectors ~wi(t) andchoosing new ~wi(0) every time to obtain Lyapunov spectrum Li averaging over theseintervals

Li = limn→∞

1

n

n∑

j=1

KiTj

. (6)


GALI k(t): Chaos and Order in Multidimensions

So, to see if an orbit is chaotic, we monitor the evolution of only one deviation vector andwait to see whether it will align with the direction corresponding to the Lmax > 0. Thisprocess, however, has notorious convergence problems and fails to provide a clearanswer especially if Lmax is very close to 0.

Thus, instead of following one deviation vector, we choose to study the mutual relation ofmany of them taken together. For example, note that the ‘wedge’ product of k such unitvectors , w1 ∧ w2 ∧ ... ∧ wk, yields the ‘volume’ of a parallelepiped formed by these k

vectors as vertices. If the orbits are chaotic they will align in the Lmax > 0-direction and thevolume would vanish . If, on the other hand, there is no such direction the vectors willremain linearly independent and the volume will be bounded away from zero .

Now, all such normalized deviation vectors wi, i = 1, 2, . . . , 2N , can be written as

wi =2N∑

j=1

wij ej , i = 1, 2, . . . , k (7)

in terms of the usual orthonormal basis of the n–dimensional Euclidean space.


The wedge product of these k deviation vectors can be written with respect to the basis eik

w1∧w2∧· · ·∧wk =∑

1≤i1<i2<···<ik≤2N

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

w1i1 w1i2 · · · w1ik

w2i1 w2i2 · · · w2ik

......

...

wki1 wki2 · · · wkik

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

ei1∧ei2∧· · ·∧eik (8)

If at least two of the normalized deviation vectors wi, i = 1, 2, . . . , k are linearlydependent , all the k × k determinants will become zero making the ‘volume’ vanish . Wedefine the norm of this quantity, ‖w1(t) ∧ w2(t) ∧ · · · ∧ wk(t)‖, as the generalizedalignment index

GALIk(t) =

∑

1≤i1<i2<···<ik≤2N

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

w1i1 w1i2 · · · w1ik

w2i1 w2i2 · · · w2ik

......

...

wki1 wki2 · · · wkik

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

2

1/2

(9)


Analytical Estimates for the GALI k(t) Indices

If at least one of the deviation vectors becomes linearly dependent on the remaining ones,GALIk(t) will tend to zero as the volume of the parallelepiped having the vectors wi asedges shrinks to zero. On the other hand, if GALIk(t) remains far from zero , this wouldindicate the linear independence of the deviation vectors.

Theoretical results for the Time Evolution of GALIExponential decay of GALI for chaotic orbitsLet us first investigate the dynamics in the vicinity of a chaotic orbit of a Hamiltonian system,with N degrees of freedom. Let σ1 ≥ σ2 ≥ · · · ≥ σ2N be the ‘local Lyapunov exponents’oscillating about their time averaged values

σ1 ≥ σ2 ≥ · · · ≥ σN−1 ≥ σN = σN+1 = 0 ≥ σN+2 ≥ · · · ≥ σ2N . (10)

which are the ‘global’ LCEs. Assuming that the σi, i = 1, 2, . . . , 2N do not fluctuatesignificantly, the evolution of any initial deviation vector ~wi may be written as

~wi(t) =2N∑

j=1

cij eσjt ej , (11)


Consequently, the matrix C of coefficients of the k normalized deviation vectorswi(t) = ~wi(t)/‖~wi(t)‖, i = 1, 2, . . . , k expressed in terms of the basis e1, e2, . . . , e2Nbecomes

C(t) = [cij ] =

s1c12|c11|

e−(σ1−σ2)t c13|c11|

e−(σ1−σ3)t · · · c12N|c11|

e−(σ1−σ2N )t

s2c22|c21|

e−(σ1−σ2)t c23|c21|

e−(σ1−σ3)t · · · c22N|c21|

e−(σ1−σ2N )t

......

......

skck2|ck1 |

e−(σ1−σ2)t ck3|ck1 |

e−(σ1−σ3)t · · · ck2N|ck1 |

e−(σ1−σ2N )t

,

(12)

with si = sign(ci1) and i = 1, 2, . . . , k, j = 1, 2, . . . , 2N .

This has been accomplished using the fact that a leading order estimate of the deviationvector’s Euclidean norm, for long enough times, is

‖~wi(t)‖ ≈ |ci1|eσ1t . (13)


The determinants appearing in the definition of GALIk (see equation (9) can be divided intwo categories depending on whether or not they contain the first column of matrix C .Indeed, we see that those that do contain the first column yield

D1,j1,j2,...,jk−1=

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

s1c1j1|c11|

e−(σ1−σj1)t · · ·

c1jk−1

|c11|e−(σ1−σjk−1

)t

s2c2j1|c21|

e−(σ1−σj1)t · · ·

c2jk−1

|c21|e−(σ1−σjk−1

)t

......

...

skckj1|ck1 |

e−(σ1−σj1)t · · ·

ckjk−1

|ck1 |e−(σ1−σjk−1

)t

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

=

=

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

s1c1j1|c11|

· · ·c1jk−1

|c11|

s2c2j1|c21|

· · ·c2jk−1

|c21|

......

...

skckj1|ck1 |

· · ·ckjk−1

|ck1 |·

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

e−[

(σ1−σj1)+(σ1−σj2

)+···+(σ1−σjk−1)]

t(14)


Thus, the time evolution of D1,j1,j2,...,jk−1is mainly determined by the exponential law

D1,j1,j2,...,jk−1∝ e

−[

(σ1−σj1)+(σ1−σj2

)+···+(σ1−σjk−1)]

t. (15)

The determinants that do not contain the first column of the transformation matrix tend tozero following an exponential law that decays faster .

Clearly, from all determinants the one that decreases the slowest is the one containing thefirst k columns of matrix C, yielding the approximation that GALIk tends to zero as

GALIk(t) ∝ e−[(σ1−σ2)+(σ1−σ3)+···+(σ1−σk)]t . (16)


The evaluation of GALI for ordered orbits

As is well–known, ordered orbits typically lie on N–dimensional tori, which can be describedby a local transformation to action–angle variables, satisfying

Ji = 0

θi = ωi(J1, J2, . . . , JN )i = 1, 2, . . . , N. (17)

These can be easily integrated to give

Ji(t) = Ji0

θi(t) = θi0 + ωi(J10, J20, . . . , JN0) ti = 1, 2, . . . , N, (18)

By denoting as ξi, ηi, i = 1, 2, . . . , N small deviations of Ji and θi respectively, thevariational equations of system (18), can be solved to yield:

ξi(t) = ξi(0)

ηi(t) = ηi(0) +[

∑Nj=1 ωijξj(0)

]

ti = 1, 2, . . . , N. (19)


This implies a linear increase of the norm of the deviation vector

‖~w(t)‖ =

1 +

N∑

i=1

N∑

j=1

ωijξj(0)

t2 +

2N∑

i=1

ηi(0)N∑

j=1

ωijξj(0)

t

1/2

, (20)

since ‖~wi(t)‖ ∝ t as t → ∞. Therefore, the normalized deviation vector ~wi(t)/‖~wi(t)‖tends to fall on the tangent space of the torus , since its coordinates perpendicular to thetorus vanish following a t−1 law.Let us now study the case of k, general, linearly independent unit deviation vectorsw1, w2, . . . , wk with 2 ≤ k ≤ 2N . Using as vector basis the set e1, e2, . . . , e2N we get:

[

w1 w2 . . . wk

]T= D(t) ·

[

e1 e2 . . . e2N

]T(21)

If no deviation vector is initially located in the tangent space of the torus, matrix D has theform

D(t) = [dij ] =1

∏km=1 ‖~wm(t)‖

· D0,k(t). (22)


where i = 1, 2, . . . , k and j = 1, 2, . . . , 2N . Recalling our earlier discussion, the norm ofvector ~wi(t) for long times, grows linearly with t as

‖~wi(t)‖ ∝ t. (23)

Hence, the matrix D assumes the much simpler form

D(t) =1

tk· Dk(t). (24)

The Case of N-Dimensional Tori

If k is lower than the dimension of the tangent space of the torus, i. e. 2 ≤ k ≤ N , the fastestincreasing determinants in this case are ∝ tk. Thus, we conclude that the contribution to thebehavior of GALIk is to provide constant terms .All other determinants introduce terms that grow at a rate slower than tk, proportional totk−m/tk = 1/tm. So we arrive at the important result

GALIk(t) ≈ constant for 2 ≤ k ≤ N. (25)


In the important case of N < k ≤ 2N deviation vectors, the fastest growing determinantshave the time evolution

Determinants ∝ t2N−k. (26)

and contribute to the time evolution of GALIk terms proportional to

GALIk(t) ∝ t2N−k/tk = 1/t2(k−N). (27)

All other determinants introduce terms that tend to zero faster than 1/t2(k−N).

The Case of Tori with Dimension Lower than N

In the case of an s–dimensional torus, the k deviation vectors eventually fall on itss–dimensional tangent space spanned by eN+1, . . . , eN+s. If we start with 2 ≤ k ≤ s

deviation vectors, the GALIk indices will oscillate about a nonzero constant . However, fors < k ≤ 2N deviation vectors, some of them will become linearly dependent and the GALIkwill tend to zero by a power law .


Next, let us turn to the case of s < k ≤ 2N − s. The fastest growing determinants are thosecontaining the s columns of the matrix Dk, having as many columns proportional to t aspossible. Since t appears at most s times and GALIk is proportional to ts/tk = 1/t(k−s), weconclude that GALIk(t) ∼ t(s−k) for s < k ≤ 2N − s.

Finally, we consider the behavior of GALIk when 2N − s < k ≤ 2N . The fastest growingdeterminants here have t appearing 2N − k times and the GALIk is proportional tot2N−k/tk = 1/t2(k−N).In conclusion, we have shown that for chaotic orbits :


while for regular orbits lying on an s–dimensional torus :

GALIk(t) ∼

constant if 2 ≤ k ≤ s

1tk−s if s < k ≤ 2N − s

1t2(k−N) if 2N − s < k ≤ 2N

. (29)


Fast Numerical Computation of the GALI k volumes:

We can write equations (7) in matrix form as:

w1

w2

...

wk

=

w11 w12 · · · w1 2N

w21 w22 · · · w2 2N

......

...

wk1 wk2 · · · wk 2N

·

e1

e2...

e2N

= A ·

e1

e2...

e2N

. (30)

Since GALIk measures the volume of the k–parallelepiped Pk having as edges the k unitarydeviation vectors wi, i = 1, . . . , k, given by:

vol(Pk) = GALIk =

√

det(A · AT) , (31)

(where (T) denotes transpose) and only the multiplication of two matrices and the squareroot of one determinant is required for its computation.


Numerical verification and applications

We shall use two examples with 2 (2D) and 3 (3D) degrees of freedom: the 2DHénon–Heiles Hamiltonian

H2 =1

2(p2x + p2y) +

1

2(x2 + y2) + x2y − 1

3y3, (32)

and the 3D Hamiltonian system :

H3 =3

∑

i=1

ωi

2(q2i + p2i ) + q21q2 + q21q3, (33)

for the parameters H2 = 0.125 and H3 = 0.09, with ω1 = 1, ω2 =√2 and ω3 =

√3.

A 2D Hamiltonian system

Let us consider first a chaotic orbit of the 2D Hamiltonian (32), with initial conditions x = 0,y = −0.25, px = 0.42, py = 0. In the figure below, we see the time evolution of L1(t) of thisorbit, which is a good approximation of the maximal LCE, σ1. Actually, for t ≈ 105, we findσ1 ≈ 0.047.


1 2 3 4 5

0.02

0.04

0.06

0.08

(a)

L 1(t)

logt0 200 400 600 800

-16

-14

-12

-10

-8

-6

-4

-2

0 (b)

slope=-1/ln(10)

slope=-21/ln(10)

slope=-41/ln(10)

GALI4

GALI3

GALI2log(GAL

Is)

t

Figure 1 :(a) L1(t) for a chaotic orbit of the 2D system. (b) The evolution of GALI2, GALI3and GALI4 for the same orbit. We find σ1 ≈ 0.047, and we know σ2 = 0 and σ3 = −σ2 andσ4 = −σ1. So, GALIk behaves as

GALI2(t) ∝ e−σ1t, GALI3(t) ∝ e−2σ1t, GALI4(t) ∝ e−4σ1t. (34)


Now, for an ordered orbit of our 2D Hamiltonian and a random choice of initial deviationvectors, we expect the GALI indices to behave as

GALI2(t) ∝ constant, GALI3(t) ∝1

t2, GALI4(t) ∝

1

t4. (35)

1 2 3 4 5-16

-14

-12

-10

-8

-6

-4

-2

0

slope=-2 slope=-4

(a)

SALI

GALI4

GALI3

GALI2

log(GAL

Is)

logt

Figure 2 : Time evolution of SALI (gray curves), GALI2, GALI3 and GALI4 for an orderedorbit. Note that the curves of SALI and GALI2 are very close to each other and thus cannotbe distinguished. Dashed lines corresponding to particular power laws are also plotted.


A 3D Hamiltonian system

Let us now study the behavior of the GALIs in the case of the 3D Hamiltonian (33). The initialconditions of the orbits of this system are defined by assigning arbitrary values to thepositions q1, q2, q3, as well as the so–called ‘harmonic energies’ E1, E2, E3 related to the

momenta through pi =√

2Ei

ωi, i = 1, 2, 3.

Chaotic orbits of 3D Hamiltonian systems generally have two positive Lyapunovexponents , σ1 and σ2, while σ3 = 0. So, for approximating the behavior of GALIs accordingto (41), both σ1 and σ2 are needed. In particular, (41) gives

GALI2(t) ∝ e−(σ1−σ2)t, GALI3(t) ∝ e−(2σ1−σ2)t, GALI4(t) ∝ e−(3σ1−σ2)t,

GALI5(t) ∝ e−4σ1t, GALI6(t) ∝ e−6σ1t.(36)

Let us consider the chaotic orbit with initial conditions q1 = q2 = q3 = 0,E1 = E2 = E3 = 0.03 of the 3D system (33) and compute σ1, σ2 for this orbit. For t ≈ 105

we find σ1 ≈ 0.03 and σ2 ≈ 0.008. Using these values as good approximations of σ1, σ2 wesee that the slopes of all GALIs are well reproduced by the analytical formulas.


1 2 3 4 50.00

0.02

0.04

0.06

(a)

L 1(t), L

2(t)

logt0 200 400 600 800 1000 1200

-16

-14

-12

-10

-8

-6

-4

-2

0 (b) slope=-( 1- 2)/ln(10) slope=-(2 1- 2)/ln(10) slope=-(3 1- 2)/ln(10) slope=-4 1/ln(10) slope=-6 1/ln(10)

GALI6

GALI5

GALI3GALI4

GALI2

log(GALIs)

t

Figure 3 : (a) The evolution of L1(t), L2(t) for the chaotic orbit of the 3D system. (b) The

evolution of GALIk with k = 2, . . . , 6. The plotted lines correspond to functions proportional

to e−(σ1−σ2)t, e−(2σ1−σ2)t, e−(3σ1−σ2)t, e−4σ1t and e−6σ1t for σ1 = 0.03, σ2 = 0.008.


Next, we consider the case of ordered orbits in our 3D Hamiltonian system . In general,the GALIs should behave as: GALI2(t) ∝ constant, GALI3(t) ∝ constant, GALI4(t) ∝1t2,GALI5(t) ∝ 1

t4, GALI6(t) ∝ 1

t6.

0 1 2 3 4 5 6

-5

-4

-3

-2

-1

0(a)

log(

L 1(t)

)

logt1 2 3 4 5

-16

-14

-12

-10

-8

-6

-4

-2

0(b)

slope=-2 slope=-4 slope=-6 GALI6

GALI5

GALI4

GALI3

GALI2

log(GAL

Is)

logt

Figure 4 : (a) L1(t) for the ordered orbit of the 3D system and (b) The GALIk with

k = 2, . . . , 6 of the same orbit. The dashed lines are the slopes of 1t2

, 1t4

and 1t6

.


Low - Dimensional Tori and Weak Diffusion

In their famous paper in 1955, Fermi, Pasta and Ulam studied energy transport in one –dimensional lattices , like the so – called β – FPU lattice described by the Hamiltonian

H =1

2

N∑

j=1

x2j +

N∑

j=0

(

1

2(xj+1 − xj)

2 +1

4β(xj+1 − xj)

4

)

= E (37)

When β is small, the dynamics can described by the linear normal modes

Qk =

√

2

N + 1

N∑

i=1

qi sinkiπ

N + 1, Pk = Qk (38)

with energies and ("phonon") frequencies

Ek =1

2

[

P 2k + ωk

2Q2k

]

, ωk = 2 sinkπ

2(N + 1)(39)

Fermi, Pasta and Ulam observed that, for low values of E, if the initial conditions are chosensuch that one or more of the low k - modes (k = 1, 2, 3, ..) are excited, one observes thefamous FPU recurrences !


This means that the energy returns periodically to the initial state, instead of leading toenergy sharing and equipartition as expected from Statistical Mechanics. Let us see whatthe GALIk give when only the k = 1 and k = 3 modes modes are excited:

900000 920000 940000 960000 980000 1000000

0.00

0.05

0.10

0.15

0.20

0.25

0.30

E = 0.29FPU FBC N=8

E3

E1

t101 102 103 104 105 106

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

E = 0.29FPU FBC N=8

GA

LI K

t

k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10

Figure 5 : FPU with 8 particles:(a) Only the E1 and E3 modes are excited and the torus is2-dimensional, since (b) only GALI2=const. and all other GALIk decay by power laws.


Next we start by putting all the energy in the 4 modes k = 1, 3, 5, 7 and observe that themotion lies on a 4–dimensional torus

900000 920000 940000 960000 980000 1000000

0.001

0.002

0.003

0.004

0.005E = 10-2FPU FBC N=8

E7

E5

E3

E1

t101 102 103 104 105 106

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

E = 10-2FPU FBC N=8

k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10

GA

LI k

t

Figure 6 : FPU with 8 particles:(a) When the E1, E3, E5 and E7 modes are excited the torusis 4-dimensional and (b) only the GALIk=const. for k = 2, 3, 4, while all others, withk = 5, . . . , 10 decay by power laws.


Finally we may put all energy in 8 modes and obtain a generic 8–dimensional torus:

900000 920000 940000 960000 980000 1000000

0.0005

0.0010

0.0015

0.0020

0.0025

E8

E7

E6

E5

E4

E3

E = 0.008FPU FBC N=8

E2

E1

t101 102 103 104 105 106

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

E = 0.008FPU FBC N=8

k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10

GA

LI K

t

Figure 7 : FPU with 8 particles:(a) When 8 modes, k = 1, 2, ..., 8, are excited the torus is8-dimensional and (b) GALIk=const.,for k = 2, . . . , 8, while the other two with k = 9, 10

decay by power laws.


Predicting the onset of weak diffusion

For initial conditions slightly off a torus leading to slow diffusive motion away fromquasiperiodicity, GALI2 predicts it quickly by falling to zero exponentially:

101 102 103 104 105 1061E-17

1E-15

1E-13

1E-11

1E-9

1E-7

1E-5

1E-3

0.1

10

k=2

GA

LI K

t

0 5000 10000 15000 20000 25000 300000.00

0.05

0.10

0.15

0.20

0.25

E1

E3

E2

E4

E5

E = 0.29FPU FBC N=8

t

Figure 8 : FPU with 8 particles:(a) The evolution of GALI2 shows already at t ≈ 500 that theorbit diffuses away from the torus and is not quasiperiodic. (b) This becomes visible in theoscillations much later , when the recurrences break down at t ≈ 14000.


−10 −8 −6 −4 −2 0 2 4 6 8 10

0

Position

Re

lativ

e A

mp

litu

de

Figure 9. Let us now consider discrete (periodic) breathers , like the one above,representing localization in configuration space , in contrast to the FPU recurrences,which are localized in Fourier space .


Quasiperiodic Breathers and their Breakdown

Finally we study a quartic Hamiltonian lattice with on site quartic potential and no lineardispersion, described by the Hamiltonian

HNL =1

2

N∑

i=1

p2i +N∑

i=1

[

1

2(qi+1 − qi)

4 +1

2q2i (1− ǫcos(ωdt))−

1

4q4i

]

(40)

with fixed boundary conditions, i.e. q0 = qN = p0 = pN = 0. We select parameter values atwhich we know that the Hamiltonian (40) has a stable discrete breather .

Now, around this (periodic) breather, due to the absence of quadratic nearest neighborinteractions –and hence no linear spectrum to generate resonances–quasiperiodicbreathers exist. GALI2 oscillates about a constant and thus identifies the presence of atorus. And since all higher order GALIk, with k > 2, decay the torus is 2–dimensional!


0 400000 800000 1200000 1600000-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

i=16 i=17 i=18

31 ParticlesH=0.07D=0.9

y(i)

t10 100 1000 10000 100000

1E-16

1E-14

1E-12

1E-10

1E-8

1E-6

1E-4

0.01

1

t-2

t-1

k=2 k=3 k=4

H=0.091D=0.9 31 Particles

GA

LI 2

t

Figure 10 :(a) The oscillations of the central three particles of this with N = 31 particles donot break down , forming a quasiperiodic breather. (b) The corresponding torus is2–dimensional, as evidenced by the fact that only GALI2 remains nearly constant, while allother GALIk, with k > 2 decrease by power laws.


However, at initial conditions further away from the periodic breather, GALI2 predicts slowchaotic diffusion by falling to zero exponentially:

1100000 1200000 1300000 1400000

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

i=16 i=17 i=18

31 ParticlesH=0.033D=0.7

y(i)

t10 100 1000 10000

1E-16

1E-14

1E-12

1E-10

1E-8

1E-6

1E-4

0.01

1

31 ParticlesD=0.7H=0.042

k=2 k=3 k=4

GA

LI k

t

Figure 11 :(a) The oscillations of the central particles of (40) with N = 31, appearquasiperiodic for very long times. (b) The solution, however, is not on a torus anddiffuses slowly away , since even GALI2 decays exponentially after about t = 5000.


Conclusions

1) We introduced the so-called Generalized Alignment Indices , GALIk, where GALIk is the‘norm’ of the ‘exterior’ or wedge product of k normalized deviation vectors and hencerepresents the ‘volume’ of a parallelepiped having as edges k > 2 initially independentdeviation vectors.

We have shown that for chaotic orbits :


while for ordered ones :

GALIk(t) ∼

constant if 2 ≤ k ≤ s

1tk−s if s < k ≤ 2N − s

1t2(k−N) if 2N − s < k ≤ 2N

. (42)

where s is the dimension of the torus and no deviation vectors are initially tangent to the

torus.


2) The GALIks has been successfully applied to:

a) Distinguish very rapidly chaotic orbits , for 2 ≤ k ≤ N , from ordered orbits (GALIkdecays exponentially or fluctuates around a constant)

b) Determine from the GALIk power law decays the dimensionality d ≤ N of thesubspace of ordered motion , identifying low - dimensional tori

c) Locate initial conditions leading to very slow diffusion of the orbits toward a largerchaotic "sea", near nonlinear normal modes of the FPU lattice and quasiperiodic breathers.

c) Efficiently chart large domains of phase space , identifying various degrees of order andchaos, by different behaviors of the indices.

d) Obtain analogous results also for 2N–dimensional symplectic maps.


References

1. Skokos, Ch. [2001], “Alignment Indices: A New, Simple Method for Determining theOrdered or Chaotic Nature of Orbits", J. Phys. A, 34, pp. 10029 – 10043.2. Skokos, Ch., Antonopoulos, Ch., Bountis, T. C. and Vrahatis, M. N. [2003], “How does theSmaller Alignment Index (SALI) Distinguish Order from Chaos?", Prog. Theor. Phys. Supp., 150,pp. 439 – 4433. Skokos, Ch., Antonopoulos, Ch., Bountis, T. C. and Vrahatis, M. N. [2004], “DetectingOrder and Chaos in Hamiltonian Systems by the SALI Method", J. Phys. A, 37, pp. 6269– 6284.4. Antonopoulos, Ch., Bountis, T. C. and Skokos, Ch., [2006], “Chaotic Dynamics ofN–Degree–of–Freedom Hamiltonian Systems”, International Journal of Bifurcation andChaos, vol.16(6), 1777-1793 , June 2006.5. Antonopoulos, Ch. and Bountis, T., [2006], “Stability of Simple Periodic Orbits and Chaosin an FPU Lattice”, PRE73, 056206, 1-8 (2006).6. T. Bountis, "Stability of Motion: From Lyapunov to N - Degree of Freedom HamiltonianSystems", [2006], "Nonlinear Phenomena and Complex Systems", vol. 9(3) ,209 -239, 2006.7. Skokos, Ch., Bountis, T. and Antonopoulos, Ch. [2007], “Geometrical properties of localdynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method”,Physica D 231, 30.


COMPLEX HAMILTONIAN DYNAMICS AND STATISTICS

Tassos BOUNTIS

Work with Chris Antonopoulos, Vassilis Basios and Thanos Ma nos


Systems (CRANS), University of Patras, GR–26500, Rion, Patras, Greece

Lecture 3:

Complex Statistics at the Edge of Chaos:

Applications to Statistical Mechanics and Galactic Dynami cs



COMPLEX HAMILTONIAN DYNAMICS AND STATISTICS – p. 1/65

Complex Hamiltonian Dynamics

Tassos Bountis and Haris SkokosSpringer Synergetics series, April 2012

Foreword by Sergej Flach

Chapter 1. Fundamental concepts of Lyapunov and Poincare

Chapter 2. Hamiltonian Systems of Few Degrees of Freedom

Chapter 3. Local and Global Stability of Motion

Chapter 4. Normal Modes, Symmetries and Stability

Chapter 5. Efficient Indicators of Stable and Chaotic Motion

Chapter 6. FPU Recurrences and the Transition from Weak to Strong Chaos

Chapter 7. Localization and Diffusion in Nonlinear 1-Dimensional Lattices

Chapter 8. Complex Statistics of Quasi-Stationary States


Contents

1. Sums of Random Variables and the Central Limit Theorem

2. Nonextensive Statistical Mechanics and q-Gaussians

3. Pdfs of Chaotic States in Area–Preserving Maps

4. Quasi–Stationary Chaotic States in Fermi Pasta Ulam Chains

5. Weakly and Strongly Chaotic Orbits in a Barred Galaxy Model

6. Conclusions



Sums of Random Variables and the Central Limit Theorem

The approach we shall follow is in the spirit of the Central Limit Theorem (CLT). In particular,we focus on chaotic regions of: (a) area–preserving maps

xn+1 = f(xn, yn)

yn+1 = g(xn, yn)(1)

and (b) N–degree–of–freedom Hamiltonian systems

dqkdt

=∂H

∂pk,

dpkdt

= − ∂H

∂qk, k = 1, 2, . . . , N (2)

and construct distributions of rescaled sums of M values of an observable ηi = η(ti),i = 1, . . . ,M , which is a linear combination of the variables (xn, yn), or (qn(t), pn(t)) of theproblem. These are viewed as independent random variables in the limit of largeM(→ ∞) and we evaluate their sums for a large number Nic of initial conditions:

S(j)M =

M∑

i=1

η(j)i , j = 1, . . . , Nic (3)


We then study the probability density functions (pdfs) of the variables S(j)M , centered

about their mean value 〈S(j)M 〉 and rescaled by their standard deviation σM

s(j)M ≡ 1

σM

(

S(j)M − 〈S(j)

M 〉)

=1

σM

(

M∑

i=1

η(j)i − 1

Nic

Nic∑

j=1

M∑

i=1

η(j)i

)

(4)

where

σ2M =

1

Nic

Nic∑

j=1

(

S(j)M − 〈S(j)

M 〉)2

= 〈S(j)2M 〉 − 〈S(j)

M 〉2. (5)

If our variables are random , or belong to uniformly ergodic regimes of deterministicsystems , when we plot the normalized distribution of probabilities P (s

(j)M ) as a function of

s(j)M , , we expect to find in the spirit of the classical CLT :

P (s(j)M ) = ae−βs

(j)2M (6)

i.e a Gaussian pdf .


However, in regimes of “weak chaos” of many conservative systems , these distributionsare well–approximated by different pdfs, the most ubiquitous of them being the q-Gaussian :

P (s(j)M ) = ae

−βs(j)2M

q ≡ a

(

1− (1− q)βs(j)2M

) 11−q

(7)

where q is the Tsallis entropy index , β is a free parameter and a a normalizationconstant . Expression (7) is a generalization of the Gaussian, since in the limit q → 1 we have

limq→1e−βx2

q = e−βx2. If 1 ≤ q < 3, (7) is normalized for an appropriate choice of a(β).

As we shall demonstrate, there are many interesting cases of conservative systems ,where chaotic orbits are strongly influenced by “stickiness” phenomena and producelong–lived quasi–stationary states (QSS), whose pdfs are well–approximated byq–Gaussians. These, however, do not always converge to a q gaussian, but evolve,through a sequence of QSS , which we seek to identify the limit t → ∞.


Nonextensive Statistical Mechanics and q-Gaussians

Multi–particle systems belong to different “universality” classes , according to theirthermostatistics. In the most widely studied Boltzmann–Gibbs class , if the system can beat any one of i = 1, 2, ...,W states with probability pi, its entropy is given by the famousformula

SBG = −k

W∑

i=1

pilnpi (8)

where k is Boltzmann’s constant, provided, of course,

W∑

i=1

pi = 1 (9)

The BG entropy satisfies the property of additivity , i.e. if A and B are two statisticallyindependent systems, the probability to be in their union is pA+B

i,j = pAi pBj and thisnecessitates that the entropy of the joint state obey

SBG(A+B) = SBG(A) + SBG(B) (10)


At thermal equilibrium , the pi that optimize the BG entropy , subject to ((9)) and a givenenergy spectrum Ei and temperature T are:

pi =e−βEi

ZBG, ZBG =

W∑

i=1

e−βEi (11)

where β = 1/kT and ZBG is the so–called partition function. For a continuum set ofstates depending on one variable , x, the optimal probability density function (pdf)corresponding to BG statistics subject to ((9)), zero mean and given variance V is, of course,

the well–known Gaussian p(x) = e−x2/2V /√2V .

Another important property of the BG entropy is that it is extensive ,

limN→∞SBG

N< ∞ (12)

i.e. it grows linearly as a function of the number of degrees of freedom N of the system. But

then, what about many physically important systems that are not extensive ?


There are many examples of non–extensive systems , like: self–gravitating systems offinitely many mass points , interacting black holes, ferromagnetic models, systems withlong range forces , in which strong correlations and power laws are dominant.

In deterministic dynamics, for instance, chaos does not always mean exponentialinstability , as there are regimes of “weak chaos” , where “stickiness phenomena” occur onor near the boundary of regions of regular motion called ‘edge of chaos’ , whereLyapunov exponents are zero and orbits separate linearly from each other.

It is for these type of situations that Tsallis proposed a different form of entropy [Tsallis,2009]

Sq = k1−∑W

i=1 pqi

q − 1with

W∑

i=1

pi = 1 (13)

depending on an index q, where i = 1, . . . ,W counts the microstates of the system

occurring with probability pi and k is the Boltzmann constant.


Just as the Gaussian distribution represents an extremal of the BG entropy ((8)), so is theq–Gaussian pdf obtained as a maximum (for q > 0) or a minimum (q<0) of the Tsallisentropy ((13)).

p(x) = ae−βx2

q ≡ a

(

1− (1− q)βx2

) 11−q

(14)

for a continuum set of states.

The Tsallis entropy is not additive , as

Sq(A+B) = Sq(A) + Sq(B) + k(1− q)Sq(A)Sq(B) (15)

and is in general not extensive . It thus offers the possibility of studying cases where the A,B subsystems mentioned above are strongly correlated.

Systems characterized by these properties are said to have complex statistics , which issignificantly different from BG systems.


Nonextensive Thermodynamics

If we have a discrete set of states j = 1, 2, ...,W , we define the escort probabilities

Pi =pqi

∑Wj=1 p

qj

(16)

Given the constraints (9) and

< Ei >q=W∑

j=1

PiEi = Uq (17)

we find that the distribution that extremizes the Tsallis entropy Sq is the q-Gaussian

pi =e−βq(Ei−Uq)q

Zq

(18)

where the partition function Zq is given by


Zq =

W∑

i=1

e−βq(Ei−Uq)q (19)

and βq is the Lagrange multiplier associated with constraint (11)

βq =β

∑Wj=1 p

qj

(20)

From the above one can prove that the fundamental thermodynamical equations havethe same form as the corresponding BG expressions with all va riables indexed by q,and β = 1/kT :

Sq = klnqZq ,1

T=

∂Sq

∂Uq, Fq = Uq − TSq = − 1

TlnqZq (21)

and lnZq = lnZq − βUq .


Pdfs of Chaotic States in Area–Preserving maps

Consider the perturbed MacMillan map, which occurs in the study of equilibriumconfigurations of Ablowitz–Ladik Discrete Nonlinear Schrodinger Equations:

xn+1 = yn

yn+1 = −xn + 2µ yn1+y2

n+ ǫyn

(22)

with parameters ǫ, µ. Since its Jacobian is 1, (22) is area-preserving.

We have analyzed the histogram of their normalized sums for a range of parameters (ǫ, µ)and have identified some generic classes of q-Gaussians , exponentials , ∼ e−k|z| orGaussians . Monitoring their time evolution for increasing number of iterations N , weobserve the occurrence of different QSS, which we present with the corresponding phasespace plots in the Figures below.

ǫ 0.2 0.5 0.9 1.2 1.6 1.8

Lmax 0.0867 0.082 0.0875 0.0513 0.0495 0.05876

Table 1: Maximal Lyapunov exponents for µ = 0.6 and ǫ = 0.2, 0.5, 0.9, 1.2, 1.6, 1.8.


-20 -10 0 10 20

1E-4

1E-3

0,01

0,1

1

=0.2=1.6

N=217Ci=5x107

N=218Ci=105

N=224Ci=106

N=225Ci=106

(q=1.5)-Gaussian ( =3)

(q=1.4)-Gaussian ( =1.3)

P(y/)

y/

Figure 1. Tendency away from a Gaussian: Dynamical and statistical behavior of chaoticorbits of the MacMillan map for parameter values µ = 1.6, and ǫ = 0.2. Left: The pdfs of thenormalized sums of iterates. Right: The corresponding phase space plot. N is the number ofiterates and Nic is the number of initial conditions randomly chosen from a square(0, 10−6)× (0, 10−6).


-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

1E-4

1E-3

0,01

0,1

1

=0.9=1.6

N=218;Nic=105

N=221;Nic=106

N=223;Nic=106

N=225;Nic=105

P(y/)

y/

Figure 2. Tendency towards a Gaussian: Dynamical and statistical behavior of chaoticorbits of the MacMillan map for parameter values µ = 1.6, and ǫ = 0.9. Left: The pdfs of thenormalized sums of iterates. Right: The corresponding phase space plot. N represents thenumber of (summed) iterates and Nic is the number of initial conditions randomly chosenfrom a square (0, 10−6)× (0, 10−6).


A. The ǫ = 0.9, (µ = 1.6) case: Towards a Gaussian

The ǫ = 0.9 case is a typical example producing time–evolving pdfs. Figures 1 shows thatthe corresponding phase space plots yield a simple chaotic region in phase space aroundtwo islands, yet the corresponding pdfs do not converge to a single distribution, rather theypass from q-Gaussians to triangular to Gaussian distributions.

There exist at least three long–lived QSS whose iterates generate pdfs passing throughdifferent shapes. Consequently, for i = 1 . . . N = 216, a QSS is produced whose pdf is a(q = 1.6)–Gaussian whose β increases with increasing N . This is most likely due to a“stickiness” effect around islands of regular motion.

Figures 3 and 4 below show phase space plots for a number of iterates N . Note that forN = 1 . . . 216, a ‘figure eight’ chaotic region is formed around two islands. But for N > 216,a more complex structure emerges, as iterates stick around new islands, and phase spacestatistics passes through a sequence of quasi-stationary s tates.


-6 -3 0 3 61E-3

0,01

0,1

1 212; Nic=106

214; Nic=105

216; Nic=105

(q=1.6)-Gaussian: =2 (q=1.6)-Gaussian: =2.5 (q=1.6)-Gaussian: =3.5

P(z/)

z/

=0.9=1.6

Figure 3. ǫ = 0.9, µ = 1.6 MacMillan map phase space plots (first panel) and thecorresponding PDFs (second panel) of the re-normalized sums as the number of iteratesi = 1 . . . N,N ≤ 1016 increases, starting from a randomly chosen initial condition in asquare (0, 10−6)× (0, 10−6).


-6 -3 0 3 61E-4

1E-3

0,01

0,1

1

223; Nic=106

225; Nic=105

226; Nic=105

(q=1.22)-Gaussian ( =1)(q=1.13)-Gaussian ( =1)

P(z/)

z/

=0.9=1.6

Figure 4. ǫ = 0.9, µ = 1.6 MacMillan map phase space for i = 1 . . . N,N ≥ 223 plotiterates, starting from a randomly chosen initial condition in a square (0, 10−6)× (0, 10−6)

and the corresponding pdfs (right panel).


Thus, more than one QSS coexist whose pdfs form a sequence of different(q 6= 1)–Gaussians. As we see in Figure 3 that this sequence of QSS for N ≤ 221 producesa distribution where central part is still well–described by a (q = 1.6)–Gaussian .

However, as we continue to iterate the map to N = 223, this (q = 1.6)–Gaussian passesthrough a superposition of states characterized by triangular distributions . From here on,as N > 223, the central part of the pdfs become q–Gaussians with q → 1 (see Figure 4) anda true Gaussian is expected in the limit (N → ∞).

B. The ǫ = 1.2, (µ = 1.6) cases: Towards a q Gaussian

Let us now analyze the cases ǫ = 1.2, whose maximum Lyapunov exponent is Lmax ≈ 0.05.In Figure 5 iterates show a diffusive behavior that spreads around islands of a higher orderresonance, as iterations reach N = 219.

The central part of the pdfs converges to a (q = 1.6)–Gaussian for N ≤ 216 (left panel ofFig. 5). Then, orbits diffuse outward and even the tail of the pdf converges to a(q = 1.6)-Gaussian (right panel of Fig. 5). For larger N , diffusion ceases at the q–Gaussianof Figure 6.


-10 -5 0 5 101E-4

1E-3

0,01

0,1

1

P(z/)

N=29;Ci=106

N=210;Ci=106

N=213;Ci=106

N=216;Ci=106

N=218;Ci=106

(q=1.6)-Gaussian ( =4.5)

z/

=1.2=1.6

CENTRALCONVERGENCE

Figure 5. Dynamical and statistical behavior of chaotic orbits of the MacMillan map forµ = 1.6, and ǫ = 1.2. On the right the phase space plot and on the left the pdfs of thenormalized sums of iterates.


-60 -40 -20 0 20 40 601E-7

1E-6

1E-5

1E-4

1E-3

0,01

0,1

1

TAILCONVERGENCE

=1.2=1.6

N=218;Ci=106

N=219;Ci=106

N=220;Ci=5x107

(q=1.6)-Gaussian ( =4.5)

P(z/)

z/

Figure 6. Tail convergence of the q–Gaussian for µ = 1.6, and ǫ = 1.2 depicting the pdfs forN > 218 numbers of iterates.


-20 -15 -10 -5 0 5 10 15 201E-5

1E-4

1E-3

0,01

0,1

1

=1.6=1.6

N=220;Ci=5.107

N=222;Ci=106

N=224;Ci=106

N=225;Ci=106

P(y/)

y/

Figure 7. Similar dynamical and statistical behavior of chaotic orbits of the MacMillan map isobserved for µ = 1.6, ǫ = 1.6, as in the ǫ = 1.2 case. Orbital diffusion to an outer chain ofislands generates pdfs of iterate sums that also converge to a true q–Gaussian as N

increases to larger and larger values.


Quasi–stationary Chaotic States in Fermi–Pasta–Ulam Hamiltonians

The FPU β− model is a one–dimensional lattice of nonlinear oscillators described by theHamiltonian

H =1

2

N∑

j=1

p2j +N∑

j=0

(

1

2(qj+1 − qj)

2 +1

4β(qj+1 − qj)

4

)

= E (23)

E being its total energy. We shall impose:1) fixed boundary conditions (fbc) : q0(t) = qN+1(t) = 0, or2) periodic boundary conditions (pbc) : qj(t) = qj+N (t), pj(t) = pj+N (t), for all t > 0.

We focus on Simple Periodic Orbits (SPOs) , where all variables return to their initial stateafter only one maximum and one minimum in their oscillation. Examples of such SPOs areNonlinear Normal Modes (NNMs) , i.e. continuations of linear normal modes of the FPUchain described by the Qq and Pq variables:

Qq =

√

2

N + 1

N∑

i=1

qi sinqiπ

N + 1, Pq = Qq (24)


These solutions are:(a) The FPU π–Mode under pbc with N even

qj(t) = −qj+1(t) ≡ q(t), j = 1, . . . , N (25)

(b) The SPO1 mode under fbc , with N odd,

q2j(t) = 0, q2j−1(t) = −q2j+1(t) ≡ q(t), j = 1, . . . ,N − 1

2. (26)

(c) The SPO2 mode under fbc , with N = 5 + 3m, m = 0, 1, 2, . . .

q3j(t) = 0, j = 1, 2, 3 . . . ,N − 2

3, (27)

qj(t) = −qj+1(t) = q(t), j = 1, 4, 7, . . . , N − 1. (28)

which are exact continuations of the q = N/2, q = (N + 1)/2 and q = 2(N + 1)/3 linearmodes respectively. Let us see what these solutions look like for some small particle chains.


Figure 8 : The π–mode of oppositely moving particles, the SPO1 mode with every otherparticle being stationary and the SPO2 mode, with one stationary particle every other two.


Let us study chaotic regions near these NNM orbits , at energies where they have justturned unstable .

(a) The FPU π–Mode for N=128 particles under pbcHere, we choose as an observable the quantity

η(t) = qN2(t) + qN

2−1

(t) (29)

using the fact that η(t) = 0 exactly at the π–mode. At energies above its first destabilizationthreshold, i.e. E > E1

u, η(t) deviates from zero and starts to explore a thin chaotic layer .We first consider the case of N = 128 and β = 1 and take as our total energy E = 0.768, atwhich the π–mode has just turned unstable.

As we see in Figures 9 and 10, as we increase the total integration time of our numericaltrajectory, the statistical distributions (red curves) approach closer and closer to aGaussian with q tending to 1 .


0.0001

0.001

0.01

0.1

-30 -20 -10 0 10 20 30

P(s

M(j) )

sM(j)

a)

0.0001

0.001

0.01

0.1

-30 -20 -10 0 10 20 30P

(sM

(j) )

sM(j)

b)

Figure 9: Plot in linear–log scale of numerical (red curve), q–Gaussian (green curve) andGaussian (blue curve) distributions for the FPU π− mode with p.b.c. for N = 128 degrees offreedom, β = 1 and E = 0.768. Panel a) corresponds to integration time tf = 105. Thenumerical fitting with a q–Gaussian gives q ≈ 1.818. Panel b) corresponds to tf = 106

using Nic = 100 and M = 100 terms in the sums. Here the fitting is with a q–Gaussian withq ≈ 1.531.


10-4

10-3

10-2

10-1

100

-30 -20 -10 0 10 20 30

P(s

M(j) )

sM(j)

c)

Figure 10: This plot corresponds to final integration time of tf = 108 using time windows oflength Nic = 1000 and M = 1000 terms in the computations of the sums. Here thenumerical distribution (red curve) has almost converged to a Gaussian (blue curve).


(b) FPU SPO1 mode for N=5 particles under fbc

We now examine the chaotic dynamics and study statistical distributions near the SPO1nonlinear mode of an FPU–β chain with N = 5 particles, under fixed boundary conditions .

The chaotic regions near this solution just above its first destabilization energy are shownin Figure 11. At first, a “figure eight” appears made by an orbit starting at a distance≈ 1× 10−7 from the SPO1 mode, see the surface of section (q1, p1) of Figure 11 (at timeswhen q3 = 0 and E = 7.4).

Starting now at a distance of ≈ 1.× 10−2 from the saddle point, a more extended chaoticregion is observed, in the form of a green “figure eight cloud” enveloping the first orbit.

Choosing even more distant initial conditions, e.g. one starting ≈ 3.4× 10−1 from the saddlepoint, a much larger chaotic region is obtained, which spreads uniformly over a muchlarger part of the available energy surface.


1.6

1.8

2

2.2

2.4

2.6

2.8

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

p 1

q1

Figure 11: (i) The “figure eight” chaotic region (blue points) for an orbit starting at adistance ≈ 1.2× 10−7 from SPO1 mode at the saddle point in the figure), (ii) a fatter“figure eight cloud” (green points) is seen starting at (≈ 1.086× 10−2) and a muchlarger chaotic region (red points) on the energy surface for an initial condition even moredistant (≈ 3.421× 10−1). N = 5 and β = 1, on the surface of section (q1, p1) computed atq3 = 0. In this figure, we have integrated our three orbits up to tf = 105 on the energysurface E = 7.4.


In the present example, we have chosen as an observable the quantity

η(t) = q1(t) + q3(t) (30)

which is equal to zero at the SPO1 orbit and becomes nonzero due to numerical errors atenergies just above the first destabilization energy E1

u of the mode.

Instead of studying 3 different initial conditions as we did in Figure 11 , we proceed inFigures 12, 13 and 14 to plot on the left panel the surface of section of a trajectory starting at≈ 1× 10−7 from the unstable SPO1) for times tf = 105, tf = 107 and tf = 108

respectively , while on the right panels we plot the corresponding pfds of the normalizedsums.

Clearly, as the integration time tf grows, our chaotic orbit eventually wanders over a moreextended domain, covering gradually a much larger part of the energy surface whentf = 108. This may also be explained by the behavior of the Lyapunov exponents of the orbit(see Figure 15).


1e-05

0.0001

0.001

0.01

0.1

1

-30 -20 -10 0 10 20 30P

(sM

(j) )

sM(j)

e)

Figure 12. Left: Surface of section of a trajectory starting from a distance of ≈ 1.2× 10−7

from the unstable SPO1 saddle and integrated for a total time tf = 105. Right: For thesame integration time, we find that the pdf representing the distribution of the sums is wellfitted by a q–Gaussian, q ≈ 2.785 with χ2 ≈ 4.05× 10−6.


1e-05

0.0001

0.001

0.01

0.1

1

-30 -20 -10 0 10 20 30P

(sM

(j) )

sM(j)

f)

Figure 13: Left: Surface of section of a trajectory starting from a distance ≈ 1.192× 10−7

from the unstable SPO1 saddle point total after time tf = 107. Right: For the sameintegration time, the statistical distribution of the sums of one chaotic component of theorbit can still be fitted–but not as well–by a q–Gaussian with q ≈ 2.483 withχ2 ≈ 6.05× 10−6.


10-5

10-4

10-3

10-2

10-1

100

-30 -20 -10 0 10 20 30

P(s

M(j) )

sM(j)

g)

Figure 14: Left: Surface of section of the same trajectory for total integration timetf = 108. Right: Final integration time tf = 108 in the computations of the sums. In thiscase it is evident that the distribution appears to converge to a Gaussian (q=1) .


1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10 100 1000 10000 100000 1e+06 1e+07 1e+08

LEs

t

d)

Figure 15: The 4 positive Lyapunov exponents of the previous solution integrated for a totaltime of t = 108. Observe the sudden jump in their magnitude at ( t ≃ 107), where the orbitescapes from the region of “weak chaos” and q–Gaussian distributions into the widerchaotic domain of “strong chaos” where the statistics is Gaussian .


(c) FPU SPO2 mode for N=5 particles under fbc

We also examined the neighborhood of the SPO2 mode, which becomes unstable atmuch smaller energies . Thus, we expect that, near SPO2, orbits will be more weaklychaotic and QSS will persist for longer times.

As Figure 16 shows, orbits now trace out a kind of “banana” shaped region (Fig. 16(a)) ina regime of very small (positive) Lyapunov exponents (Fig. 16(b)). Here, the normalizedsum pdfs, up to tf = 1010!, converge to a function that is close to a q–Gaussian andnever deviates towards a Gaussian. As we see in Figure 16(a), the dynamics near SPO2“sticks” to a type of quasiperiodic torus , at least up to tf = 108.

In Figure 16(b), we have plotted the four positive Lyapunov exponents up to tf = 109. Notethat, although they are all seen to decrease towards zero, at about tf > 109, the largestexponent shows a tendency to converge to a very small positiv e value of about 10−7,indicating that the orbit is “weakly chaotic”.


-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

p1

q1

a)

10-10

10-8

10-6

10-4

10-2

100

101 102 103 104 105 106 107 108 109

LE

s

t

b)

Figure 16: (a) The dynamics near SPO2 “sticks” to a quasiperiodic torus–like structure, atleast up to tf = 108. The weakly chaotic nature of the motion is shown in (b), where wehave plotted the four positive Lyapunov exponents up to tf = 109. Note that, although theyall decrease towards zero, at about tf > 109, the largest exponent shows a tendency toconverge to a very small positive value .


10-5

10-4

10-3

10-2

10-1

100

-40 -30 -20 -10 0 10 20 30 40

P(s

M(j) )

sM(j)

e)

10-5

10-4

10-3

10-2

10-1

100

-40 -30 -20 -10 0 10 20 30 40

P(s

M(j) )

sM(j)

c)

Figure 17: Left panel: The distribution of the normalized sum pdf of the orbit starting nearSPO2, for a total integration time tf = 106. Right panel: Remarkably at integration timetf = 1010 the pdf has converged to a shape whose central part resembles a q–Gaussianwith q ≈ 2.769, but turns down at the boundaries having a temptingly smooth form thatwe have not yet been able to describe analytically .


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

1 20 40 60 80 100 120

En

n

(b)

0

0.005

0.01

0.015

0.02

0.025

1 20 40 60 80 100 120

En

n

(c)

Figure 18: When does energy equipartition occur? Left: Plotting the instantaneous on

site energy En along an N = 128 FPU chain with p.b.c., at t = 107 we observe a chaotic

breather preventing equipartition . Right: At time t = 108, the chaotic breather has

collapsed and energy equipartition has set in . [Cretegny et al., Physica D 121, 109–126

(1998)].


10-4

10-3

10-2

10-1

100

-30 -20 -10 0 10 20 30

P(s

M(j) )

sM(j)

(c)

0

0.5

1

1.5

2

2.5

3

105 106 107 108

q-e

ntr

op

ic p

ara

me

ter

t

(d)

Figure 19: Left Panel: At t = 107, the distribution describing the statistics of thechaotic breather is close to a q-Gaussian with q ≈ 1.6. Right Panel: The approach toenergy equipartition is indicated by the time evolution of t he q index , whose values onthe average fall significantly closer to 1 for t > 5× 107.


Weakly and Strongly Chaotic Orbits in a Barred Galaxy Model

The motion of a test particle in a 3D rotating model of a barred galaxy is governed by theHamiltonian:

H =1

2(p2x + p2y + p2z) + V (x, y, z)− Ωb(xpy − ypx). (31)

The bar rotates around its z–axis (short axis), while the x–direction is along the major axisand the y along the intermediate axis of the bar. Ωb is the pattern speed of the bar and H

is the total energy of the orbit in the rotating frame of reference (Jacobi constant). Thecorresponding equations of motion are:

x = px +Ωby, y = py − Ωbx, z = pz , (32)

px = −∂V

∂x+ Ωbpy, py = −∂V

∂y− Ωbpx, pz = −∂V

∂z.

The potential V of our model consists of three components: (1) A Miyamoto-Nagai disc:

VD = − GMD√

x2 + y2 + (A+√z2 +B2)2

. (33)


(2) A bulge, modeled by a Plummer sphere whose potential is:

VS = − GMS√

x2 + y2 + z2 + ǫ2s, (34)

where ǫs is the scale-length of the bulge and MS is its total mass.(3) A triaxial Ferrers bar, with density ρ(x):

ρ(x) =

ρc(1−m2)2 ,m < 1

0 ,m ≥ 1, (35)

where ρc = 10532π

GMB

abcis the central density, MB the total mass of the bar and

m2 =x2

a2+

y2

b2+

z2

c2, a > b > c > 0, (36)

a, b and c being the semi-axes. The corresponding potential is:

VB = −πGabcρc

n+ 1

∫ ∞

λ

du

∆(u)(1−m2(u))n+1, (37)


where m2(u) = x2

a2+u+ y2

b2+u+ z2

c2+u, ∆2(u) = (a2 + u)(b2 + u)(c2 + u), n is a positive

integer (with n = 2 for our model) and λ is the unique positive solution of m2(λ) = 1, outsideof the bar (m ≥ 1), and λ = 0 inside the bar. We use the parameter values G=1, Ωb=0.054(54 km · sec−1 · kpc−1), a=6, b =1.5, c=0.6, A=3, B=1, ǫs=0.4, MB=0.1, MS=0.08,MD=0.82. The units are: 1 kpc (length), 1000 km · sec−1 (velocity), 1 Myr (time),2× 1011M⊙ (mass). The total mass G(MS +MD +MB) is set equal to 1.

Realistically, the maximal time of integration of the orbit s is T=10000Myr (10 billionyrs), corresponding to a time of the order of one Hubble time. The question therefore is:Can we use pdfs to identify the “ weakly” vs. “strongly” chaot ic nature of the orbitsbased only on the small set of data provided by the integratio n over only T = 10000?.

We test first the 2-degree-of-freedom case:1) Strongly Chaotic Orbit 1: H = E = −0.3, (x, y, px, py) = (0,−0.625,−0.314512,−0.24)

2) Weakly Chaotic Orbit 2: H = E = −0.36, (x, y, px, py) = (0,−0.625,−0.002,−0.24)

and then apply our methods to a more realistic 3-degree-of-freedom example


Figure 20: The Poincare Surface of Section (PSS) (y, py) with px > 0, x = 0 for theChaotic Orbit 1 above of the 2DOF Hamiltonian system for final integration time tf = 10000

(left panel) and tf = 100000 (right panel) for the same data.


Figure 21: Pdfs of η = x+ y for the the Chaotic Orbit 1 above: Plot in linear-log scale ofnumerical (red curve), q-Gaussian (green curve) and Gaussian (blue curve) distributions.Left panel: Final integration time tf = 10000, with Nic = 4000 time windows and M = 50

terms in the sums. Numerical fitting with a q-Gaussian gives q ≈ 1.095 with χ2 ≈ 0.0003.Right panel: tf = 100000, Nic = 20000 and M = 10, with q ≈ 0.97 and χ2 ≈ 0.00033.


Figure 22: Maximum Lyapunov exponent evolution up to tf = 100000 for the the ChaoticOrbit 1 of the 2DOF Hamiltonian system (red curve) and for the Weakly Chaotic Orbit 2 ofthe same system (green curve).


Figure 23: Left panel: The Poincare Surface of Section (PSS) (y, py) with px > 0, x = 0 for

the the Weakly Chaotic Orbit 2 of the 2DOF Hamiltonian system for integration time

tf = 10000 (left panel) and tf = 100000 (right panel) for the same data.


Figure 24: Pdfs of η = x+ y for the the Weakly Chaotic Orbit 2 : Plot in linear-log scale ofnumerical (red curve), q-Gaussian (green curve) and Gaussian (blue curve) distributions.Leftpanel: Final integration time tf = 10000, Nic = 4000 and M = 50 terms. Fitting with aq-Gaussian gives q ≈ 3.52 with χ2 ≈ 0.00074. Right panel: tf = 100000, Nic = 25000 andM = 50. Here, the numerical fitting gives q ≈ 3.539 with χ2 ≈ 0.00057.


Figure 25: Left panel: Projections for the Chaotic Orbit 1 of the 3DOF Hamiltoniansystem for final integration time tf = 10000 in x, y and x, z (right panel) . Initial conditions:(x, y, z, px, py , pz) = (0.5875, 0, 1.291670, 0, 0, 0). H = E = −0.2852654501087481.


10-3

10-2

10-1

100

-20 -15 -10 -5 0 5 10 15 20

P(s

M(j) )

sM(j)

Figure 26: Pdf plots for the Chaotic Orbit 1 of the 3DOF Hamiltonian system in linear-logscale of numerical (red curve), q-Gaussian (green curve) and Gaussian (blue curve)distributions. Left panel: η = z and tf = 10000 using Nic = 4000 and M = 50 terms.Here, the numerical fitting gives q ≈ 1.25 with χ2 ≈ 0.00017. Right panel: η = x+ y,tf = 100000, Nic = 10000 and M = 100 and fitting gives q ≈ 0.95 with χ2 ≈ 0.00029.


Figure 27: Maximum Lyapunov exponent evolution up to tf = 100000 for the WeaklyChaotic Orbit 2 of the 3DOF Hamiltonian system (green curve) and the Chaotic Orbit 1of the same system (red curve).


Figure 28: Projections for the Weakly Chaotic Orbit 2 of the 3DOF Hamiltonian system

for final integration time tf = 10000 in x, y (left panel) and x, z (right panel). Initial

conditions: H = E = −0.2792149022676664,

(x, y, z, px, py , pz) = (2.35, 0, 0.08883, 0, 0.133330, 0).


10-2

10-1

-30 -20 -10 0 10 20 30

P(s

M(j) )

sM(j)

e)

10-2

10-1

-20 -15 -10 -5 0 5 10 15 20

P(s

M(j) )

sM(j)

Figure 29: Pdf plots for the Weakly Chaotic Orbit 2 of the 3DOF Hamiltonian system inlinear-log scale of numerical (red curve), q-Gaussian (green curve) and Gaussian (bluecurve) distributions. Left panel: η = z and tf = 10000, with Nic = 4000, M = 50 terms.Right panel: η = x+ y, tf = 100000, Nic = 10000 and M = 100. In this case thenumerical fitting with a q-Gaussian gives q ≈ 2.464 with χ2 ≈ 0.00101.


1

2

3

104 105 106

<q>

M

t

a)

1

2

3

104 105

<q>

M

t

a)

1

2

3

104 105 106

<q>

Mt

b)

1

2

3

104 105

<q>

M

t

b)

Figure 30: The time evolution of the 〈q〉M -entropic parameter of the Weakly Chaotic Orbit2 of the 3DOF system (31) for η = x+ y, Nic = 20000 and M = 50, 100, . . . , 450, 500. Thedashed lines corresponds to one standard deviation from the average entropic parameter.The inset is a zoom-in up to tf = 105. Panel b): Same as in panel a) but for η = z. Theseresults have been verified by Fourier spectra calculations.


QSS and Dynamical Phase Transitions in a Microplasma System

We now turn our attention to another N degree of freedom Hamiltonian system of verydifferent type than the FPU models, as it is characterized by long range interactions ofthe Coulomb type and study statistically the dynamics during the transition from“crystalline–like” to “liquid–like” phase (the so called “melting transition”) for smallenergies and also from “liquid–like” to “gas–like” phase at higher energies. We considera microplasma of N ions of equal mass m = 1 and electric charge Q in a Penning trap withelectrostatic potential

Φ(x, y, z) = V02z2 − x2 − y2

r20 + 2z20(38)

and constant magnetic field along the z direction with a vector potential of the form

A(x, y, z) =1

2(−By,Bx, 0). (39)

The system is described by the Hamiltonian

H =N∑

i=1

1

2m(pi − qA(ri))

2) +QΦ(ri)

+∑

1≤i<j≤N

Q2

4πǫ0rij(40)


Here ri is the position of the ith ion, rij is the Euclidean distance between the ith and jthions and ǫ0 is the vacuum permittivity. The ions are subjected to a harmonic confinementin the z direction with frequency

ωz =

√

4QV0

m(r20 + 2z20)(41)

while, in the perpendicular direction, they rotate with frequency ωc = QB/m. Thus, in aframe rotating around the z axis with Larmor frequency ωL = ωc/2, the ions are subjected to

a harmonic confinement with frequency ωx = ωy =

√

ω2c

4− ω2

z

2in the direction

perpendicular to the magnetic field. In the rescaled time τ = ωct, position R = r/a and

energy H = Hmω2

ca2 with a =

(

Q2

4πǫ0mω2c

) 13 , the Hamiltonian (40) describing the motion of

the N ions takes the form

H =N∑

i=1

[1

2P2

i

]

+N∑

i=1

[(1

8− γ2

4

)

(X2i + Y 2

i ) +γ2

2Z2i

]

+∑

i<j

1

Rij= E (42)


where E is the total conserved energy of the system, Ri = (Xi, Yi, Zi) andPi = (PXi

, PYi, PZi

) are the positions and momenta in R3, Rij is the Euclidean distance

between different ions i, j given by

Rij =√

(Xi −Xj)2 + (Yi − Yj)2 + (Zi − Zj)2 (43)

and γ = ωz/ωc. The ions perform bounded motion under the condition

0 < |γ| < 1√2. (44)

The trap is called prolate if 0 < |γ| < 1√6

, isotropic if |γ| = 1√6

and oblate if 1√6< |γ| < 1√

2.

Thus, the motion is quasi 1–dimensional in the limit γ → 0 and quasi 2–dimensional inthe limit γ → 1/

√2. The Z direction is a symmetry axis and hence, the Z component of the

angular momentum LZ =∑N

i=1 XiPYi− YiPXi

is conserved. We suppose, from now onLZ = 0) and that the motion is studied in the Larmor rotating frame.


Recent results demonstrate the occurrence of dynamical regime changes when thesystem is in a prolate quasi 1–dimensional configuration ( γ = 0.07). More specifically, inthe lower energy regime , a transition from a “crystalline–like” to a “liquid–like” behavior isobserved, called the “melting phase” . It appears that there is no clear “macroscopic”approach for identifying and studying the “melting” process of the microplasma system indetail.

However, using the Smaller Alignment Index (SALI) method to study the local dynamics,we discovered that there exists an energy range Emt of weakly chaotic behavior , i.e.E ∈ Emt = (2, 2.5), where the positive Lyapunov exponents are very small and SALIexhibits a stair–like decay to zero with varying decay rates, due to the presence of long lived“sticky” orbits near the boundaries of resonance islands.

Thus, we set γ = 0.07 and consider a prolate trap giving rise to quasi 1–dimensionalmotion for the case of N = 5 ions . For these parameters, the minimum energy for the ionsto start moving around their equilibrium positions is E0 ≈ 1.8922.


We first study the “melting transition” , as the energy of the Hamiltonian (42) increasesabove E0, from the viewpoint of probability distributions associated with chaotic trajectories.Based on the results of the previous sections, we expect to find q–Gaussians with 1 < q < 3

in the vicinity of Emt where the positive Lyapunov exponents λi, i = 1, . . . , 3N are quitesmall compared to the maximum values they can attain. This is also expected since in theinterval Emt the orbits stick to the boundaries of islands, at the so called “edge of chaos”[Tsallis, 2009].

Indeed, in Figure 31, we apply our statistical analysis to the microplasma system, takingNic = 1000, M = 1000, tf = 2× 107 as integration time for our orbits and using as anobservable the quantity η(t) = X1(t), i.e. the first component of the position R1 of the firstion. We find that, in the energy range of the “melting transition”, the values of the entropicparameter of the q–Gaussian distribution (14) exhibiit a maximum well above q = 1,indicating that the statistics is not Gaussian. Rather, a q–Gaussian is detected, whichpersists up to about E = 4.5.


0

0.5

1

1.5

2

2.5

3

10

q-en

trop

ic p

aram

eter

E

Figure 31: Plot of the q–entropic parameter as a function of the energy E of the

microplasma Hamiltonian (42) for γ = 0.07 (prolate trap) and N = 5 ions. In this plot, we

have used Nic = 1000 and M = 1000. We have also plotted the line at q = 1 for reference

to the entropic parameter of the Gaussian distribution.


Finally, we study a second transition from “liquid–like” to “gas–like” behavior, where thesystem is “weakly chaotic” at higher energies, where the biggest Lyapunov exponentsdecrease towards zero according to the equation [see Gaspard, 2003]

LE1 ∼⟨

N2

R3ij

⟩

∼ N(lnT )1/2

T 3/4(45)

where T is defined as T = kBTmω2

ca2 , with T being the temperature of the system. This

formula represents the asymptotic power law decay of the biggest Lyapunov exponent forsufficiently high energies (well above its peak at E ≈ 6), see Figure 32(a).

As Figure 32(b) shows, the q indices are well above unity for 30<E<200 indicating that inthis range a significant dynamical change occurs. The energy increase drives the system toa more organized state favoring the kinetic part rather than the Coulomb part and leading toa regime characterized by few rare particle encounters, much like in an “ideal gas”.


0.0001

0.001

0.01

0.1

10 100

LE

1, L

E2

E

a)

0

0.5

1

1.5

2

2.5

3

10 100

q-e

ntr

op

ic p

ara

me

ter

E

b)

Figure 32: Panel a) Plot of the two biggest positive Lyapunov exponents (LE1 with red colorand LE2 with green color) as a function of the energy E in log–log scales. We observe that,after their peak values at E ≈ 6, both of them decay to zero according to the formula (45).Panel b) Plot in log–linear scales of the q–entropic parameter as a function of E for γ = 0.07

(prolate trap) and N = 5 ions. In this plot, we have used Nic = 1000 and M = 1000.


Conclusions

1. At “weakly chaotic regimes” located at the boundaries of islands of quasiperiodicmotion of Hamiltonian systems , probability density functions (pdfs) of sums of chaoticvariables, are well approximated by q–Gaussians .

2. These q–Gaussians are in fact Quasi–Stationary States (QSS) which last for longtimes and often pass through different stages to exponential and finally to Gaussianform .

3. In some cases, however, it is possible to find that the orbits converge to a specific QSS ,whose pdf is close to a q–Gaussian , at least over its central part.

4. QSS approximated by q–Gaussians can be used to identify weakly from stronglychaotic orbits in models of barred galaxies over realistically short time intervals , whereother methods, like Fourier analysis and Lyapunov exponents are not very helpful.

5. These results are also observed in low–dimensional conservative maps andmulti-dimensional Hamiltonian systems independent of the number of degrees offreedom .


References

1. Tsallis, C. [2009], “Introduction to Nonextensive Statistical Mechanics: Approaching aComplex World”, Springer, New York.

2. Antonopoulos, C., Bountis, T. and Basios, V. [2010], “Quasi–Stationary States inMultidimensional Hamiltonian Systems”, PRE 81, 016211.

3. Ruiz Lopez G., Bountis, T. and Tsallis, C. [2011], “Time–Evolving Stastistics of ChaoticOrbits in Conservative Maps in the Spirit of the Central Limit Theorem”, Intern. J. Bifurc.Chaos, Vol. 22 (9), pp. 12502.

4. Bountis, T. and Skokos, H. [2012], Complex Hamiltonian Dynamics, Springer Verlag SynergeticsSeries, Springer Verlag, Berlin.

5. Bountis, T., Manos, Th. and Antonopoulos Ch [2012], “Complex Statistics in HamiltonianBarred Galaxy Models”, Celestial Mechanics and Dynamical Astronomy, 113, Issue 1 (2012),Page 63-80.


6. Leo, M. and Leo, R. A. and Tempesta, P. [2010], “Thermostatistics in the Neighborhood ofthe π–mode Solution for the Fermi–Pasta–Ulam β System: From Weak to Strong Chaos”, J.

of Stat. Mech.: Th. & Exp., 04, pp. 04021.

7. Antonopoulos, Ch., Bountis, T. and Skokos, Ch., [2006],“Chaotic Dynamics ofN–Degree–of–Freedom Hamiltonian Systems”, International Journal of Bifurcation andChaos, vol.16(6), 1777-1793 , June 2006.

8. Antonopoulos, Ch. and Bountis, T. , [2006], “Stability of Simple Periodic Orbits and Chaosin an FPU Lattice”, PRE73, 056206, 1-8 (2006).


COMPLEX HAMILTONIAN DYNAMICS AND STATISTICS

T. Bountis1, Ch. Antonopoulos2, H. Skokos3, H. Christodoulidi1 and C. Tsallis4

1Department of Mathematics, University of Patras, Patras, Greece2Institute for Complex Systems and Mathematical Biology, University of Aberdeen

3Department of Physics, University of Thessaloniki, Greece4Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro-RJ, Brazil

Lecture 4:

Chaotic Diffusion and the Role of Long Range Interactions




Contents

1. Nonextensive Statistical Mechanics and q-Gaussians

2. Statistics of Subdiffusion in a Disordered KG Chain

3. The FPU β− Model With 1/rα Long Range Interactions

4. Different Dynamics for 0 < α < 1 and α > 1

5. Different Statistics for 0 < α < 1 and α > 1

6. A “Phase Diagram” Separating BG from q-Statistics

7. Conclusions



Complex Dynamics and Statistics in a Disordered KG Chain

One fundamental issue in condensed matter physics is the study of conductivity ofelectrons in solids . What happens in realistic situations when disorder is present in thecrystal due to impurities or defects?

How does disorder affect conductivity? This question was first answered by PhilipAnderson, who showed that if disorder is large enough the diffusive motion of theelectrons will eventually stop (Anderson localization) .

Anderson’s analysis, however, is based on a linear model that can be represented inone dimension by the time-dependent Schrödinger equation

i∂ψl

∂t= ǫlψl − ψl+1 − ψl−1, (1)

where ǫl are the random on-site energies, drawn from an uncorrelated uni formdistribution in [−W/2,W/2], W parametrizes the disorder strength and ψl is the complexwave function associated with lattice site l.


So what is the effect of nonlinearity on the localization properties of wave packets indisordered systems?

Many studies have shown that an initially localized wave pac ket spreadssubdiffusively and chaotically for moderate nonlinearities. For weak nonlinearities, onthe other hand, wave packets have been shown to “freeze” over the available integrationtime, thereby resembling Anderson localization.

In fact, it was conjectured by Johansson, Aubry et al. 2010 that these states will belocalized for long enough times on some Kolmogorov-Arnold- Moser (KAM) torus ,since every particle’s motion tends to become linear in that limit.

Based on our statistical analysis , we will argue that KAM localization does not occurand chaotic diffusion persists for infinitely long times.


In particular, we study statistically diffusive motion in a disordered Klein Gordon (KG)one–dimensional lattice of N = 1000 particles, treated extensively in the literature, withHamiltonian

HKG =N∑

l=1

p2l2

+ǫl

2x2l +

1

4x4l +

1

2W(xl+1 − xl)

2 = E, (2)

where xl and pl are the positions and momenta on site l, and ǫl are chosen uniformly fromthe interval

[12, 32

].

We examined two different regimes:

1) The low energy (sub–diffusive regime)

2) The higher energy (self–trapping regime) ,

using two representative examples of energies, E = 0.4 and E = 1.5 and integrated theKG chain using a fourth order Yoshida’s symplectic integrat or .


We analyze normalized energy distributions zν ≥ 0 defined by

zν ≡ Eν/∑

µ

Eµ (3)

where ν =∑

ν νzν and Eν is the νth normal mode energy. We then compute the timeevolution of the second moment

m2 =∑

ν

(ν − ν)2zν , (4)

and the participation number

P = 1/∑

ν

z2ν , (5)

Starting with very narrow packets , the wave diffuses always chaotically , both in the lowenergy (diffusive) regime, as well as in the higher energy (self–trapping) regime, with thesecond moment following the law m2 ∼ tα with α ≈ 1/3 for long times .


Our statistical analysis showed that weak chaos converges quickly to strong chaos , fortimes up to 109 and thus we conclude that the dynamics does not relax onto aquasiperiodic KAM torus and diffusion spreads chaotically for arbitrarily long times .

We plot the q-Gaussian pdfs and focus on the time evolution of the q entropicparameter for a class of observables , starting from the central particle (i.e. η1 = x500) andtaking into account more and more sites , i.e.:

η1 = x500

η5 = x498 + . . .+ x502

η9 = x496 + . . .+ x504

η19 = x491 + . . .+ x509

η29 = x486 + . . .+ x514

η39 = x481 + . . .+ x519

η1000 = x1 + . . .+ x1000


In Figure 1 we show two representative examples of pdfs with different q entropic indices forthe low energy subdiffusive case, i.e. E = 0.4. In panel (a) we plot the numericaldistribution (dashed curve) for the observable η1 computed in the time interval [0, 108] andfind that it is well fitted by a q-Gaussian distribution (solid thick curve) withq1 = 0.993± 0.009.

On the other hand, panel (b) which is the same plot as a), for the observable η29, reveals aclear q-Gaussian distribution, over nearly four decades on the vertical axis, withq29 = 1.22± 0.01.

Next, we present the pdfs for the self-trapping case E = 1.5 in Figure 2 keeping everythingelse the same as in Figure 1. We see that in this case not only the entropic index q1 butalso q29 is even closer to the q = 1 value of a Gaussian compared to that of Figure 1.


10-4

10-3

10-2

10-1

-20 -10 0 10 20

P(sM(j))

sM(j)

a)q1=0.993

10-4

10-3

10-2

10-1

-30 -20 -10 0 10 20 30

P(sM(j))

sM(j)

b)q29=1.22

Figure 1. Panel a : Plot of the numerically computed pdf (dashed curve) for the observableη1 in the time interval [0, 108] with q1 = 0.993± 0.009 taken from fitting with a q-Gaussiandistribution in solid thick. Panel b : Similar plot of the numerically computed pdf (dashed) forthe observable η29 and a time interval [0, 108] with q29 = 1.22± 0.01. In both panels,N = 1000 and E = 0.4 that corresponds to the subdiffusive case .


10-4

10-3

10-2

10-1

-20 -10 0 10 20

P(sM(j))

sM(j)

a)q1=1.062

10-4

10-3

10-2

10-1

-40 -30 -20 -10 0 10 20 30 40

P(sM(j))

sM(j)

b)q29=1.08

Figure 2. Panel a : Plot of the numerically computed pdf (dashed curve) for η1 in the timeinterval [0, 108] with q1 = 1.062± 0.008 fitted by a q-Gaussian distribution (solid thickcurve). Panel b : Plot of the pdf (dashed curve) for η29 and the time interval [0, 108] fittedby a q-Gaussian distribution (solid thick curve), with q29 ≈ 1.08± 0.01. In both panels, weuse N = 1000 and E = 1.5 belongs to the self-trapping regime .


Next, in panel a) of Figure 3, in the subdiffusive case ( E = 0.4), the central 5 to 29particles initially perform a weakly chaotic motion (although the q1-q29 later decaytowards 1). On the other hand, if we study more particles with η39, the motion is morechaotic since the q index now tends more quickly to 1 at t = 109, while for all particles (i.e.for η1000) strong chaos becomes even clearer as q1000 tends to 1 even more rapidly.

These results suggest that the behavior of the central part of the lattice is more weaklychaotic , while the whole lattice behaves in a strongly chaotic way . This is also shown inpanel b) of Figure 3, where the three largest Lyapunov exponents initially decrease towardszero, but after t = 105 they jump to higher values and then decrease with a slope smallerthan 1. Recently it was found that the maximum Lyapunov exponent λ1 decreases asλ1 ∝ t−0.25. This behavior is also seen in Figure 3 b).

From the results of Figure 3 b) it is evident that the evolution of these exponents isdetermined by the maximum Lyapunov exponent λ1, as all of them show similar behaviors.As we see from panel c) of Figure 3, the expected behavior of the second moment, i.e.m2 ∝ t1/3, is also well reproduced by our numerical results.


0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

108 2.5 108 5 108 109

t

a)q1q5q9

q19q29q39

q1000

10-810-710-610-510-410-310-210-1100101

101 102 103 104 105 106 107 108 109

t

b)λ1λ2λ3

t-1

t-0.25

100

101

102

103

104

101 102 103 104 105 106 107 108 109

m2

t

c)t1/3

Figure 3. Panel a : Plot of the q entropic indices q1, q5, q9, q19, q29, q39 and q1000 forN = 1000 and E = 0.4 for the subdiffusive case . Panel b : Plot of the corresponding threelargest Lyapunov exponents λ1, λ2, λ3, and of t−1, t−0.25 to guide the eye. Panel c : Plot ofthe second moment m2 in time together with t1/3 to guide the eye.


By contrast, in Figure 4, where the same study is repeated for the higher en ergyE = 1.5 of the self-trapping case, the dynamics is somewhat differe nt . Panel a) showsthat all q entropic indices of Figure 3 are now much closer to 1 , even those of the centralparticles.

Comparing the three largest Lyapunov exponents in the two cases, we see that at the higherenergy of the self-trapping case (which corresponds to stronger nonlinearity) they jump tohigher values at about t = 104, i.e. one order of magnitude, and do so earlier than in thecase of the lower energy of the subdiffusive case.

After that point, the Lyapunov exponents start decaying to zero but a bit faster than the one(∝ t−0.25) observed in the subdiffusive case of Figure 3. We note again that m2 growsagain in time as t1/3 as can be evidenced in panel c) of Figure 4.


0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

108 2.5 108 5 108 109

t

a)q1q5q9

q19q29q39

q1000

10-8

10-6

10-4

10-2

100

102

101 102 103 104 105 106 107 108 109

t

b)λ1λ2λ3

t-1

t-0.25

10-1

100

101

102

103

104

101 102 103 104 105 106 107 108 109

m2

t

c)t1/3

Figure 4. Panel a : Plot of the q entropic indices q1, q5, q9, q19, q29, q39 and q1000 for

E = 1.5 of the self-trapping . Panel b : Time plot of the three largest Lyapunov exponents

λ1, λ2, λ3, and of t−1, t−0.25 to guide the eye. Panel c) : Plot of the corresponding second

moment m2 in time together with t1/3 to guide the eye.


The FPUβ− Model With 1/rα Long Range Interactions

Let us now focus on the importance of introducing long range interactions to a class ofFermi-Pasta-Ulam (FPU) β−models , whose Hamiltonian is given by

H =1

2

N∑

n=1

p2n +1

2

N∑

n=0

(xn+1 − xn)2 +

+b

4N

N∑

n=0

N+1∑

m=n+1

(xn − xm)4

|n−m|α= U(N) (b > 0; α ≥ 0) , (6)

with fixed boundary conditions , i.e. x0 = xN+1 = p0 = pN+1 = 0. The quartic part ofthe potential energy scales with N as follows

N(N,α) ≡1

N

N∑

i=0

N+1∑

j=i+1

1

(j − i)α=

1

N

N∑

i=0

N + 1− i

(i+ 1)α(7)

The N scaling is introduced in the Hamiltonian so as to make the kinetic as well as both

parts of the potential energy extensive (i.e. proportional to N ) for all values of α.


There are two important limits in the extremal cases:

(i) The long range limit α→ 0: Each particle interacts equally with all others independentlyof the distance between them and

(ii) The short range limit α→ ∞: Only interactions with nearest neighbors apply,recovering exactly the Hamiltonian of the FPU-β model.

In the present study we have implemented long range interactions only in the quarticpart of the potential in (6).We have found, however, that if this type of scaling is alsointroduced in the quadratic terms the results are very simil ar.

As initial conditions for integrating our trajectories, we always take zero positions andmomenta drawn randomly from a uniform distribution.


Different Dynamics for 0 < α < 1 and α > 1

We begin by computing the largest Lyapunov exponent λmax for different values of α, Nand specific energies u = U(N)/N . In Figure 5 we have plotted the λmax versus thesystem size N for different α values, ranging from 0 to 10. The critical value α = 1, asfound by other researchers, clearly distinguishes between two different regimes:

i) For α ≥ 1 the Lyapunov exponent λmax tends to stabilize at a finite and positive valueas N increases. So, for the system with short–range interactions in the thermodynamic limitλmax depends only on ǫ and α.

ii) For α < 1 the Lyapunov exponent is observed to decrease with system size asN−κ(α), where the dependence of the exponent κ(α) on α is shown in the inserted panel ofFigure 5.

We therefore expect that the system with short–range interactions tends to a BGequilibrium in the thermodynamic limit , while long–range dynamics is associated witha weaker form of chaos .


10 100 1000 100000.15

0.2

0.25

0.3

0.35

0.40.450.5

0.550.6

0.650.7

0.0 0.4 0.8 1.20.00

0.05

0.10

0.15

_

FPU 10 5 4 1 0.8 0.6 0.4 0.2 0

N

b=10, U(N)/N=9

periodic boundary condiitons =0)__

Figure 5: Maximal Lyapunov exponent for increasing N and various values of α, whenU(N)/N = 9, and b = 10 calculated at t = 106 for fixed boundary conditions. Note, asshown in the insert, that λmax ∝ N−κ(α).


Different Statistics for 0 < α < 1 and α > 1

Plotting now momentum distributions of orbits, we find in the case of long–rangeinteractions a quasi–stationary state (QSS) is formed with a q–Gaussian probabilitydensity function (pdf) of the momenta , with 1 < q < 3.

The histograms shown on the left panel of Figure 6 , for α = 0.7 are well fitted by theq–Gaussian pdf :

P (p) = P (0)[1 + β(1− q)(pP (0))2]1/(1−q), q ≥ 1. (8)

with q = 1.249. This figure was produced for two time intervals (105 − 5× 105) and(4× 105 − 8× 105), for which q is nearly constant.

For times longer than t = 106, q slowly decreases as a power law in time and tends tothe value 1, which explains why this is a QSS . In Figure 6(b), on the other hand, thedistribution for the interaction with α = 1.4 follows from the beginning a pureGaussian pdf (q → 1 in (8)) with β = 0.043.


-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 3010-6

10-5

10-4

10-3

10-2

10-1

100

101

t [105,5*105] t [4*105,8*105] q=1.249, =0.069 q=1, =0.069

P(p

)/P

(0)

pP(0)

=0.7, b=10, =4, N=8192, U(N)/N=9

-30 -25 -20 -15 -10 -5 0 5 10 15 20 25 3010-6

10-5

10-4

10-3

10-2

10-1

100

101

t [105,5*105] q=1, =0.043

P(p

)/P

(0)

pP(0)

=1.4, b=10, =4, N=8192, U(N)/N=9

Figure 6: Momentum distributions for the system with N = 8192 and: (a) α = 0.7 in twotime intervals, (b) α = 1.4.


The q–dependence on α in Figure 7 clearly shows that the transition from q–statistics toBG–statistics is evident as α exceeds 1 . The data of Figure 7 is averaged over severalrealizations. Increasing N we observe that the q values shift upward somewhat but theoverall qualitative picture remains the same.

To check the robustness of our results, we have computed the q-generalized kurtosis ofthe distribution defined as follows:

κq(q) =

∫∞

−∞dp p4[P (p)]2q−1/

∫∞

−∞dp [P (p)]2q−1

3[∫

∞

−∞dp p2[P (p)]q/

∫∞

−∞dp [P (p)]q

]2 (9)

Using the q values found in Figure 7, we show in Figure 8 that the numerical dependenceof q-kurtosis on q compares very well with the analytical curve κq(q) = (3− q)/(1 + q)

obtained by substituting the q-Gaussian pdf (8) in Eq. (9).


0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

1.0

1.1

1.2

1.3

1.4

1.5

103 1044.55.05.56.06.57.0

2048 4096 8192 16384

b=10, U(N)/N=9

q

t [100000,500000]

(qin

f-q)-1

N

=0

Figure 7: α-dependence of the index q for b = 10 and U(N)/N = 9 averaged over 4independent realizations when N is 2048, 4096, 8192 and 2 realizations for N = 16384, alltaken in the time interval t ∈ [5 · 105, 9 · 105]. Inserted panel shows the line(q∞ − q)−1 = 1.76 logN − 0.9, for the data of the main figure with α = 0.


1.0 1.1 1.2 1.3

0.7

0.8

0.9

1.0

=0

=2

q

q

=0.7

BG

q(q)=(3-q)/(1+q)

Figure 8: q-dependence of the q-kurtosis κq for typical values of α, together with theanalytical prediction κq = (3− q)/(1+ q) (blue curve) for the data of Figure 7 and N = 8192.


A “Phase Diagram” Separating BG from q-Statistics

Figure 9(b) displays the crossover between the two regimes in the form of a “phasediagram” , in which a straight line fit of the data separates the two “phases” in the 1/N vs.1/tγc plane. Each point in the graph of Figure 9(b) corresponds to a value of t = tc

representing the maximum time that q remains constant, after which q tends to the BG valueq = 1 following a power law, as shown in Figure 9(a).

For high nonlinearity strength b = 10 the line separating the two domains has a big slopeand the region of BG statistics is rather limited in the upper part of the diagram.However, when we decrease the nonlinearity to b = 2 the slope of the boundarydecreases and the BG domain becomes wider .

As a final remark, we point out the nonuniformity of the (N, t) → (∞,∞) limit implied bythe diagram of Figure 9(b). Clearly, in the limN→∞ limt→∞ ordering it is the q = 1 behaviorthat prevails, while in the limt→∞ limN→∞ ordering it is the q > 1) statistics that becomesdominant.


105 106 107

0.1

0.2

0.3

0.40.5

tc

q-1

t

N=4096, =0.7, b=10, =2, U(N)/N=9

0.0 2.0x10-8 4.0x10-80.0

5.0x10-4

1.0x10-3

8192

b=21024

2048

(q >1) -dominated region (q -Gaussians)

(q

=1)

-dom

inate

d

regio

n (G

auss

ians)

=0.7, U(N)/N=9

1/N

1/tc

4096

b=10

Figure 9: (a) Evolution of q − 1 in double logarithmic scale . Each point corresponds totime averaged results over a window 2 · 105. Note that the QSS lasts for times up to tc. (b)Crossover frontiers between Gaussian and q-Gaussian thermostatistics.


A “Universal” Picture?

As we saw in Fig. 17(b), for high nonlinearity strength b = 10 the line separating the twodomains has a large slope and the region of BG statistics is somewhat limited. However,when we decrease the nonlinearity to b = 2 the slope of the boundary decreases andthe BG domain prevails over q-statistics.

The crossover frontier is approximatively given by

1

N∼ D(α, u)

bδ

tγc(10)

and may be plotted as a single straight line , yielding a “universal picture” of thenonextensive statistics valid for alpha < 1 (see Figure 10 below), with D = 2.3818× 104,δ = 0.27048, and γ = 1.365.

Thus, for α < 1, our evidence suggests that the system follows q-statistics and formsquasi-stationary states (QSS) , in a “weakly” (non–uniformly ergodic) subspace of the fullphase space, where it lives for a very long time, until it eventually enters a domain of“strong” chaos and uniform ergodicity .


0.0 2.0x10-8 4.0x10-80.0

5.0x10-4

1.0x10-3

b=10 b=2

8192

1024

2048

(q >1) -dominated region (q -Gaussians)

(q =1) -dominated region (Gaussians)

=0.7, U(N)/N=9

1/N

b /tc

4096

Figure 10: The crossover frontier for α < 1 may be plotted by a single straight line1/N = D(α, u)bδ/tγc . It divides quasi-stationary states (QSS) that live for long times in“weakly” chaotic (non–uniformly ergodic) subspaces and obey nonextensive statistics, untilthey eventually enter a domain of “strong” chaos (uniform ergodicity) and BG statistics.


Conclusions

1. In the subdiffusive case ( E = 0.4), the central particles initially perform weaklychaotic motion . On the other hand, if we study more particles, the motion is morechaotic since the q index now tends more quickly to 1 at t = 109. The expected behaviorof the second moment, i.e. m2 ∝ t1/3, is well reproduced by our numerical results.

2. For the higher energy E = 1.5 of the self-trapping case , the dynamics is somewhatdifferent. All q entropic indices are now much closer to 1 , even those of the centralparticles. Here also m2 in time as t1/3

3. In the presence of all to all interactions of the form 1/rα, the long range case with0 < α < 1 is characterized by QSS with decaying Lyapunov exponents and q–statistics .For the short range α > 1 Lyapunov exponents converge to positive values and BGstatistics sets in .

4. There appears to be a “phase diagram” separating the two thermostatistics in a1/N,∝ 1/tγ plane, with a “universal” boundary for fixed α and energy density. In thelimN→∞ limt→∞ ordering we find BG statistics , while in the limt→∞ limN→∞ orderingit is the q > 1 statistics that prevails .


References

1. Antonopoulos, Ch., Bountis, T., Skokos, Ch. and Drossos, L. [2014] ,“ComplexStatistics and Diffusion in Nonlinear Disordered Particle Chains”, to appear in special issueof Chaos, ed. by Georg Gottwald.

2. Christodoulidi, H., Tsallis, C. and Bountis, T. [2014] “Fermi-Pasta-Ulam model withlong-range interactions: Dynamics and thermostatistics”, submitted to PRL.

3. Bountis, T. and Skokos, H. [2012] , “Complex Hamiltonian Dynamics”, SpringerSynergetics series, Springer Verlag, Berlin.

4. Tsallis, C. [2009] , “Introduction to Nonextensive Statistical Mechanics: Approaching aComplex World”, Springer, New York.

6. Anteneodo C. and Tsallis C. [1998] , “Breakdown of the exponential sensitivity to theinitial conditions: Role of the range of the interaction”, Phys. Rev. Lett. 80, 5313.

5. Tsallis, C. [2014] , “An introduction to nonadditive entropies and a thermostatisticalapproach to inanimate and living matter”, Contemporary Physics, published online: 30 April2014.


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