chapter4.pdf

Chapter 4

The Classical Delta-Kicked Harmonic

Oscillator

4.1 Introduction

The test system that we propose in order to experimentally investigate quantum chaos is thethe delta-kicked harmonic oscillator. In this chapter we go into considerable detail to explainthe properties of the classical delta kicked harmonic oscillator, in order to provide context,and to give us something to compare the expected quantum dynamics to. The delta-kickedharmonic oscillator has has been extensively investigated in a series of papers by Zaslavskyand co-workers [1, 2, 3], and is also dealt with in depth in his book [4]. What follows is mostlyan amalgam of these various publications; certain aspects have been elaborated upon, andothers deemed less immediately relevant have been omitted.

The most extensively studied test system of all is of course the delta-kicked rotor, coveredextensively in [5]; there have even been experiments on an equivalent system in the quantumregime, carried out in atom-optical systems by the group of Raizen [6]. The delta-kickedharmonic oscillator offers the advantage that when dealing with cold atoms or ions, whereexperiments on quantum chaotic systems could be (and have been [6]) carried out, thereis essentially always some kind of trapping potential, which is harmonic to some degree ofaccuracy. As this potential is present in any case, it is convenient to simply take it intoaccount, and the delta-kicked harmonic oscillator has some interesting properties on its own.

4.2 Derivation of a Kick to Kick Mapping

We consider a point particle held in a harmonic potential, periodically kicked by a cosinepotential. The classical Hamiltonian function of this delta-kicked harmonic oscillator is givenby

H =p2

2m+

m2x2

2+ K cos(kx)

n=

(t n), (4.1)

41

42 CHAPTER 4. THE CLASSICAL DELTA-KICKED HARMONIC OSCILLATOR

where x, p is the momentum, m is the mass of the particle, is the harmonic frequency,k = 2pi/ is the wavenumber of the cosine potential (where is the wavelength). The mostimportant parameter is the kick strength K, which has the dimension of action (kgm2s1);the delta function has the dimension of a frequency, where is the time between kicks.

Hamiltons equations of motion for this system are:

dx

dt=

p

m, (4.2)

dp

dt= m2x + kK sin(kx)

n=

(t n). (4.3)

Clearly, between kicks, the time evolution is simply that of a conventional harmonic oscilla-tor, given by

x(t) = x(0) cos(t) +p(0)

msin(t), (4.4)

p(t) = p(0) cos(t)mx(0) sin(t). (4.5)

We now consider Eq. (4.3), the equation of motion for the momentum, integrating over theneighbourhood of the time when some arbitrary kick occurs n , to get

p(n + ) p(n ) = m2 n+

n

dtx(t) + kK

n+n

dt sin[kx(t)]

n=

(t n)

= m2 n+

n

dtx(t) + kK sin[kx(n)]. (4.6)

As we let 0, the remaining integral shrinks to nothing, leaving us with

p(n+) = p(n) + kK sin[kx(n)]. (4.7)

We can now combine this with the equations of motion for the harmonic oscillator to derivea mapping from just before one kick to just before the next. We introduce the convenientshorthand xn, pn to stand for x(n

), p(n).

xn+1 = xn cos() +1

m[pn + kK sin(kxn)] sin(), (4.8)

pn+1 = [pn + kK sin(kxn)] cos()mxn sin(). (4.9)

4.3. RESCALING TO DIMENSIONLESS VARIABLES 43

4.3 Rescaling to Dimensionless Variables

4.3.1 Dimensionless x and p.

We can rescale the variables x and p to dimensionless quantities in order to reduce thenumber of free parameters:

x =kx

2, (4.10)

p =kp2m

. (4.11)

We also let t = t, = , so that we have a dimensionless time. This means that thekick to kick mapping can be rewritten to give

xn+1 = xn cos() + [pn + sin(

2xn)] sin(), (4.12)

pn+1 = [pn + sin(

2xn)] cos() xn sin(), (4.13)where = k2K/

2m, and we have dropped the primes for convenience. It is now apparent

that we have exactly two free parameters: the dimensionless kicking strength , and thedimensionless kick to kick period . We can also write a dimensionless rescaled Hamiltonianfunction in terms of the rescaled dimensionless x and p: H = Hk2/2m2,

H =p2

2+

x2

2+

2

cos(

2x)

n=

(t n), (4.14)

where we will drop the prime in future.The dynamics produced by the mapping described in Eqs. (4.12,4.13) can be grouped

into two broad categories: one where /2pi is a rational number, which is to say that theparticle is kicked a rational number of times per harmonic oscillator period; and where /2piis an irrational number. We will concentrate on the case where /2pi is rational, whichhas some interesting particular properties. Sample stroboscopic Poincare sections of thedynamics described by the mapping of Eqs. (4.12,4.13) for /2pi = 1/4, 1/5, 1/6, and 1/7are displayed in Figs. (4.1,4.2,4.3,4.4).

The initial condition in each of the cases displayed in Figs. (4.1,4.2,4.3,4.4) are all un-stable, iterated over 40000 kicks. There is obviously a high degree of rotational symmetryin the trajectories taken, which may seem incompatible with the idea of chaotic dynamics.The symmetry is essentially rotational in phase space however, and the unpredictable partof the point particles dynamics is more in how it moves in and out through phase space,i.e. what the value of

x2 + p2 is. The symmetric structure brought out by these unstable

dynamics is called a stochastic web, as there is an interconnected web of channels of unstabledynamics spread through all of phase space.

Note also that in the cases of /2pi = 1/4 and 1/6 displayed in Figs. (4.1,4.3) thereappears to be a translational symmetry in the phase space structure described by the plottedtrajectory. The significance of this will be dealt with in what follows, in particular inSection 4.7.


80 60 40 20 0 20 40 60 80

60

40

20

0

20

40

60

x

p

Figure 4.1: /2pi = 1/4, = 0.8

100 50 0 50 100

60

40

20

0

20

40

60

x

p

Figure 4.2: /2pi = 1/5. = 0.8

4.3. RESCALING TO DIMENSIONLESS VARIABLES 45

80 60 40 20 0 20 40 60 8060

40

20

0

20

40

60

x

p

Figure 4.3: /2pi = 1/6. = 0.8

100 50 0 50 100

80

60

40

20

0

20

40

60

80

x

p

Figure 4.4: /2pi = 1/7. = 0.7


4.3.2 Phase Variable

The mapping described in Eqs. (4.12,4.13) can be rewritten as a single equation in terms ofa single complex variable and its complex conjugate, where = (x + ip)/

2:

n+1 =

[n + i

2

sin(n +

n)

]ei . (4.15)

This more compact description is often advantageous when investigating symmetry proper-ties of this system.

4.4 Perturbation Expansion

From now on we consider only = 2pir/q, where r/q is an irreducible rational. Bearing thisin mind, from the mapping described by Eq. (4.15) we can determine an expression for thevalue of at a time q kicks (= r oscillation periods) later:

n+q = n + i2

q1j=0

sin(n+j +

n+j)ei2pijr/q. (4.16)

From this one can directly determine expressions to describe the position and momentum atthis time:

xn+q = xn q1j=0

sin(

2xn+j) sin(2pijr/q), (4.17)

pn+q = pn +

q1j=0

sin(

2xn+j) cos(2pijr/q). (4.18)

As can be seen, this means that for = 0, the position and momentum have cycled back totheir original values, which they continue to do, every q kick periods.

We now consider the case of small perturbations to the harmonic oscillator dynamics, i.e.small . Taking Eq. (4.16) and keeping terms up to first order in only, we express n+qpurely in terms of n, as

n+q = n + i2

q1j=0

sin(nei2pijr/q + ne

i2pijr/q)ei2pijr/q, (4.19)

which can be written in terms of x and p as

xn+q = xn q1j=0

sin{

2[xn cos(2pijr/q) + pn sin(2pijr/q)]} sin(2pijr/q), (4.20)

pn+q = pn +

q1j=0

sin{

2[xn cos(2pijr/q) + pn sin(2pijr/q)]} cos(2pijr/q). (4.21)

4.5. CANONICAL TRANSFORMATION 47

This mapping possesses a q-fold rotational symmetry in phase space. This can be see asfollows. We substitute n with n = ne

i2pil/q [this is equivalent to replacing xn with yn =xn cos(2pi/q) pn sin(2pi/q) and pn with sn = xn sin(2pi/q) + pn cos(2pi/q)]. Thus, we have

n+q =

[n + i

2

q1j=0

sin(nei2pi(jrl)/q + ne

i2pi(jrl)/q)ei2pi(jrl)/q

]ei2pil/q. (4.22)

We have as the imaginary argument to all the exponentials inside the square brackets 2pi(jrl)/q. Now, ei2pijr/q has exactly q possible values, all of which are represented in the sumin Eq. (4.16). If we add in an l/q to the argument, this clearly still holds for the sum inEq. (4.22). The point of this is that, because of the cyclic nature of the variable terms beingsummed over, both sums contain the same terms, and we can replace one sum with another.Thus

n+q =

[n + i

2

q1j=0

sin(nei2pijr/q + ne

i2pijr/q)ei2pijr/q

]ei2pil/q, (4.23)

from which we can immediately see that if n = nei2pil/q, then n+q = n+qe

i2pil/q, whichdescribes the q-fold rotational symmetry in phase space. As this is a result coming froma perturbation expansion around , this is only an approximate symmetry, most valid forsmall . This symmetry can be expressed in terms of x and p as

yn = xn cos(2pil/q) pn sin(2pil/q) yn+q = xn+q cos(2pil/q) pn+q sin(2pil/q), (4.24)sn = pn cos(2pil/q) + xn sin(2pil/q) sn+q = pn+q cos(2pil/q) + xn+q sin(2pil/q), (4.25)

where (y + is)/

2 = . This explains the observed rotational symmetry of the phase spacestructure shown by the Poincare sections of Figs. (4.1,4.2,4.3,4.4).

4.5 Canonical Transformation

In the previous section we saw that there is a rotational q symmetry that makes itselfapparent every q kicks. This motivates us to examine the dynamics of rotating variables.Thus, we now wish to make a canonical transformation into a rotating reference frame [7],using the new variables X and P , defined by

X = x cos(t) p sin(t), (4.26)P = p cos(t) + x sin(t). (4.27)

Note that the new variables X and P coincide with x and p whenever t = 2pin, where n ,and that q kick periods corresponds to t = 2pir. Thus, if one observes the variables justbefore every kick, the two pairs of variables coincide every q kicks, and only every q kicks.

In the absence of kicks, these transformations have the effect of fixing any initial conditionin X, P phase space, for all time. Similarly, when kicks are periodically added, there will beno movement through X, P phase space between kicks; movement will only occur when a


kick is applied. The hope is that examining the transformed Hamiltonian will bring out init some underlying structure, fixed in X, P space, with the rotational symmetry observed inthe perturbation expansion. As the evolution of x and p coincides with that for X and Pevery q kicks, this should tell us something about the averaged dynamics of x and p over atime period of q kicks.

The transformed Hamiltonian H is given by [7]

H = H +F

t, (4.28)

where

xF (x, X) = p, (4.29)

XF (x, X) = P. (4.30)

We thus need to express p and P exclusively in terms of x and X. With a little algebra, itis easy to see that

p =x cos(t)X

sin(t), (4.31)

P =xX cos(t)

sin(t). (4.32)

Substituting Eqs. (4.31) into Eqs. (4.29) and integrating, we see that

F (x, X) =x2 cos(t)/2 xX

sin(t)+ f(X). (4.33)

Substituting this expression for F (x, X) with Eq. (4.32) back into Eq. (4.30), we arrive at

xsin(t)

+

Xf(X) = xX cos(t)

sin(t)(4.34)

which implies directly that

Xf(X) = X cot(t). (4.35)

We thus see immediately that f(X) = X2 cot(t)/2 [plus an arbitrary constant, which can beseen to be irrelevant on examining Eq. (4.28)], and that therefore

F (x, X) =x2 + X2

2cot(t) xXcosec(t). (4.36)

We now need to take the partial derivative with respect to time of F (x, X):

tF (x, X) = x

2 + X2

2cosec2(t) + xXcosec(t) cot(t). (4.37)

4.6. AVERAGED HAMILTONIAN 49

If we substitute in x = X cos(t) + P sin(t), we can write this expression terms of X and Ponly. After some algebra, we arrive at

tF (x, X) = X

2

2 P

2

2. (4.38)

As x2 + p2 = X2 + P 2, we can now rewrite Eq. (4.28) in terms of X and P as

H =2

cos{

2[X cos(t) + P sin(t)]}

n=

(t n), (4.39)

and we have the transformed Hamiltonian, written in terms of the canonically conjugaterotating variables X and P .

4.6 Averaged Hamiltonian

4.6.1 Derivation

We substitute the following identity for the train of delta kicks

n=

(t n) =q1j=0

m=

[t (mq + j) ] (4.40)

into Eq. (4.39), and get

H =2

q1j=0

m=

[t (mq + j) ] cos(

2{X cos[(mq + j) ] + P sin[(mq + j) ]}). (4.41)

Considering now explicitly the case where = 2pir/q, this simplifies down to

H =2

q1j=0

cos{

2[X cos(2pijr/q) + P sin(2pijr/q)]}

m=

[t (mq + j)2pir/q]. (4.42)

The infinite train of delta kicks is periodic in time, and can thus be expanded as a Fourierseries,

m=

[t (mq + j)2pir/q] = 12pir

n=

ein2pij/qeint/r. (4.43)

We substitute this back into Eq. (4.42), and after rearranging somewhat, the Hamiltoniancan be expressed as

H = Hr/q + Vr/q, (4.44)

Hr/q =

2pir

2

q1j=0

cos{

2[X cos(2pijr/q) + P sin(2pijr/q)]}, (4.45)

Vr/q =

pir

2

q1j=0

cos{

2[X cos(2pijr/q) + P sin(2pijr/q)]}

n=1

cos[n(t/r 2pij/q)]. (4.46)


All of the time dependence in H is in Vr/q, and we note that over a time of q kick periods(i.e. t = 2pir), the average value of Vr/q is zero. We thus call Hr/q the averaged Hamiltonian,as it is all that is left over after this averaging procedure.

4.6.2 Interpretation

The averaged Hamiltonian Hr/q clearly has a q-fold rotational symmetry through phasespace, in the same sense as that described in Section 4.4, i.e. if we replace X with Y =X cos(2pil/q) P sin(2pil/q) and P with S = P cos(2pil/q) + X sin(2pil/q), Hr/q remainsunchanged. This rotational symmetry can be seen graphically in Figs. (4.5,4.6,4.7,4.8,4.9),where we have displayed contour plots of rHr/q for q = 1 7. Note that except for a factorof two, q = 1 or 2 gives the same result, as does q = 3 or 6, and so we do not plot Hr/qspecifically for q = 2 or 6.

A number of things become clear upon examining the plots for Hr/q. Firstly, certain of theHr/q display a translational, or crystal, symmetry, corresponding to that seen for the Poincaresections where q = 4 and 6 in Figs. (4.1,4.3). This takes place when q qc = {1, 2, 3, 4, 6}.Given that it is known that it is only possible to combine a translational with a rotationalsymmetry when q qc [8], equivalent to tiling the plane without gaps with regular q-gons,it is inevitable that this be the case. When q / qc, rHr/q displays a quasicrystal symmetry[2, 3]. Secondly, in the cases q = 3, 4, 6 there is a network of interconnected separatricescovering all of phase space. The contour plots of Hr/q show an energy surface; thus thisseparatrix net describes an interconnected region of constant energy extended throughoutphase space.

One would therefore expect point particles whose dynamics are governed by Hr/q startingon this separatrix net, to travel freely along this net through a wide area of phase space. Inthe case of q / qc, shown in Figs. (4.8,4.9) for the examples q = 5 and 7, there is no suchan interconnected network, but one can see that there are many closed loops which lie veryclose together, and that it would take only a small disturbance to connect them, and wewould again have a structure spread throughout phase space.

Such a disturbance is provided for by Vr/q for small . Remember that Hr/q is theaveraged Hamiltonian, and that perturbations due to Vr/q average out to zero over any givencycle of q kicks (a q cycle). At any one time during a q cycle there are of course non-zerocontributions to H from Vr/q, although if is small, one expects these contributions to beslight. We can thus say that the stochastic webs observed in Figs. (4.1,4.2,4.3,4.4) are basedaround the separatrix nets or web skeletons defined by Hr/q. We emphasize that this is validfor small only; if this is the case then the dynamics of a point particle are mostly accountedfor by Hr/q, but small perturbations due to Vr/q link nearly adjacent loops in phase spacetogether in the case that q / qc, meaning that a point particle can diffuse out through phasespace along the stochastic web, and in general these perturbations cause the stochastic webto have a finite thickness.

Those a little more expert may notice this derivation appears to contradict the KAM [9]theorem, which essentially states that if an integrable system is perturbed so as to becomenon-integrable, then the KAM tori are mostly expected to be similar, apart from some


15 10 5 0 5 10 15

10

8

6

4

2

0

2

4

6

8

10

x

p

Figure 4.5: Contour plot of Hr/q when q = 1 or 2.

15 10 5 0 5 10 15

10

8

6

4

2

0

2

4

6

8

10

x

p

Figure 4.6: Contour plot of Hr/q when q = 3 or 6.


15 10 5 0 5 10 15

10

8

6

4

2

0

2

4

6

8

10

x

p

Figure 4.7: Contour plot of Hr/q when q = 4.

50 0 50

30

20

10

0

10

20

30

x

p



80 60 40 20 0 20 40 60 8060

40

20

0

20

40

60

x

p


distortion, to the invariant tori of the integrable system. If the plots shown in Figs. 4.64.9are supposed to approximate the phase space structure for the limit of small , this is clearlynot true. Even in this limit the harmonic oscillator invariant tori (exact circles in this case)are all crossed over with the separatrix net. The KAM theorem has some rather hedgingconditions however. In particular there should be no degeneracy of the frequencies of theorbits of the unperturbed system. The phase space orbits of the harmonic oscillator clearlyall have exactly the same frequency, an extreme case of degeneracy which means that theKAM theorem is no longer valid. In the case of rational kicks, i.e. = 2pir/q there is aresonance between the kicking frequency and the frequency of every single periodic orbit inphase space. Consequently, nearly every invariant torus is destroyed, in the cases of q = 3,4 or 6 with infinitesimal . Arnold diffusion [10] takes place along the separatrix net whichwould be impossible for a one dimensional system subject to the KAM theorem.

4.6.3 Equations of Motion

We now show explicitly that the dynamics produced by Hr/q coincide exactly with theperturbation expansion to first order in described by Eqs. (4.20,4.21). To do this we takeHamiltons equations of motion for Hr/q considered in isolation. As Hr/q is time independent,the equations of motion for X and P must be integrable, and we can in fact find an exact


solution.

dX

dt=

2pir

q1j=0

sin{

2[X cos(2pijr/q) + P sin(2pijr/q)]} sin(2pijr/q), (4.47)

dP

dt=

2pir

q1j=0

sin{

2[X cos(2pijr/q) + P sin(2pijr/q)]} cos(2pijr/q). (4.48)

From this we can immediately see that

cos(2pijr/q)dX

dt= sin(2pijr/q)dP

dt. (4.49)

We now take the second time derivative of X,

d2X

dt2=

2pir

q1j=0

sin{

2[X cos(2pijr/q) + P sin(2pijr/q)]} sin(2pijr/q)

[dX

dtcos(2pijr/q) +

dP

dtsin(2pijr/q)

](4.50)

which, because of Eq. (4.49), is equal to zero. If we differentiate Eq. (4.49) with respect totime, we immediately see that this in turn also immediately implies that d2P/dt2 = 0. Thederivatives dX/dt and dP/dt must then both be constant, and we conveniently choose toevaluate these constants at the initial time t = 0. The variables X and P thus evolve likeX(t) = X(0) + dX/dt|t=0t and P (t) = P (0) + dP/dt|t=0t, giving

X(t) = X(0) 2pir

q1j=0

sin{

2[X(0) cos(2pijr/q) + P (0) sin(2pijr/q)]} sin(2pijr/q)t, (4.51)

P (t) = P (0) +

2pir

q1j=0

sin{

2[X(0) cos(2pijr/q) + P (0) sin(2pijr/q)]} cos(2pijr/q)t. (4.52)

When considering t = 2pir, i.e. q kick periods, this reduces to

X(2pir) = X(0) q1j=0

sin{

2[X(0) cos(2pijr/q) + P (0) sin(2pijr/q)]} sin(2pijr/q), (4.53)

P (2pir) = P (0) +

q1j=0

sin{

2[X(0) cos(2pijr/q) + P (0) sin(2pijr/q)]} cos(2pijr/q). (4.54)

As we are considering t = 2pir, this means that X and P coincide with x and p. Making thissimple substitution, along with x(0), p(0) = xn, pn and x(2pir), p(2pir) = xn+q, pn+q, we findthat we have reproduced Eqs. (4.20,4.21). Thus the averaged Hamiltonian Hr/q produces thedynamics of the delta kicked harmonic oscillator, when the time evolution is taken to onlyfirst order in the kicking parameter , which is to say that Hr/q produces the dynamics ofthe harmonic oscillator undergoing small perturbations due to a rational fraction of cosinekicks every oscillator period.

4.7. TRANSLATIONAL SYMMETRY 55

4.7 Translational Symmetry

We have already observed that in Hr/q, which gives the dynamics up to first order in , thereis a translational or crystal symmetry for a certain subset of cases where = 2pir/q, namelywhere q qc = {1, 2, 3, 4, 6}. In other words, there is a translational or symmetry in thephase space, expressed over a time period of q kicks. We will now see that this translationalsymmetry is in fact exact, for any value of .

To do this we return to the phase variable notation introduced in Section 4.3.2. Again us-ing Eq. (4.16), we will determine the conditions for such a translational symmetry, expressedas

n = n + n+q = n+q + , . (4.55)Substituting this into Eq. (4.16) results in

n+q + = n + + i2

q1j=0

sin(n+j +

n+j)ei2pijr/q, (4.56)

from which it can immediately be inferred that

q1j=0

sin(n+j +

n+j)ei2pijr/q =

q1j=0

sin(n+j +

n+j)ei2pijr/q. (4.57)

This in turn provides the necessary condition

n+j +

n+j = n+j +

n+j + 2pilj j, lj . (4.58)Taking this condition, Eq. (4.15) for n gives

n+1 =

[n + + i

2

sin(n +

n)

]ei2pir/q, (4.59)

which states that if n = n + , then n+1 = n+1 + ei2pir/q. This can be carried out

iteratively, so that defining j = n+j n+j, we can state that j = ei2pijr/q. Using this,Eq. (4.16) for n+q can be simplified to

n+q = n + + i2

q1j=0

sin(n+j +

n+j + j +

j )ei2pijr/q. (4.60)

Our condition for translational symmetry is thus reduced to j +

j = 2pilj, which can bere-expressed as

pilj = +

2cos(2pijr/q) i

2sin(2pijr/q). (4.61)

It is immediately apparent that l0 = ( + )/2pi, and so this can be simplified to

lj = l0 cos(2pijr/q) i

2pisin(2pijr/q). (4.62)


If we now let j = q/2 m or (q m)/2, depending on whether q is even or odd, we canadd or subtract lj+ and lj. Bearing in mind that cos is symmetric and sin antisymmetricaround = npi, we get

cos(2pij+r/q) =lj+ + lj

2l0 , (4.63)

This implies that cos(2pi/q) , and it is known that this can only be true if q qc ={1, 2, 3, 4, 6} [11]. It is then easy to see that cos(2pijr/q) k, r , the only possiblevalues being 0, 1/2, 1. Thus there is is indeed a translational symmetry in phase space,for q qc only, as expected. It should be emphasized that this is an exact symmetry, for anyvalue of , unlike the rotational q symmetry. There are an infinite number of values of forwhich this translational symmetry applies, the values of which can be determined from thefollowing formulae derived in the same way as Eq. (4.63):

+

picos(2pijr/q) , (4.64)

i

pisin(2pijr/q) . (4.65)

From this, one can easily determine the allowed symmetry preserving displacements in po-sition and momentum x and p, where (x + ip)/

2 = : for q = 1 or 2, x =

2pin;

for q = 4, x =

2pin, p =

2pim; and for q = 3 or 6, x =

2pin, p =

3/2pim,where in each case n, m . One can also think of these equations as determining the sizeof the basic cell of the crystal structure, where the dynamics in equivalent locations in anytwo cells are also equivalent, when viewed over a time period of q kicks. A point particlemay of course move from one cell to another, in which case the dynamics of other equivalentpoint particles would dictate that they move in lockstep into corresponding cells.

There is thus an exact crystal symmetry in the mapping of Eq. (4.16) for q qc. Adirect implication of this is that we can confine ourselves to studying the dynamics of pointparticles initially in just one cell. There are also important implications in the case of thequantum delta-kicked harmonic oscillator.

4.8 Poincare Sections

Our test example from now on will be the case where r/q = 1/6. There is thus one remainingfree parameter, the dimensionless kick strength . We investigate the effect of varying thisby studying Poincare sections for from -0.2 to -4, displayed in Figs. (4.104.18). Thesign of does make a difference to the overall phase space dynamics, it so happens that inthis case for negative the transition to chaos is more rapid.

In each case the same set of initial conditions was chosen, and each initial condition wassubjected to 10000 kicks. We show a region of phase space based around the central islandstable for small , with parts of the next circle of stable islands also depicted.

As expected, the most exact rotational symmetry is visible in Fig. (4.10), where =0.2 is the smallest studied. The stochastic web appears very thin, and while present, the

4.8. POINCARE SECTIONS 57

5 0 5

3

2

1

0

1

2

3

x

p

Figure 4.10: Poincare section of the delta-kicked harmonic oscillator when = 0.2.

5 0 5

3

2

1

0

1

2

3

x

p



5 0 5

3

2

1

0

1

2

3

x

p

Figure 4.12: Poincare section of the delta-kicked harmonic oscillator when = 1.

5 0 5

3

2

1

0

1

2

3

x

p



5 0 5

3

2

1

0

1

2

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5 0 5

3

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1

0

1

2

3

x

p



5 0 5

3

2

1

0

1

2

3

x

p

Figure 4.18: Poincare section of the delta-kicked harmonic oscillator when = 4; noobvious structure remains.

phase space appears to be dominated by stable dynamics, as signified by the many closedcurves, bound inside cells which are either hexagonal or triangular. One can clearly see thecorrespondence of the contour plot ofH1/q shown in Fig. (4.6) to Fig. (4.10). In Fig. (4.11) wesee that the stochastic web has substantially thickened, and the stable islands within the webare now significantly distorted. In Fig. (4.12) we see bifurcations of the stable islands, and inFig. (4.13) we see how the boundaries between the newly formed sub-islands become filledwith chaotic dynamics. In Fig. (4.16) the central stable cell has bifurcated at the origin (andcorrespondingly so in the next outer ring of cells). In Fig. (4.17) these cells have drasticallyshrunk, and in Fig. (4.18) global chaos reigns.

There is an important difference between the central stable island and the surroundingrings of islands. The closed curves observed inside the central cell describe stable dynamicswhich really are confined within that cell. In the case of the next ring of corresponding stablecells, the closed curves describe stable dynamics which are confined within that ring of cells;point particles hop anticlockwise from one cell in that ring to the next. For this reason wesay that the stable orbits in the central cell are based around a stable fixed point of orderone, whereas the orbits of the next outer ring are based around a stable fixed point of ordersix. These fixed points make up a periodic orbit.


References

[1] G. M. Zaslavski, M. Yu. Zakharov, R. Z. Sagdeev, D. A. Usikov, and A. A. Chernikov,JETP Lett. 44, 451 (1986); G. M. Zaslavski, M. Yu. Zakharov, R. Z. Sagdeev,D. A. Usikov, and A. A. Chernikov, Sov. Phys. JETP 64, 294 (1986); A. A. Chernikov,R. Z. Sagdeev, D. A. Usikov, M. Yu. Zakharov, and G. M. Zaslavsky, Nature 326, 559(1987); V. V. Afanasiev, A. A. Chernikov, R. Z. Sagdeev, and G. M. Zaslavsky, Phys.Lett. A 144, 229 (1990).

[2] A. A. Chernikov, R. Z. Sagdeev, and G. M. Zaslavsky, Physica D 33, 65 (1988).

[3] A. A. Chernikov, R. Z. Sagdeev, D. A. Usikov, and G. M. Zaslavsky, Computers Math.Applic. 17, 17 (1989).

[4] G. M. Zaslavsky, Physics of Chaos in Hamiltonian Systems (Imperial College Press,London 1998).

[5] L. E. Reichl, The Transition to Chaos In Conservative Classical Systems: QuantumManifestations (Springer-Verlag, New York 1992).

[6] J. C. Robinson, C. Bharucha, F. L. Moore, R. Jahnke, G. A. Georgakis, Q. Niu,M. G. Raizen, and B. Sundaram, Phys. Rev. Lett. 74, 3963 (1995); J. C. Robin-son, C. F. Bharucha, K. W. Madison, F. L. Moore, B. Sundaram, S. R. Wilkinson,and M. G. Raizen, Phys. Rev. Lett. 76, 3304 (1996); B. G. Klappauf, W. H. Oskay,D. A. Steck, and M. G. Raizen Phys. Rev. Lett. 81, 1203 (1998); B. G. Klappauf,W. H. Oskay, D. A. Steck, and M. G. Raizen Phys. Rev. Lett. 81, 4044 (1998).

[7] H. Goldstein, Classical Mechanics (Addison-Wesley, Reading 1980).

[8] L. Fejes Toth, Regular Figures (Pergamon, Oxford 1964).

[9] A. N. Kolmogorov, Dokl. Akad. Nauk. SSSR 98, 527 (1954); V. I. Arnold, Russ. Math.Survey 18, 9, 85 (1963); J. Moser, Nachr. Akad. Wiss. Gottingen II, Math. Phys. Kl. 18,1 (1962); see also V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York 1978).

[10] V. I. Arnold, Sov. Math. Doklady 5, 581 (1964), reprinted in Hamiltonian DynamicalSystems, edited by R. S. MacKay and J. D. Meiss (Adam Kilger, Bristol 1986).

[11] C. W. Curtis and I. Reiner, Theory of Finite Groups and Associative Algebras (Wiley-Interscience, New York 1962).

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