Dynamical Chaos and Non-Equilibrium Statistical Mechanics: From Rigorous Results to Applications in Nano-Systems August, 2006

Pseudo-Chaotic Orbits of Kicked OscillatorsJ. H. Lowenstein, New York University

Outline

1. One-dimensional oscillators driven by impulsive kicks, periodic in x and t Resonant sin(x) kick amplitude Hamiltonian and equations of motion Orbits in phase space Poincaré section Crystalline and quasi-crystalline structure in phase space Chaos in a stochastic web: stretching, folding, and branching Normal and anomalous transport Accelerator modes and sticky orbits Pseudo-chaos: branching without stretching and folding 2. Resonant sawtooth oscillator with quadratic irrational parameter

Models with diffusive, super-diffusive, sub-diffusive, ballistic long-t behavior

3. Resonant sawtooth oscillator with cubic irrational parameter

Infinitely many invariant components; multi-fractal structure; long-t behavior

4. Nonresonant sawtooth oscillator Invariant measure of aperiodic orbits; numerical explorations

Hamiltonian and Equations of Motion

H(x,p) = (p2 + x2 ) + F(x) S d( t- 2prn )

r = rotation number = # kicks per natural period t = 2p (Resonant case: r rational)

12 n

x = = p

p = - = - x - F'(x) S d( t- 2prn )n6

66

6H

Hx

p

Free oscillation for fraction r of a natural period, followed by momentumshift p --> p + Dp, Dp = -F'(x) = "kick amplitude"

F(x) = F(x + 2p)

Initial choice: F(x) = K cos(x), Dp = K sin(x)

Example: K = 0.8, quasi-periodic orbit

Animation

.

.

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Quasi-periodic Phase-Space Orbit and Poincaré Section

-6 -4 -2 2 4

-6

-4

-2

2

4

6

Stroboscopic view in phase space: plot (x,p) just before each kick.

x cos 2pr sin 2pr x p -sin 2pr cos 2pr y + K sin x( (() = )) = y + K sin x

-x( ) Poincare map

r = 1/4K= 0.8

Chaotic Orbit and Poincaré Section

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

-10 -5 5 10

-10

-5

5

10

17 kick periods 425 kick periods

10000 kick periods

Stochastic Web with Fivefold Symmetry

Fixed points of W5 form a planar quasi-crystal.

r = 1/5

K = 0.8

106 itns.

1

2

1'

1"

2"

2'

web map mod 2p

a b c

def

a'

b'

c' d'

e'

f '

a"

b"

c" d"

e"

f"

Local map: ( x , y ) ( y + K sin x, -x ) mod 2p

0

0

0

origin

Folding the 4-Fold Web Map into the Fundamental Cell, [0, 2p)2

Phase Portrait of Local Stochastic Web Map, K=0.8

Phase Portrait near a Saddle Point, K = 0.8

0

0.05

0.1

-0.05

-0.1

-0.153 3.05 3.1 3.15 3.2 3.25

3.12 3.14 3.16 3.18

-0.08

-0.09

-0.1

-0.11

Phase Portrait Zoom, K= 0.8

0 1 2 3 40

1

2

3

4

3.05 3.1 3.15 3.2

3.05

3.1

3.15

3.2

stable

manifold

unstable manifold

Chaos in the Stochastic Web

Chaos: separation of nearby initial points increases on avg. as eLt, L= Lyapunov exponent

Mechanism for chaos in the stochastic web: stretching and folding near saddle point branching near saddle point (right-left choice at each saddle leads to "random walk" on square lattice of saddle points)

Normal and Anomalous Diffusion

Web orbit, K = 0.1

For typical values of K, the chaotic orbits of W4 in the stochastic web proceed to infinity with mean-square distance from the initial pointsatisfying

< x2 + y2 > ~ D t for t 8

D = diffusion constant (quasi-linear theory: D=K2/2)

More general power law behavior:

< x2 + y2 > ~ D' tm for t 8

diffusion: m = 1super-diffusion: m > 1sub-diffusion: m < 1

Pseudo-Chaos

Suppose an orbit spends large amounts of time in the vicinity of the boundaryof the stochastic web, for example near an island of stability, where stretching/folding is absent and the effective Lyapunov exponent is near zero. Suchmotion can generate complex geometric structures (fractals) and anomalouslong-time behavior. This is pseudo-chaos.

Zaslavsky, Nyazov, Edelman : pseudo-chaos in 4-fold web map for special kick parameters. Island-around-island self-similarity.

G. M. Zaslavsky and B. A. Niyazov, Physics Reports 283 (1997) 73-93G. M. Zaslavsky, M. Edelman, and B. A. Niyazov, Chaos 7 (1997) 159-181

Dana : pseudo-chaos in 3-fold web map for special kick parameters.super-diffusion associated with accelerator modes.

I. Dana, Phys. Rev. E 69, 016212 (2004)

J.H.L., Poggiaspalla, Vivaldi : kicked oscillator with sawtooth kickamplitude. Zero Lyapunov exponent. Branching withoutstretching. All aperiodic orbits are "sticky".

J. H. Lowenstein, G. Poggiaspalla, and F. Vivaldi, Dynamical Systems 20 (2005) 413-451

G. M. Zaslrusky, B.A. Niyuxx I Physics Rrporis 283 (1997) 73-93

(b) -moo U OCOO

Fig. 1. Typical trajectory with long flights for the web-map of the four-fold symmetry after 2 x 10h iterations (K = 6.35):

(a) full trajectory; (b) magnification of the central part of the trajectory.

exists for almost all values of K although a time needed to observe the asymptotics may be too large (3.3).

For the value

Kan = 6.349972, (3.4)

78 G. M. Zusluvsky, B.A. Niyaxw I Physics Reports 283 (1997) 73-93

-0.25

-1.00

LOG10 @/Dql 1

0 10 20 30 40

K

Fig. 2. Diffusion coefficient 9 normalized to the quasilinear theory coefficient Qsl= K2/2 for different values of K,

obtained after averaging over 2500 trajectories for each value of K with 0.5 x lo5 iterations for each trajectory.

it is found that the diffusion is anomalous with the exponent

p= 1.26, (3.5)

whereas for K,, = 6.25 we get the normal diffusion with p = 1. The accuracy in both cases is a few percents. Let us look at Fig. 3 where a set of trajectories is plotted on the torus for (u, u), mod 2x, for the anomalous case (3.4). The distribution of points for the map (2.5) is highly uniform except for the four small domains which belong to the accelerator modes (2.11). It seems, at first glance, that the contribution of the islands should be small and comparable with their size, but it is not the case and the explanation of the influence of islands is given in the next section.

4. Self-similar stickiness

How the islands affect the transport process can be qualitatively described in the following way: Let us first consider the case of uniform distribution in the phase space, similar to the Sinai billiard system, as an example. Let us fix a domain of a finite size and fairly good boundaries. We assume the existence of a characteristic time that a particle needs to escape the domain. Typically, for the Sinai’s billiard this time is proportional to the domain’s volume and it does not depend on where the domain is taken. Such kind of a “uniformity” does not exist for systems with islands. Consider a system with an island in the phase space, and surround the island by a ring of such a width so as to include a set of islands smaller than the central one. The ring forms a boundary layer, from which

G.M. Zuslnvsky, B.A. Niyazovl Physics Reports 283 (1997) 73-93 81

V

(a) 4.4

P

1.2

(b) I,

, x 5

Fig. 5. Magnification of islands in Fig. 3: (a) boundary islands chain (BIC) of the 1st generation; (b) one trajectory of 2 x lo6 iterations displays the 1st generation BIC; (c) magnification of an island from (b); (d) magnification of an island from (c).

Let &,, be a value of K which corresponds to the ml-BIC of the e-order accelerator mode. One can say that the number ml of the BIC exists if

C(K,k): zn115B•@zn1G~zn!#, ~4!

where B is the matrix (K11,1;K,1) and G(z)5kg(x)3(0,1). For 24,K,0, one can easily verify thatB5Q21AQ, whereA5(cosa, sina;2sina, cosa) with

2 cosa5K12 ~5!

and Q5(k,2k/2;0,1/2) with k[ tan(a/2). Thus, the com-position Q+C(K,k)+Q21 is precisely a web map~3! with w5(u,v)5Q•z and F(w)5Q•G(Q21

•w). Explicitly, F(w)5(k/2)g(v1u/k)(2k, 1). We denote by C5@2p,p!3~2`,`! the cylindrical phase space forFs

(K,k) and defineCC

(K,k) as the map~4! with xn11 ~in first equation! takenmodulo S. Clearly, if we restrict ourselves to initial condi-tions z0PC, CC

(K,k)5Fs(K,k) . The image ofC underQ is an

oblique cylinderCa , i.e., the strip bounded by the linesv56p2u/k. We define the web map ‘‘moduloCa’’ as

Fw5Fw mod Ca : wn115A•@wn1F~wn!#2mw, ~6!

wherew5Q•(2p,0)5(2pk,0) andm is the unique integersuch thatwn11P Ca . The following exact relation thenholds for all orbits with initial conditionsz0PC ~or w0PCa):

Fs(K,k)5CC

(K,k)5Q21+Fw+Q. ~7!

For integerK523,22,21 @corresponding, by relation~5!, to q[2p/a53, 4, 6, respectively#, it is easy to showthat the orbit structure ofC(K,k) is periodic with unit cellT25@2p,p)2. This implies a similar periodicity forFw

5Q+C(K,k)+Q21 with unit cell Ta25Q•T2. In fact, Fw has

crystalline symmetry~triangular, square, hexagonal forq53, 4, 6, respectively! @8,9#. One can then expect the exis-tence of an extended chaotic orbit having this symmetry andforming a ‘‘stochastic web.’’ This web, which has been ob-served for particular mapsFw @8,9#, encircles the torusTa

2 intwo independent directions. This implies global chaos for themap ~6! in Ca and, due to Eq.~7!, also forFs

(K,k) in C.For nonintegerK, the mapsC(K,k) and Fw exhibit no

periodicity and there is no simple relation between the orbitstructures ofFs

(K,k) and Fw . In the k50 case, theoreticalarguments and numerical evidence@6# strongly indicate thatfor irrational q the torusT2 can be partitioned into two re-gions having nonzero area:~a! A connected pseudochaoticregion ~zero Lyapunov exponent!. ~b! An apparently denseset of elliptic islands whose boundaries do not cross or touchthe discontinuity linex52p. Because of the last fact, thepseudochaotic region encirclesT2 in both thex andp direc-tions, implying global pseudochaos inC. This will generallyturn into global weak chaos when a small perturbationkg(x)is applied. Figure 1~b! shows an example fora5p(A521)/2 (K'22.7247) andk50.15.

Accelerator-mode fixed points~AFPs! of map ~1! satisfyp12p052p j andx15x05x12p112p j 8 for integersj Þ0and j 8. One can choosep050 and, forFs

(K,0) ~k50!, we getx052p j /K. Sincex0P(2p,p), one must havej 561 and24,K,22. The latter results remain essentially un-

changed for sufficiently smallk. The AFPs are surroundedby relatively large AIs~see Fig. 1! and a strong superdiffu-sion ~large m! was always observed numerically for24,K,22. In what follows, we shall study in detail the caseof K523 with g(x)5sinx on the basis of relation~7!. Fig-ure 2 shows the stochastic web ofFw (q53) for k50.8. The‘‘cylinder’’ Ca is the oblique strip bounded by the paralleldashed lines. Together with these lines, the two horizontaldashed segments define the unit cellTa

2 in which there ap-pears, up to the transformationQ in Eq. ~7!, the chaoticregion in Fig. 1~a!. The j th unit cell in Ca , j 52`, . . . ,`,contains one ‘‘hexagonal’’ island Hj and two ‘‘triangular’’islands, Lj and Rj . The hyperbolic~x! pointsa, b, d, andelie on the boundaries ofCa while c and f are insideCa . Thepointsa, b, c are equivalent, moduloTa

2 , to d, e, f, respec-tively. It is interesting to see first how the AFPs emergeaccording to~7!. The islands Hj , L j , Rj are invariant underFw

3 and their centers CHj , CLj , CRj are fixed points ofFw3

which are rotated clockwise bya52p/3 underFw . Then,denotingFw by ° and moduloCa by ⇒, it is clear fromFig. 2 that CL21°CL0°CL08⇒CL1°CL18⇒CL2. In gen-

eral, we see thatFw(CHj )5CHj , Fw(CLj )5CLj 11, andFw(CRj )5CRj 21, so that CLj and CRj correspond to theAFPs and Lj and Rj correspond to the AIs.

Extensive numerical observations indicate that for smallka chaotic orbit is always a random sequence of three types ofmotion: ~a! A ‘‘H motion,’’ bounded inp, sticking around theboundary of Hj . ~b! A ‘‘flight’’ in the positive or ~c! negativep direction, accompanied by stickiness to the boundary of Ljor Rj , respectively. Remarkably, a flight was always foundto be a quasiregularsequence of steps. An example of such aflight, interrupted by a H motion, is shown in Fig. 3 fork50.3. Very long, uninterrupted flights~of at least 106 itera-tions! were usually observed fork<0.3. The steps in theflights clearly leave their fingerprints in the strong superdif-fusion of ^pn

2/2&, at least for smalln ~see inset in Fig. 3!.

FIG. 2. Stochastic web of the web mapFw related by Eq.~7! tothe standard map defined in the caption of Fig. 1~a!. See text fordetails.

ITZHACK DANA I. Dana, Phys. Rev. E 69, 016212 ~2004!

016212-2

This steplike structure will now be explained using relation~7!. The notationa(L j ) ~and similar notation for other xpoints and islands! will indicate a web~chaotic! point stick-ing to the Lj boundary very close toa and insideCa ; if thispoint is outsideCa , it will be denoted bya(L j ). For smallk.0, Fw is almost a clockwise rotation bya52p/3 whilethe motion of web points underFw

3 is a slow drift in thedirections of the stable and unstable manifolds, indicated byarrows in Fig. 2. Since the drift velocity vanishes near xpoints, a cycle such ase(L21)° f (L0)°d(L08) can repeatmany times. This cycle will then drift to the cyclec(L21)°a(L0)°b(L08) which can also repeat many times.The cyclese f d andcab give the ‘‘horizontal’’ part of a step,wherep is bounded around its values at the x points~see Fig.4!. This part containsl cycles (l 566 in Fig. 4!, where l isthe largest integer such that all pointsFw

i @e(L21)#, i50, . . . ,3l 21, lie insideCa , so that no moduloCa has tobe taken in the cycles. The ‘‘vertical’’ part of a step is due tothe following process. The lastcab cycle is followed by thecycle c(L21)°a(L0)°b(L08), whereb(L08) is outsideCa .

Taking then the moduloCa , b(L08)⇒e(L1), we see that

a(L0) is mapped intoe(L1) by Fw . Next, Fw@e(L1)#5 f (L2). Thus, sincec is equivalent tof, the cyclecae isactually an ‘‘acceleration’’ cyclef ae with Fw

3 @ f (L21)#5 f (L2). The set $a(L0)% of all points a(L0) which aremapped into pointsb(L08) by Fw is the acceleration spot~AS! in unit cell j 50. As shown by the inset in Fig. 4, theAS touches the linex52p (x850), corresponding to thelower boundary ofCa . The cycle f ae will repeat r times(r 56 in Fig. 4!, wherer is the largest integer such that allpoints Fw@ f (L3i 21)# mod Ta

2 , i 50, . . . ,r 21, lie in theAS. After r f ae cycles, one leaves the AS by crossing theline x52p and arrives toa(L0) which is equivalent tod(L08). Thus, Fw@ f (L3r 21)# is equivalent tod(L08), not to

a(L0). The horizontale f d cycles of the next step then start.This completes the analysis of one step.

Figure 4 shows a strong trapping near six islands in theAS. In fact, by considering a large number of steps in verylong orbits, we found thatr assumes only two values fork50.3: r 56 with probability P'0.95 and r 57 with P'0.05. The value ofl ranges betweenl 553 andl 582. Thisrange should be associated with the continuation of the trap-ping, outside the AS, near the island chain to which the sixAS islands belong. Ask decreases, the steplike structure ofthe flights becomes more pronounced and the high regularityof the vertical parts of the steps, i.e., the values ofr, contin-ues to be observed. Fork50.2, l 5882136 andr 511 (P'0.99) or r 512 (P'0.01). Fork50.1, l 51502259 andr 529 (P'0.93) orr 530 (P'0.07).

In conclusion, our study of theK523 case indicates thatthe global superdiffusion of weak chaos is basically differentin nature from the superdiffusion observed in the usual stan-dard and web maps@3,5#. In the latter systems, the AIs are‘‘tangle’’ islands @4#. These islands born in a strong-chaosregime and are fundamentally different from normal islands,e.g., resonance or web islands, which continue to exist in anintegrable limit. Since a tangle island lies inside the lobe of aturnstile@4#, it causes the acceleration of chaotic orbits stick-ing all around its boundary. On the other hand, relation~7!implies that the AIs forK523 are essentially normal webislands folded back into the cylinder. As a consequence, onecan have a situation that a chaotic orbit sticking to the AIboundary is acceleratedonly at tiny acceleration spots, insharp contrast with the case of tangle islands. The resultingsteplike structure of the chaotic flights is gradually assumedalso by elliptic flights with initial conditions approaching theAI boundary from inside the AI. The basic origin of both the

FIG. 3. Chaotic flight, interrupted by a H motion, for K523,k50.3, andg(x)5sinx. The inset shows ln(^En&) vs ln(n) ~dottedline!, whereEn5pn

2/2, the average & is over an ensemble of 106

initial conditions well localized around (x50, p52p), andnmax

512 000; the linear fit~solid line! has slopem'1.96.

FIG. 4. Magnification of one step in the chaotic flight shown inFig. 3. The horizontal part of the step~dots! consists ofe f d cyclesfollowed by cab cycles ~total of 198 points!. The vertical part~squares! consists of six acceleration cyclesf ae ~18 points!. Theinset shows the acceleration spot$a(L0)% in unit cell j 50 using thevariablesx85(x1p)3105 andp85(p1p)3105.

GLOBAL SUPERDIFFUSION OF WEAK CHAOS PHYSICAL REVIEW E69, 016212 ~2004!

016212-3

Kicked Oscillator with Sawtooth Kick Amplitude

xy

xy

lx - y x

l(x mod 1) - y x

( ) mod 1( )( )

( )( )

K

W

=

=

l = 2 cos(2p r)

r = irrational rotation number

Example: l = 1/2

y

l

0 1 2-1-2-3

( mod 1)l x

Chaotic-Looking Orbit

(x,y) = (0.02522, 0.02522)

Local orbit50,000 itns.

Movie:

Global orbit,5,0000,000 x 4 itns.

Dynamical Chaos and Non-Equilibrium Statistical Mechanics: From Rigorous Results to Applications in Nano-Systems August, 2006

Pseudo-Chaotic Orbits of Kicked OscillatorsJ. H. Lowenstein, New York University

Outline

1. One-dimensional oscillators driven by impulsive kicks, periodic in x and t Resonant sin(x) kick amplitude Hamiltonian and equations of motion Orbits in phase space Poincaré section Crystalline and quasi-crystalline structure in phase space Chaos in a stochastic web: stretching, folding, and branching Normal and anomalous transport Accelerator modes and sticky orbits Pseudo-chaos: branching without stretching and folding 2. Resonant sawtooth oscillator with quadratic irrational parameter

Models with diffusive, super-diffusive, sub-diffusive, ballistic long-t behavior

3. Resonant sawtooth oscillator with cubic irrational parameter

Infinitely many invariant components; multi-fractal structure; long-t behavior

4. Nonresonant sawtooth oscillator Invariant measure of aperiodic orbits; numerical explorations