Stickiness in Hamiltonian systems: from sharply divided to hierarchical phase space

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by  Eduardo G. Altmann

Stickiness in Hamiltonian systems: from sharply divided to hierarchical phase space
Eduardo G. Altmann,1,∗ Adilson E. Motter,2,3 and Holger Kantz1
1Max Planck Institute for the Physics of Complex Systems,
N¨othnitzer Strasse 38, 01187 Dresden, Germany
2CNLS and Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
3Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA
(Dated: February 4, 2008)
WeinvestigatethedynamicsofchaotictrajectoriesinsimpleyetphysicallyimportantHamiltonian
systems with non-hierarchical borders between regular and chaotic regions with positive measures.
We show that the stickiness to the border of the regular regions in systems with such a sharply
divided phase space occurs through one-parameter families of marginally unstable periodic orbits
6 and is characterized by an exponent γ = 2 for the asymptotic power-law decay of the distribution
0 of recurrence times. Generic perturbations lead to systems with hierarchical phase space, where
0 the stickiness is apparently enhanced due to the presence of infinitely many regular islands and
2 Cantori. In this case, we show that the distribution of recurrence times can be composed of a sum

n of exponentials or a sum of power-laws, depending on the relative contribution of the primary and
a secondary structures of the hierarchy. Numerical verification of our main results are provided for
J area-preserving maps, mushroom billiards, and thenewly defined magnetic mushroom billiards.

4
PACSnumbers: 05.45.-a

]
D
I. INTRODUCTION In Hamiltonian systems, the border between a regu-
C
lar and chaotic region often presents a complex hierar-
.
n chicalstructure ofKolmogorov-Arnold-Moser(KAM) is-
Hamiltonian systems usually exhibit divided phase
i landsandCantori. CantoriareinvariantCantorsetsthat
l space,where regularandchaoticregionscoexist. Anim-
n work as partial barriers to the transport close to KAM
portant property of chaotic trajectories in divided phase
[ islands [4, 9]. Although many properties of this struc-
spacesistheintermittentbehaviorwithsporadicallylong
1 periods of time spent near the border of regular regions ture are well understood, their consequences to the dy-
v [1]. Because of this stickiness and the ergodicity of the namics are still a matter of intense study [2, 5, 6, 10].
8 Even in the simplest case of two-dimensional systems, a
chaotic regions, even small islands can have a large ef-
0 number of non-equivalent models have been proposed to
fect on global properties of the system, such as trans-
0
describe the stickiness of chaotic trajectories. Meiss and
1 port [2] and decay of correlations [1]. The stickiness can
Ott introduced a Markov-tree model that accounts for
0 be quantified in terms of the distribution P(T) of re-
thehierarchicalstructureandpredictsascalingexponent
6 currence times T of a typical trajectory to a pre-defined
0 recurrence region, usually taken away from regular is- γ =1.96 [4]. Chirikov and Shepelyansky used renormal-
n/ lands. For fully chaotic hyperbolic systems, the recur- ization arguments at the breakdown of the golden mean
torus to predict a universal exponent γ = 3 [5]. Za-
i rence time distribution (RTD) decays exponentially [3],
l slavsky and co-workersapplied different renormalization
n while for Hamiltonian systems with divided phase space
arguments to the case of self-similar island chains, ob-
: the RTD has been argued to decay approximately as a
v power law P(T) ∼ T−γ′ for large T, where γ′ is a scal- taining simple relations between γ and the scaling prop-
i
X ing exponent [1, 2, 4, 5, 6]. For power-law decay, the erties of these chains [2]. There is also strong evidence
of other stickiness mechanisms in generic Hamiltonian
r cumulative RTD is given by
a systems [11, 12, 13, 14]. The effects described in these
∞ previous works typically coexist and are responsible for
Q(τ)≡ P(T)∼τ−γ, (1) finite-time numerical estimates of γ lying in the inter-
X
T=τ val 1.5 ≤ γ ≤ 2.5 [6]. However, because the conver-
genceinHamiltoniansystemscantakeanarbitrarilylong
′
where γ = γ −1. We say that a system has the prop- time,ingeneralitisnotevenclearwhetherthe RTDap-
erty of stickinessif Q(τ) decays atleast as slowlyas τ−γ proximates a power-law distribution in the asymptotic
for some γ > 0. The existence of a finite mean recur- limit. ThisslowconvergencehasinspiredMotterandco-
rence time implies γ > 1 [2]. Experimental evidence of workers to introduce a new model that accounts for the
stickiness has been observed, for example, in the trans- effectsoftheCantoristructureatfinitetimes[15]. While
port of particles advected by fluid flows [7] and in the the generalasymptotic behaviorremainsunresolved,the
fluctuations of conductance in chaotic cavities [8]. insightprovidedbythestudyofclassesofcomprehensible
Hamiltonian systems is of fundamental importance.
Inthispaper,weinvestigateamechanismforthestick-
∗Electronicaddress: [email protected] iness ofchaotic trajectoriesin Hamiltoniansystems with
2
non-hierarchicalborders between regular and chaotic re- A. Piecewise-linear maps
gions when both regions can have positive measure. Ex-
amples of systems with such a sharply divided phase Consider two-dimensionalarea-preservingmaps of the
space include piecewise-linear area-preserving maps [16, form
17]andmushroombilliards[18,19]. InHamiltoniansys-
tems, it is a commonsense statement to relate the stick- yn+1 =yn+Kf(xn) mod 1, (2)
iness to the presence of hierarchical structures of KAM xn+1 =xn+yn+1 mod 1,
islands and Cantori. While this statement by itself is
where K is a parameter that controls the nonlinearity.
not wrong, here we show that regular islands with non-
For f(x )=sin(2πx ), Eq. (2) corresponds to the stan-
hierarchical borders also stick. Our results are valid for n n
dard map, which has served as a prototype of Hamilto-
both zero- and positive-measure regular islands, and in-
nian system in numerous studies of stickiness in hierar-
clude as particular cases previous findings for systems
chical phase space [1, 2, 5, 6, 24].
not exhibiting KAM islands, such as the Sinai and sta-
However, for f(x ) defined as a piecewise-linear func-
diumbilliards[20,21,22,23]. Wefindthatthestickiness n
tion of the interval x ∈ [0,1], the phase space of map
near non-hierarchicalbordersoccurs due to the presence n
(2) can be sharply divided in the sense that regular and
of one-parameter families of marginally unstable peri-
chaotic regions are separated by a simple curve [25]. As
odic orbits (MUPOs). Based onthe analysisof MUPOs,
shown in Refs. [25, 26, 27], the form and distribution of
we show that the recurrence time does follow a power-
the regular regions in general depend on the function f
law distribution and that the scaling exponent is γ = 2
and on the parameter K. In the case of hierarchical dis-
in two-dimensional sharply divided phase spaces, irre-
tributionofislands,ithasbeenshownthatthestickiness
spective of other details of the system. We also study
ofchaotictrajectoriesleadstoanomalousdiffusioninthe
the properties of generic perturbations of these systems.
extended phase space of these maps [28].
Based on numerical simulations of mushroom billiards
To quantify the stickiness in the case of sharp border,
perturbed by magnetic fields, we observe that the per-
we consider two simple examples with a single regular
turbations generate hierarchical structures of KAM is-
island:
lands and Cantori of the same nature of those observed
(i) The first example is obtained for
in typical Hamiltonian systems. The perturbation of
Hamiltonian systems with sharply divided phase space
f(x )=1−|2x −1|, (3)
n n
thusrepresentsanewroutetoHamiltoniansystemswith
hierarchical phase space. Previously considered routes andwascalledcontinuoussawtoothmapinRef.[17]. The
start either from fully integrable or fully chaotic config- phasespaceofthismapisshowninFig.1(a)forK =3/2.
urations. The onset of hierarchicalstructures introduces ItwasarguedinRef.[17]thatinthiscaseasingleregular
oscillations in the RTD, which we show to be related to island exists [Fig. 1(a), triangular region]. As we show,
the relative contribution of the primary and secondary the absence of other islands and the ergodicity of the
structures of the hierarchy. chaotic region do not rule out the possible existence of
zeromeasuresetsofMUPOsinthechaoticregion. Inthis
The paper is organized as follows. In Sec. II, we an-
paper we use the acronym MUPOs to refer to periodic
alyze the stickiness in sharply divided phase spaces. In
orbits in contact with the chaotic component that have
Sec. III, we consider the effect of the hierarchical struc-
zeroLyapunovexponentsandrealeigenvalueswithmod-
tures when a system with sharply divided phase space is
ulus one. In sharply divided phase spaces,we regardthe
perturbed. Discussion and conclusions are presented in
borders of regular islands as families of MUPOs when-
the last section.
ever they are periodic. For example, for the continuous
sawtoothmap, the followingsets andtheir imagescorre-
spondtoone-parameterfamiliesofperiod-threeMUPOs:
{x = 1/6,1/6 ≤ y ≤ 1/3}, {x = 1/3,1/3 ≤ y ≤ 2/3},
and {x = 1/2,1/2 ≤ y ≤ 1}, where the latter is at the
II. SHARPLY DIVIDED PHASE SPACE border of the island. These families of MUPOs corre-
spond to the straight lines in Fig. 1(a).
(ii) A second example of sharply divided phase space is
We study the stickiness of chaotic trajectories in two-
obtained for
dimensional Hamiltonian systems with non-hierarchical
borders between the regions of chaotic and regular mo- −x if 0≤x <1/4,
n n

tion. As examples, we consider piecewise-linear area- f(x )= −1/2+x if 1/4≤x <3/4, (4)
n  n n
preservingmapsandmushroombilliardsinSecs.IIAand 1−x if 3/4≤x ≤1,
n n
IIB, respectively. In contrast to the previously consid- 
ered stadium and Sinai billiards, in these systems both as considered in Ref. [16]. The properties of map (2)-(4)
the chaoticandregularregionsofthephasespacehavea with K = 2 are essentially the same of the continuous
positive measure. The scaling exponent γ =2 is derived sawtooth map with K = 3/2 [Fig. 1(b) vs. Fig. 1(a)].
in Sec. IIC. Inparticular,the phasespaceofmap(2)-(4)hasasingle
3
cases the cumulative RTD is best approximated by a
power law with scaling exponent γ =2.
Wehavefoundsimilarresultsforotherpiecewise-linear
area-preservingmapswithpolygonalislands,whichwere
obtained from the maps considered above for different
choices of the parameter K and from a different map
studied in Ref. [25]. There are cases when the results
describedinthissectionandthetheoryofSec.IICdonot
apply, suchas when the regularislands are elliptical and
the outermost torus is quasi-periodic. Nevertheless, we
observedthat inmany of these casesthe exponentγ =2
also fits the power-law tails of the numerically obtained
RTD.
FIG. 1: (Color Online) (a) Phase-space portrait of the con-
tinuous sawtooth map for K = 3/2. The dots correspond
to 104 iterations of a chaotic trajectory and the blank re-
gion corresponds to the regular island. The straight lines
representthreedifferentfamilies ofperiod-threeMUPOs(see
text), and the different symbols correspond to specific MU-
POs in one of these families. (b) Phase-space portrait of
map (2)-(4) for K = 2. We plot −0.5 ≤ y ≤ 0.5 in (a) and
−0.5 ≤ x,y ≤ 0.5 in (b) for visualization convenience. (c)
From bottom to top, the cumulative RTDs of the maps con-
sidered in (a) and (b) (multiplied by a factor 10 for clarity). FIG.2: Phase-spaceportraitofthecontinuoussawtoothmap
The upper curve is a straight line with slope γ = 2. Inset: for K =3/2 showing theinitial conditions of thetrajectories
′
distanceofachaotictrajectorytothetheborderoftheisland that remain inside the dashed triangle for at least n = 1,
in (a) during an event with recurrence time T =335. 2, 4 and 1000 iterations of the map, respectively. The inner
triangle corresponds to the regular island. We plot 0 ≤ x ≤
1.5 and −1≤x≤1 for visualization convenience.
regularisland. Theonlyrelevantdifferenceisthatinthis
case there is no other family of MUPOs apart from the
border of the island.
In the stickiness of a chaotic trajectory,the trajectory B. Mushroom billiards
approaches a family of MUPOs and follows a nearly pe-
riodic dynamics for a long period of time before leaving Billiards can be used as simple models in the study of
the neighborhood of the MUPOs [inset of Fig. 1(c)]. In Hamiltonian systems. Recently, Bunimovich introduced
Fig. 2, we show the trajectories that stick to the MU- the so called mushroombilliards [19], which are billiards
POs at the border of the regular island and remain in that have a single regular and a single chaotic ergodic
′
the neighborhood of the island after n = 1, 2, 4 and region. Atypicalmushroombilliardisdefinedbyasemi-
1000 iterations of the continuous sawtooth map. In circle (hat) placed on top of a rectangle (foot), as de-
general, the closer to the island the longer it will take picted in Fig. 3(a). The phase space is described by the
for the trajectory to leave. This stickiness is properly normalized position x on the boundary of the billiard
quantified in terms of the RTD for a recurrence region and angle θ ∈[−0.5,0.5]with respect to the normalvec-
taken apart from islands. We have performed numeri- torrightafterthe specularreflection. Theregularregion
cal simulations for two different configurations present- corresponds to the orbits in the hat of the mushroom
ing sharply divided phase space: the continuous saw- thatnevercrossthedashedcircleofradiusr inFig.3(a).
tooth map with K = 3/2 [Fig. 1(a)] and map (2)-(4) Theborderbetweentheregularandchaoticregionofthe
with K = 2 [Fig. 1(b)]. As shown in Fig. 1(c), in both mushroombilliardisthereforenon-hierarchical,asshown
4
in fig 3(b). (a) (b)
0.5
Mushroom billiards have two different classes of MU-
POs,asillustratedinFig.3(a). Oneofthemcorresponds
to orbits bouncing between the parallel walls in the foot MUPO 0.3
) 2θ
ofthemushroom. SimilarMUPOsarealsofoundinmany R 0.1
r
other billiards with parallel walls, such as the Sinai and θ
stadium billiards [20, 21, 22, 23]. The other and more −0.1
MUPO
h
interesting class of MUPOs corresponds to periodic or-
bits in the chaotic region that never leave the hat of the −0.3
mushroom. Inapreviousstudy[18],wehaveshownthat
x=0 −0.5
there is usually a complex distribution of these MUPOs 0.3 0.4 0.5 0.6 0.7
(c) x
closetotheregularregionandincontactwiththechaotic 100
component. The border of the regular island can be re- 10−1
garded as MUPOs of this class. We have found similar
10−2
MUPOs in other billiards with circular component, such
10−3
asAanrneluelvaarnbtiplloiainrdtsco[2n9c]e.rningthestickinessinmushroom (τ)Q1100−−54 101−010 γ=2
bRiTllDiaridssaigsatihnaγt t=he2s,craelginagrdelxespsonoefnttheofcothnetroculmpaurlaatmivee- 10−6 10−2 T=300
ter 0 < r/R ≤ 1, as shown in Fig. 3(c). In this case, 10−7 10−3
the whole foot of the mushroom is taken as the recur- 10−8 0 200 40n0 600 800
rence region in order to avoid the trivial parallel wall 100 101 102 103 104
τ
MUPOs. The injection and escape mechanism of the
chaotic trajectories near the island are slightly different
fromthoseobservedin the continuoussawtoothmapbe- FIG. 3: (a) Typical mushroom billiard, defined by the geo-
cause in mushroom billiards there are escaping regions metricparameters(r,R,h). TwoMUPOsareshownindotted
tangent to the island and the injection and escape oc- lines. (b)Phase-space representation of thesemi-circular hat
of the mushroom billiard with r/R = 0.5. (c) From bot-
cur in a single iteration [inset of Fig. 1(c) vs. inset of
tom to top, RTDs for r/R = 0.6, 0.75, and 0.5 (multiplied
Fig. 3(c)] [18]. However,from a more fundamental point
by a factor 2 for clarity). The upper curve is a straight line
of view, the stickiness is remarkablysimilar in these sys-
with slope γ = 2. Inset: distance of a chaotic trajectory
tems because in both cases the stickiness is mediated by
to the border of the island during an event with recurrence
MUPOs and the RTD has an exponent γ = 2. Alto-
time T =300.
gether,this suggeststhe possible existence of a universal
scenario for the stickiness of chaotic trajectories in sys-
tems with dividedphase space,asconsideredin the next a one-parameter family of MUPOs of period q. For
section. concreteness, we assume that the family of MUPOs is
{x ≤x≤x , θ =θ }. The phase space of map M(x,θ)
i f 0
issketchedinFig.4(a)andapossibleconfigurationspace
in Fig. 4(b). The following analysis does not depend on
C. Scaling exponent: Theory
whetherthe MUPOsareinthe chaoticseaoratthe bor-
der of a regular island.
We now derive the scaling exponent γ = 2 for the cu-
Consider small perturbations of a MUPO (x ,θ ):
0 0
mulative RTD of two-dimensional systems with sharply
′ ′ ′
divided phase space. Our theory applies to the class of (i) If (x,θ )=(x0+ǫx,θ0) and xi ≤x ≤xf, another
systems presenting one-dimensional families of MUPOs. periodic orbit of the set of MUPOs is obtained. In
This includes as particular cases some systems without this case Mq(x′,θ0)=(x′,θ0), that shows that the
regular islands, such as the stadium and Sinai billiards, perturbation neither grows nor shrinks.
whosecumulativeRTDs areknowntobe governedbyan ′ ′
(ii) If (x,θ ) = (x ,θ + ε), the perturbation in the
0 0
exponent γ = 2 [22, 23]. Most importantly, our results
θ direction does not grow. On the other hand, in
also apply to systems with mixed phase space, such as
the x direction the trajectory is not strictly peri-
thoseconsideredinthe previoussections. The theoryre-
odic anymore and there is a displacement δx every
mainsvalidwhentheorbitatborderoftheregularregion period q: Mq(x ,θ′)=(x +δx,θ′).
0 0
is quasi-periodicandthe first-escaperegionis tangentto
the border. That is, if there are trajectories arbitrarily Both effects (i) and (ii) have to be taken into account
close to the island that move away in one or few time when a generic perturbation is considered:
steps, such as in the mushroom billiards [18]. Mq(x′,θ′)≡Mq(x +ǫ ,θ +ε)=(x′+δx,θ′). (5)
0 x 0
The essential features of the systems considered in
the previous sections are captured by an area-preserving After q iterations, the same arguments used above
′ ′ ′ ′
map M(x,θ) defined on the torus and that contains for (x,θ ) apply to (x +δx,θ ). We thus see that the
5
′
perturbedtrajectoryfollowsthedynamics(5),remaining as a constant. From Eqs. (7) and (8), we obtain γ =2
tr
ataconstantdistanceεfromthefamilyofMUPOs,until for the power-law exponent of the distribution of escape
it travels ∆x = x −x reaching the end x = x of the times, or γ = 1 for the cumulative distribution. This
f 0 f tr
family of MUPOs (see Fig. 4) [30]. We note that Eq.(5) description is not valid when recurrences are calculated,
impliesalineargrowthofthe perturbationintime,what because the initial conditions are chosen inside the re-
is consistent with the marginal instability of the fixed currenceregionandthus away fromthe MUPOs. Inthis
point that forbids exponential growth of perturbations. case, the convergence of p(ε) to the invariant measure is
The displacement δx is related to the difference be- much slower (algebraically for the stadium billiard [23])
tween the frequency of the perturbed and unperturbed and for any finite time p(ε)→0 for ε→0. However, we
orbits, and can therefore be approximated linearly as show in the Appendix that the scaling exponent for this
second case can be derived from the first: the power-law
δx=Dε, (6) exponentincreasesby +1whenthe initialconditions are
taken away from the MUPOs, i.e., γ =γ +1 (see also
tr
inthe limit ofsmallε. Forthe continuoussawtoothmap Refs. [5, 23, 31, 32]). In systems with a one-parameter
with K = 3/2, one obtains D = 6. In the case of bil- family of MUPOs, this leads to the asymptotic expo-
liards with parallel walls D =2l, where l is the distance nent γ = 2 for the cumulative RTD, in agreement with
between the walls. For MUPOs in circular like billiards, our numerical results.
suchasmushroomandannularbilliards,D =2qR,where Since every family of MUPOs contributes with the
R is the radius of the circle. same exponent γ = 2 asymptotically, the exponent does
not depend on the possible presence of other families of
(a) (b) MUPOs in addition to the one at the border of the reg-
θ=θ+ε’0 (x0,θ’) Mqδ(xx0,θ’) MUPO l(x0,θ0=0) x0 uiisnlaotrhbesiselcraovnneddtsin.inuoImnudsuessehadwr,otooamoltahbrigmleliaanprudamsnb[d1e8nr]o,onafeosimnthatelhrlenfacumamsieblieeorsf
θ ε ∆x δx map(2)-(4),andthescalingexponentisthesameineach
of these cases.
) 2θ’=2ε ∆x
chaos It would be interesting to verify the generality of the
θ0
(x,θ) exponent γ = 2 [33], for example, by investigating other
0 0 MUPOs of period q
x systemspresentingsharplydividedphasespace[34]. The
regular island f
x x stickinessthroughMUPOsdescribedaboveresemblesthe
0 x f
mechanismunderlyingstickinessandanomalousdiffusion
FIG. 4: (a) Illustration of the dynamics of a perturbed in one-dimensional maps with marginally unstable fixed
MUPO (x0,θ0 +ε) in the phase space (see text). (b) The points [35]. Here we have considered two-dimensional
corresponding dynamics in the configuration space of a bil- systems and we believe that similar results hold true in
liard with parallel walls (q=2). higher dimensional Hamiltonian systems.
Thetime aperturbedtrajectorytakestoreachx and
f
escape from the dynamics (5) is given by III. HIERARCHICAL PHASE SPACE
∆x 1
T = ∼ (7) Nowweconsiderperturbationstosystemswithsharply
δx ε divided phase space. This leads us to the problem of
stickinessinHamiltoniansystemswiththeusualcomplex
for small ε. In what follows, we see that this time is
hierarchy of infinitely many KAM islands and Cantori.
equivalent to the recurrence time if the initial condi-
As a model system we consider the mushroom billiard
tions are chosen properly. Relation (7) shows that the
perturbed by a magnetic field, a system that we refer to
smaller the perturbation the longer the time the trajec-
as the magnetic mushroom billiard and that allows for a
tory takes to escape. The asymptotic distribution of es-
directcomparisonbetweenthe effectsofhierarchicaland
cape times P(T) as a function of the distribution of per-
non-hierarchicalborders.
turbations p(ε) is given by
p(ε)
P(T)= ∼p(ε)ε2, withε∼1/T. (8) A. Perturbation of non-hierarchical borders
|dT/dε|
Thedistributionp(ε)depends onthe choiceofthe initial Acommonfeatureofthesystemsconsideredinthepre-
conditions. vioussectionsis that theirdynamics is piecewisesmooth
For instance, choosing the initial conditions in the andpresentsabruptchanges. Theseabruptchanges,gen-
neighborhood of the family of MUPOs leads to a rapid erated by non-smooth functions f in map (2) and sharp
convergence of p(ε) to the invariant measure of the sys- corners in the mushroombilliard, are responsible for the
tem. In this case, p(ε) can be asymptotically regarded creation of sharply divided phase spaces. Generic per-
6
turbationsofthesesystemsareexpectedtosmoothdown billiards have both integrable and chaotic regions in the
the dynamics and introduce hierarchies of KAM islands phase space and both effects are expected to take place.
and Cantori. Examples of such perturbations include to Moreinterestingly,mushroombilliardsalsohaveMUPOs
smoothen functions (3) or (4) in the case of piecewise- that are expected to undergo a transformationwhen the
linearmapsandsoftenthewallsinthecaseofmushroom systemis perturbed. Indeed, because the eigenvaluesas-
billiards. In the case of a billiard with chargedparticles, sociated to these orbits are real and have modulus 1,
we can also perturb the system with a magnetic field, as arbitrarily small perturbations are expected to generate
studied below. elliptic or saddle points in the neighborhood of the reg-
Consider the mushroom billiard studied in Sec. IIB ular island of the unperturbed billiard. These effects of
subject to uniform transverse magnetic field B and con- the magneticfieldinthe mushroombilliardareshownin
sider the dynamics of charged particles within this bil- Fig.5, wherea representativemagnificationofthe phase
liard. Due to the Lorentz force, the charged particles space at the border of chaos is shown for different val-
move on circular orbits. We choose the charge of the ues of L. The hierarchyof KAMislands and Cantoriare
particles and orientation of the magnetic field such that clearlyvisible, providingevidence that the complete pic-
the trajectories are oriented counter-clockwise and have tureofHamiltonianchaosisobtainedinmagneticmush-
radius room billiards.
1
L∝ , (9) 10−1
B
10−2
whichisusedasacontrolparameter. Thisparameterhas
to be compared with the geometric scales of the billiard 10−3 L=50 γ=1
definedinFig.3(a)(inoursimulationsweuseR=2and
10−4
r =1). The unperturbedmushroombilliardcorresponds
to L=∞. 10−5
τQ() 10−6 γ=2 L=100
L=∞
10−7
10−8
10−9
10−10
101 102 103 104 105 106
τ
FIG.6: CumulativeRTDsforthemagneticmushroombilliard
with r/R = 0.5 and different values of the magnetic field.
From bottom to top the lines represent: a power law with
γ =2,thenumericalresultsforL=∞(shifteddownwardby
two decades for clarity), L = 100 (shifted downward by one
decade) and L=50, and a power law with γ =1.
The emergence of complex structures of KAM islands
in the phase space influences the stickiness, as shown in
Fig.6forRTDofmagneticmushroombilliardswithL=
100 and L = 50. Comparing these distributions with
those of the unperturbed system (L = ∞), we note the
presence of fluctuations around a slower power-law ten-
dency (γ < 2). This result indicates that, as intuitively
FIG.5: Magnificationofthephase-spaceportraitofthemag- expected,ahierarchicalbordersticksthetrajectoriesina
netic mushroom billiard at the border between the chaotic moreeffectivewaythananon-hierarchicalborder. While
and regular regions for r/R = 0.5 and various values of the the presence of a single family of MUPOs in a hierarchi-
magnetic field.
cal phase space would be enough to guarantee γ ≤ 2,
we observe that, generically, all the families of MUPOs
Previous works on magnetic billiards [36] have shown disappear. One could expect that the outermost torus
that the curvature of the trajectories often leads to the of a regular island, which is marginally unstable, could
creation of KAM tori [37, 38] in fully chaotic systems play the role of the MUPOs described in Sec. II. How-
andchaoticregions[39]inintegrablesystems. Mushroom ever, there is usually an infinite number of Cantori that
7
accumulatenearthe islandinvalidating relations(5) and
(6) and thus the derivation of the exponent γ = 2. In
the next section we study carefully the effect of such a
hierarchical border on the stickiness.
B. Hierarchical phase-space scenarios
We now investigatethe originofthe oscillationsinthe
RTDs shown in Fig. 6.
We focus initially on the parameter L = 50. For this
parameter, many KAM tori are destroyed but the chain
ofislandsandCantoriarestillclearlyvisibleinthephase
space,asshowninFig.7. The differentdensity ofpoints
seen in Fig. 7(a) is related to the presence of chains of
islands and Cantori acting as partial barriers [9] to the
transport in the θ direction. In order to associate the
presenceofthesebarrierstotheRTD,westudythemin-
imum distance between the trajectory and the main is-
landbeforethetrajectoryleavestheneighborhoodofthe
island and visits the recurrence region. In our simula-
tions we use the minimum collision angle θ of the tra-
jectory as a measure ofthe distance because the barriers
mimic the originaltoriand haveapproximatelyconstant
θ, and we take the foot of the mushroom billiard as the
recurrence region. The fraction of events that have a
minimum angle θ is defined as g(θ)dθ = ηθ/η, where ηθ FIG.7: AnalysisofthemagneticmushroombilliardwithL=
is the number of recurrencesthat havea minimum angle 50: (a) phase-space magnification at theborder of chaos; (b)
in the interval [θ,θ +dθ] and η is the total number of fractiong(θ)ofrecurrencesthathaveθastheirminimalangle;
recurrences. Numerical results for g(θ) with L = 50 are (c) the RTD of all trajectories (upper solid curve) and the
shown in Fig. 7(b). The function g(θ) goes to zero at RTDsof the trajectories in regions (1)-(5) of (b) (lower solid
the angles that correspond to the position of the barri- curves). Thelower curvesin(c) aredividedby10for clarity.
ers because the trajectories that manage to pass a bar-
rier quickly spread throughout the next chaotic region.
From the behavior of g(θ) in Fig. 7(b), we can resolve 5 chain of coupled hyperbolic systems, where each hyper-
different regions limited by these barriers. To associate bolic system models the area of the phase space limited
these regions with the RTD, we label all the recurrence bysuccessiveCantori. Oneofthestrengthsofthismodel
events from (1) to (5) according to the number of re- isthatitpredictsnotonlytheasymptoticbehaviorofthe
gions the trajectory penetrates before returning to the non-hyperbolic dynamics around KAM islands but also
recurrence region. The RTD of each of these groups of the finite-time dynamics assessable in numerical simula-
recurrenceeventsareshowninFig.7(c). TheRTDofall tions and experiments. The model predicts that the sur-
the events correspondsto the sum of these partialRTDs vivalprobabilityofparticlesintheneighborhoodofKAM
andisshowninthe same figure(upper solidcurve). The islands fluctuates around a power law and is composed
partial RTD of each region (1)-(5) presents a relatively of a sum of exponentials associated to the Cantori. Our
peaked maximum followed by an exponential decay. Ac- resultsinFig.7showthatthisbehaviorisindeedpresent
cordingly, most of the orbits that have the same recur- in real Hamiltonian systems. As shown below, this pic-
rencetimeT penetratethesamenumberofbarriers[note ture changes when secondary structures of the hierarchy
the logarithmicscaleinFig.7(c)]. Theseresultsindicate are relevant. This more generalstickiness scenario is ob-
that, for T <106, the stickiness is dominatedby the pri- served in the mushroom billiard for larger values of the
mary chain of barriers around the main regular island, magnetic field (e.g., L=10).
that is, the contribution of barriers associated to sec- In Fig. 8, we show the same as in Fig. 7 for the pa-
ondaryislandsisnegligible. Theseresultsalsoshowthat rameter L = 10. The effect of the primary barriers is
the oscillations observed in the RTD around the power- still important, as shown in Fig. 8(b) where these bar-
law behavior are intrinsically associated to the presence riers correspond to zeros of g(θ). However, as shown
of the barriers in the phase space. in Fig. 8(c), the partial RTDs corresponding to regions
These stickiness properties agree well with the predic- (1)-(5) exhibit a power-law rather than an exponential
tionsofthemodelproposedbyMotteret al. inRef.[15]. decay. For instance, the RTD of trajectories belonging
In that paper, the hierarchy of Cantori is modeled by a to region (2) exhibits an approximate power-law decay
8
IV. CONCLUSIONS
Wehavestudiedthestickinessofchaotictrajectoriesin
Hamiltonian systems with sharply divided phase space,
which are characterized by non-hierarchical borders be-
tween the regions of chaotic and regular motion. The
stickinessoccursthroughthe approachtoone-parameter
families of MUPOs in contact with the chaotic region.
The main characteristics of this stickiness scenario are
theexponentγ =2forthepower-lawdecayofthecumu-
lative RTD and the long intervals of regular motion at a
constant distance from families of MUPOs. Dynamical
systemsdescribedbythisscenarioincludemushroombil-
liardsandvariouspiecewise-lineararea-preservingmaps.
Genericperturbationsappliedtosystemswithsharply
divided phase space destroy the MUPOs and introduce
hierarchies of regular islands and Cantori. We believe
that these perturbations can serveas a new paradigmto
the study of stickiness in generic Hamiltonian systems.
Using as an example mushroom billiards perturbed by
a transverse magnetic field, we characterize two differ-
ent scenarios of stickiness in the presence of perturba-
tions. For small perturbations, the stickiness is domi-
nated by the primary chain of Cantori, which work as
FIG. 8: (Color Online) Analysis of the magnetic mushroom partialbarrierstothetransportaroundthemainregular
billiard with L=10. (a) Phase-spacemagnification with two island. In this case, the RTD is composed of a sum of
typical sticking trajectories with recurrence time T ≈ 8 104:
exponentialdistributionsassociatedtothe probabilityof
trajectory1sticksneartheupperislandandtrajectory2fills
crossing each of these barriers. For increasing perturba-
thechaoticregion. (b)Fractiong(θ)ofrecurrencesthathaveθ
tions, the primary barriers weaken while the secondary
astheirminimalangle. (c)TheRTDofalltrajectories(upper
islands and the correspondingsticking regionsgrow. For
solid curve) and the RTDs of the trajectories in regions (1)-
(5)of(b)(seelegend). Thelowercurvesin(c)aredividedby largeperturbations,thestickinessofthesecondarystruc-
10 for clarity. tures becomes relevant and the exponential components
ofthe RTD areconvertedthemselvesinto power-lawdis-
tributions. This providesdirectevidence ofthe effects of
Cantori structures at finite times, in strong support of
themodelintroducedinRef.[15]anditsgeneralizations.
thatmakestheserecurrenceeventsdominantnotonlyfor
small times (10<T <500) but also for very large times TheasymptoticbehavioroftheRTD,whichhasbeena
(T ≈105). Ontheotherhand,theRTDofeventsassoci- matterofconsiderablerecentdebate [5,6],cannotbe re-
ated to region(4) does not dominate the (total) RTD at solved alone by numerical experiments. Our simulations
any time. The slower decay of the RTD of region (2) is suggestthatthehierarchicalstructuresenhancethestick-
a consequenceof the stickiness to the chainof secondary iness of non-hierarchical borders, what would lead to an
islands shown at the top of Fig. 8(a). In this figure, upper bound 2 for the scaling exponent γ. This upper
we show two representative trajectories with recurrence bound is guaranteed when the phase space has one or
time T ≈ 8104. The first (trajectory 1) penetrates only more families of the MUPOs described in Sec. II. How-
tworegionsandsticksto asecondaryisland. The second ever, in general, this numerical evidence of upper bound
(trajectory2)penetrates fiveregionsandapproachesthe shouldbe takenwithcautionbecauseonecannotneglect
mainisland. Inthe contextofstochasticmodels [24,40], thepossibilitythatthehierarchicalstructureswillreduce
asymptotic effects ofsecondaryislands canbe accounted the stickinessfor asymptoticallylargetimes. For general
for by the Markov-tree models [4]. In a deterministic Hamiltonian systems, even the question of whether the
framework,thefullhierarchyofislandscanbeaccommo- oscillationsintheRTDvanishasymptotically,givingrise
datedwithinachainmodelofnon-hyperbolicsystemsor toawelldefinedpower-lawexponent,isaproblemyetto
atreemodelofhyperbolicsystems,whicharestraightfor- be settled. Our results provide an answer to this ques-
ward generalizations of the model introduced by Motter tion for an important class of Hamiltonian systems with
et al. [15]. sharply divided phase space.
9
Acknowledgments putes the time T the trajectory takes to return to that
region. If the trajectoryis followedfor a long time t, the
The authors thank G. Cristadoro, S. Denysov, and R. cumulative RTD is
Klages for helpful discussions. E. G. A. was supported
N
τ
by CAPES (Brazil) and DAAD (Germany). A. E. M. Q(τ)= , (11)
N
was supported by the U.S. Department of Energy under
Contract No. W-7405-ENG-36. where N is the number of recurrences with time T ≥τ
τ
and N is the total number of recurrences observed in
time t. The relation between the times in Eq. (10) and
Appendix the number of recurrences in Eq. (11) is given by [5]
Considerauniformdistributionofinitialconditionsin t ∼N hTi, (12)
the neighborhoodofa stickingregionofthe phasespace, t ∼N τ, (13)
τ τ
as usually studied in problems of transient chaos, and
consider the time it takes for the corresponding trajec- where hTi is the average recurrence time. Altogether,
tories to escape to a pre-defined region away from the this leads to
sticking region. The distribution S(τ) of escape times
longer than τ is proportional to the measure µ(τ) of the S(τ)
Q(τ)∼ . (14)
region of the phase space to which the trajectories stick τ
for a time longer than τ. Due to the ergodicity,
In particular, if the escape times follow a power-law dis-
t tribution S(τ) ∼ τ−γtr, then the cumulative RTD is
τ
S(τ)∝µ(τ)= , (10) Q(τ)∼τ−γ where
t
wheretτ isthe totaltimespentinsidethe stickingregion γ =γtr+1. (15)
and t is the total observation time.
On the other hand, in the study of recurrence prob- An equivalent relation was obtained in Ref. [32] for
lems, one usually initializes a single trajectory in a re- chaotic scattering problems, another example where the
currence region away from the sticking region and com- trajectoriesare initialized awayfromthe sticking region.
[1] C.F.F.Karney,PhysicaD8,360(1983);B.V.Chirikov [16] K.-C. Lee, Physica D 35, 186 (1989).
and D. L.Shepelyansky,Physica D 13, 395 (1984). [17] J. Malovrh and T. Prosen, J. Phys. A: Math. Gen., 35,
[2] G. M. Zaslavsky, Physics Reports 371, 461 (2002). 2483 (2002).
[3] E. G. Altmann, E. C. da Silva, and I. L. Caldas, Chaos [18] E. G. Altmann, A. E. Motter, and H. Kantz, Chaos 15,
14, 975 (2004). 033105 (2005).
[4] J.D.MeissandE.Ott,Phys.Rev.Lett.55,2741(1985); [19] L. A. Bunimovich, Chaos 11, 802 (2001); ibid. 13, 903
Physica D 20, 387 (1986). (2003).
[5] B.V.ChirikovandD.L.Shepelyansky,Phys.Rev.Lett. [20] L. A. Bunimovich, Comm. Math. Phys. 65, 295 (1979);
82, 528 (1999). L. A. Bunimovich and Y.Sinai, ibid. 78, 479 (1980).
[6] M.Weiss,L.Hufnagel,andR.Ketzmerick,Phys.Rev.E [21] F. Vivaldi, G. Casati, and I. Guarneri, Phys. Rev. Lett.
67, 046209 (2003); Phys. Rev.Lett. 89, 239401 (2002). 51, 727 (1983); K.-C. Lee, ibid. 60, 1991 (1988).
[7] T. H. Solomon, E. R. Weeks, and H. L. Swinney, Phys. [22] P. Gaspard and J. R. Dorfman, Phys. Rev. E 52, 3525
Rev.Lett. 71, 3975 (1993). (1995).
[8] R. Ketzmerick, Phys. Rev. B 54, 10841 (1996); A. S. [23] D.N.Armstead,B.R.Hunt,andE.Ott,PhysicaD193,
Sachrajda,R.Ketzmerick,C.Gould,Y.Feng,P.J.Kelly, 96 (2004).
A.Delage, and Z.Wasilewski, Phys.Rev.Lett.80, 1948 [24] R. B. White, S. Benkadda, S. Kassibrakis, and G. M.
(1998). Zaslavsky, Chaos 8, 757 (1998).
[9] R.S.Mackay,J. D.Meiss, and I. C. Percival, Physica D [25] M. Wojtkowski, Commun. Math. Phys. 80, 453 (1981).
13, 55 (1984). [26] P. Ashwin, Phys. Lett.A 232, 409 (1997).
[10] P. Grassberger and H. Kantz, Phys. Lett. A 113, 167 [27] R. Adler, B. Kitchens, and C. Tresser, Erg. Theor. &
(1985). Dyn.Syst., 21, 959 (2001).
[11] G. M. Zaslavsky, Physica D 168-169, 292 (2002). [28] J. H. Lowenstein, K. L. Kouptsov, and F. Vivaldi, Non-
[12] V.Rom-KedarandG.M.Zaslavsky,Chaos9,697(1998). linearity17371(2004);J.H.Lowenstein,J.Phys.: Con-
[13] D.delCastilloNegrete,J.M.Greene,andP.J.Morrison, ference Series 7 68 (2005).
Physica D 91, 1 (1996). [29] N. Saitˆo, H. Hirooka, J. Ford, F. Vivaldi, and G.H.
[14] O.Barash andI.Dana,Phys.Rev.E71,036222 (2005). Walker, Physica D 5, 273 (1982).
[15] A. E. Motter, A. P. S. de Moura, C. Grebogi, and [30] If the MUPOs are at the border of a regular island, x
f
H.Kantz, Phys.Rev.E 71, 036215 (2005). represents either the vertex of a polygonal island [see
10
Fig.1(a,b)]or,asinthecaseofcircular-likebilliards,the [35] G.ZumofenandJ.Klafter,Phys.Rev.E47,851(1993).
point of tangency between the first escaping region and [36] M.RobnikandM.V.Berry,J.Phys.A:Math.Gen.18,
theisland (see Ref. [18]). 1361 (1985).
[31] J.D. Meiss, Chaos 7, 139 (1997). [37] W. Breymann, Z. Kov´acs, and T. T´el, Phys. Rev.E 50,
[32] A.S.Pikovsky,J.Phys.A:Math.Gen.25,L477(1992). 1994 (1994).
[33] Anexponentγ =2fortheRTDisalsoobtainedchoosing [38] L. G. G. V.Dias da Silvaand M. A.M. deAguiar, Eur.
the recurrence region inside KAM islands, as shown in: Phys. J. B 16, 719 (2000).
N. Buric, A. Rampioni, G. Turchetti, and S. Vaienti, J. [39] O. Meplan, F. Brut, and C. Gignoux, J. Phys.A:Math.
Phys. A: Math. Gen. 36, L209 (2003); H. Hu, A. Ram- Gen. 26, 237 (1993).
pioni, L. Rossi, G. Turchetti, and S. Vaienti, Chaos 14, [40] J.D.Hanson,J.R.Cary,andJ.D.Meiss,J.Stat.Phys.
160 (2004). 39, 327 (1985).
[34] S.DeBi`evre,P.E.Parris,andA.Silvius,PhysicaD208,
96 (2005).