Poincare recurrences and transient chaos in systems with leaks

1.2 MB·English

by  Eduardo G. Altmann

Poincar´e recurrences and transient chaos in systems with leaks
Eduardo G. Altmann1,2 and Tama´s T´el3
1Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
2Northwestern Institute on Complex Systems, Northwestern University, Evanston, Illinois 60208, USA
3Institute for Theoretical Physics, E¨otv¨os University, P.O. Box 32, H-1518 Budapest, Hungary
In order to simulate observational and experimental situations, we consider a leak in the phase
space of a chaotic dynamical system. We obtain an expression for the escape rate of the survival
probability applying the theory of transient chaos. This expression improves previous estimates
based on the properties of theclosed system and explains dependencies on the position and size of
the leak and on the initial ensemble. With a subtle choice of the initial ensemble, we obtain an
equivalence to the classical problem of Poincar´e recurrences in closed systems, which is treated in
9 thesame framework. Finally, we show how our results apply to weakly chaotic systems and justify
0 asplitoftheinvariantsaddleinhyperbolicandnonhyperboliccomponents,related,respectively,to
0 theintermediate exponential and asymptotic power-law decays of thesurvival probability.
2

n
I. INTRODUCTION ponential decay of both p and p
a r e
J
0 pr,e(t)∼e−γr,et, (1)
Thenumericalandtheoreticalstudy ofdynamicalsys-
2
for large enough t = {T,n}. Exponentials are the sig-
tems can resemble observational and experimental situ-
nature of strong chaotic properties as seen, e.g., in the
] ations if the escape of an initial ensemble of trajecto-
D divergence of nearby initial conditions, in the decay of
ries through a leak placed in the otherwise closed phase
C space is considered. The idea of introducing a leak in correlations, and in the convergence to equilibrium dis-
tributions. The exponent γ is the key quantifier of
. a closed chaotic dynamical system to generate transient r,e
n system specific characteristics and will be considered in
chaos was first suggested by Pianigiani and Yorke [1].
i
l Laterthis problemwasdiscussedindetail inthe context detail in this paper.
n
The possibility of comparing a system with leak to its
[ offractalexitboundaries[2],ofgeometricalacoustics[3],
of quantum chaos [4], of controlling chaos [5], of reset- correspondingclosedsystemshowsnumerousadvantages

2 incomparisontonaturallyopenedsystems. Forinstance,
ting in hydrodynamical flows [6, 7], of leaked Hamilto-
v the rate γ in Eq. (1) can be estimated from the prob-
nian systems [8], of astronomy[9] and of cosmology[10]. r,e
5 ability of escaping or returning. For small leaks, this
The subject has recently received a renewed attention
8
probability is obtained by computing the closed systems
7 [11, 12, 13, 14, 15, 16, 17]. Part of this renewed inter-
naturalmeasureµofatypicalleakregionI,anditfollows
3 est comes from recent developments of quantum chaos,
that [5, 24, 27]
. that started to consider carefully the effect of measure-
8
0 mentdevices,absorption,andotherformsofleakingboth 1
8 theoretically and experimentally [11, 12, 13, 14]. It is by γr =γe =µ(I)= for 1≫µ(I)>0, (2)
hti
0 nowclearthatleakingdynamicalsystemsprovidesaway
v: to ’peeping at chaos’ [16], and can be used as a kind of where hti denotes the mean recurrence or escape times,
i chaotic spectroscopy [4]. hTi or hni.
X
Despite its simplicity, relation (2) is of little practical
From a theoretical point of view, leaking systems pro-
r relevancesincethelimitµ(I)→0isneverachievedinei-
a vide an interesting bridge between open and closed dy-
ther numerical or experimental applications [27, 28, 29].
namical systems. In closed systems the phase space co-
Here we do not restrict ourselves to this limit and find
ordinates remain confined, and a rigorous mathematical
that, apart from being unrealistic, it masks different in-
approach based on attractors and ergodic components
teresting relations. We perform a formal description
can be employed [18]. An important quantity is the dis-
through the theory of transient chaos, i.e., in terms of
tribution p (T) of the first Poincar´e recurrence times T
r
a nonattracting chaoticset [18] andthe conditionally in-
[19, 20, 21, 22, 24, 25] to a preselected region of the
variantmeasure[1,30],andobtaindistinctrelationsforγ
phasespace. AsoriginallyproposedbyChirikovandShe-
and1/htiasa function ofthe positionandsize ofthe re-
pelyansky[25](seealso[23]),p (T)isausefulanalyserof
r
gion I and of the initial ensemble. Relation (2) does
the entiredynamics. Incontrast,inopensystemstrajec-
not hold, but we show that γ = γ ≡ γ remains valid.
toriesmayleavethe regionofinterest(e.g.,tendto±∞) r e
This is based on a full correspondence with the recur-
and the method of transient chaos is usually employed
renceproblemobtainedforaspecialinitialensembleand
[18, 26]. In this case, the dynamics is characterized by
settingtheleak to be equal to the recurrenceregion of the
the escape time distribution p (n) (the derivative of the
e closedsystem. Wehaveexploredthemoststrikingconse-
survival probability).
quences of this unified treatment from the point of view
Chaotic dynamics usually leads to an asymptotic ex- of Poincar´e recurrences in a previous short paper [31].
2
Here, after describing the relationship between the two T ≥1,istheprobabilityoffindingT ≡T inaninfinitely
i
problems with additional details, we focus on the escape long trajectory. The cumulative version can be written
problem and consider the recurrence problem as a par- as P (τ) = ∞ p (T). The mean recurrence time is
r T=τ r
ticular, yet important, case. We also use this connection defined as
P
to apply and adapt a method to determine γ [27] based
N ∞
only on the escape probabilities for short times. This 1
hTi≡ lim T ≡ Tp (T).
is a main advantage over genuine open systems where N→∞N i r
long-term periodic orbits are needed to determine γ . Xi=1 TX=1
e
In systems showing weak chaos, as e.g., Hamiltonian Duetotheergodicityofthemeasureµ,thespecificchoice
systems with mixed phase space, the exponential decay of the initial point ~x ∈ I is irrelevant for p (T). In-
0 r
law (1) experiences a cross-overfor longer times towards stead of following a single trajectory, p (T) can also be
r
anasymptoticpower-lawbehavior[23,24,32]. Weargue, obtained by using an ensemble of trajectories chosen in-
however, that on intermediate times a decay rate γ can side I according to the invariant density ρ . A conse-
µ
be well defined quence of ergodicity is Kac’s lemma [19]:
pr,e(t)∝ te−−αγt ffoorr tin>tertm.ediate t<tc, (3) hTi= 1 , (4)
c µ(I)
(cid:26)
We interpret our results based on an effective splitting which is valid for any recurrence region I.
of the chaotic saddle into hyperbolic and nonhyperbolic The recurrence time distribution of an uncorrelated
components. random process with fixed return probability µ follows
The paperisorganizedasfollows. InSec.II wereview a binomial distribution [27]
the main properties of Poincar´e recurrence times. The
µ
correspondingresultsforescapetimes appearinSec.III, p (T)=µ(1−µ)T−1 = eln(1−µ)T. (5)
r
together with the dependence of γ and hti on I. The 1−µ
equivalence between recurrence and escape problems is
Therelaxationrateγ definedinEq.(1)isthusgivenfor
carefully discussed in Sec. IV. In Sec. V numerical il- r
such a random process by
lustrations of the results are provided for paradigmatic
dissipative models. In Sec. VI, the nonhyperbolic case is
γ∗ =−ln(1−µ). (6)
investigated using an r
area-preserving map. Relations to periodic orbits and For smallrecurrenceprobabilityµ→0, γ∗ →µ, andthe
r
an efficient method to determine γ are described in
Poisson distribution p (T) = µexp(−µT) is obtained.
r
Sec. VII. Finally, conclusions and discussions appear in
Identifying µwith µ(I) (recurrenceprobabilityequals to
Sec. VIII, where the case of higher order recurrences is
the natural measure of the recurrence region), we see
also investigated.
that a Poisson process is the simplest process leading to
Eq. (2). Indeed, the Poisson distribution is proved to
describe deterministic hyperbolic systems(see references
II. POINCARE´ RECURRENCES IN CLOSED
in [41]).
SYSTEMS
We study next the distribution of recurrence times
p (T) for generic chaotic systems. In agreement with
r
Poincar´e recurrences to a certain region of the phase
Eq. (1), we write
space have played an important role since the founda-
tion ofthe kinetic descriptionof noneqilibriumprocesses irregular for 1<T <T∗,
p (T)≈ (7)
[21, 24]. We define the problem in the context of low- r gre−γrT for T ≥T∗.
dimensional chaotic maps that can be dissipative [33] or (cid:26)
Hamiltonian [24, 25]. Implicit is the view that the approach toward the expo-
Consideradiscretetimechaoticsystem~x =M(~x ) nentialdistributionforT →∞doesnotoccuruniformly,
n+1 n
definedonaboundedphasespace~x∈Γ. Letthenatural but that after some short time T∗ the distribution is
ergodicmeasureontheinvariantchaoticset(e.g.,strange practically very close to an exponential [27]. This will
attractoror chaoticsea)be denoted by µ andits density be theoretically justified in the next section and numeri-
by ρ . We define the recurrence region as a subset I ⊂ callyillustratedinSec.V. Weareinterestedinthedecay
µ
Γ with µ(I) > 0. As stated above, we do not restrict rate γ , which typically deviates from the random esti-
r
ourselves to the unrealistic limit µ(I) → 0 [27, 28, 29]. mate(6),andalsointheshorttimeirregularfluctuations
ThePoincar´erecurrencetheoremensuresthatforalmost for T < T∗, where T∗ is of the order of hTi and will be
all initial conditions ~x ∈ I, there are infinitely many better interpreted in Sec. IV. These short time irregular
0
time instants n=n ,n ,... such that Mn(~x )∈I. The oscillations[denotedasirregularinEq.(7)andequations
1 2 0
first recurrencetimes aredefined as T =n −n , with hereafter]dependstronglyonthechoiceoftherecurrence
i i i−1
i ≥ 1 and n ≡ 0 (if the points remains in I at the ith region I. For instance, if a periodic orbit of (short) pe-
0
iterate T = 1). The recurrence time distribution p (T), riod p is present inside I, there is a higher probability of
i r
3
recurrence at T = p. As shown in Ref. [27] and will be [18,26]. Thetime n∗,belowwhichorbitsescapewithout
e
discussed in Sec. VII, normalization and Eq. (4) imply approaching the saddle, can be estimated by the inverse
that the short-time fluctuations uniquely determine the of the negative Lyapunov exponent λ′.
deviation of γ from γ∗. Therefore, for different recur- A rigorous description of open systems is possible in
r r
rence regions I of the same natural measure µ(I) [while terms of the conditionally invariant measure [1, 30, 35,
hTi is fixed and given by (4)], the value of γ depends 36], hereafter called c-measure. A measure µ is said to
r c
strongly on the position of I [27]. be conditionally invariant if
µ (M˜−1(E))
c
III. ESCAPE IN OPEN SYSTEMS µ (M˜−1(A)) =µc(E), (10)
c
A. General relations for transiently chaotic systems for any E ⊂ A. In words: the c-measure is not invari-
ant under the map: µ (M˜−1(E)) 6= µ (E), but is pre-
c c
By now, transient chaos has a well established theory served under the incorporation of the compensation fac-
[18, 26]. In this case, the discrete system ~x =M˜(x ) torinthe denominatorof(10). Thecompensationfactor
n+1 n
is defined in an unbounded phase space Γ˜. A region of µ (M˜−1(A)) < 1 corresponds to the c-measure of the
c
interest A ⊂ Γ˜ (non-trivial dynamics) is defined in such set remaining inside A, i.e., the trajectories that do not
a way that trajectories may leave A [M˜(A)*A] but do escape in one iteration. The c-measure of the region of
not returnto it [M˜(Γ˜\A)∩A≡∅]. An initial ensemble interest is µc(A) = 1. The c-measure has a density ρc,
whichis unique and the only attractorfor smoothinitial
is started according to a smooth density ρ (~x), with ~x∈
0
densities ρ . Taking into accountthe exponential escape
A. The escape time distribution p (n) is given by the 0
e
expressedin (8), one sees that convergenceis only possi-
fraction of trajectories that leave A at time n ≥ 1. For
ble if [1]
chaotic systems it can be written as
p (n)= irregular for 1<n<n∗e, (8) γe =−ln(µc(M˜−1(A))). (11)
e (cid:26)gee−γen for n≥n∗e,
wheren∗ correspondstoashortconvergencetimeandγ Consideringthe evolutionofρ0 throughanevolutionop-
e e erator,thePerron-Frobeniusoperator[20,30],exp(γ )is
is the escape rate. The survival probability inside A up e
totimenisgivenbyP (n)= ∞ p (n′),andisalsoan thelargesteigenvalueandρc isthecorresponding(right)
e n+1 e eigenvector. The c-measure describes the escape process
exponential distribution with the same γ as in Eq. (8).
e
The average escape time, hniP≡ ∞np (n), also called of the system and therefore ρc is concentrated on the
e 1 e saddle and along its unstable manifold [30].
the lifetime ofchaos,is usually estimatedby the recipro-
P Numerically, ρ is obtained multiplying ρ at each it-
cal of γ [18, 26]. Note, however, that the exact average c 0
e
eration by a constant factor which, after many itera-
is, in general, different from 1/γ and depends (just like
e
n∗) on ρ . tion, is necessarily equal to exp(γc). In practice, we it-
e 0
erate a large number of trajectories and we obtain ρ by
The ergodic theory of transient chaos explains the c
renormalizing the surviving trajectories at a long time.
above properties by the existence of a non-attracting
µ (M˜−1(A)) in (11) [or 1−µ (I) in (14) below] is ob-
chaoticset. Ininvertiblesystemsthissetis,e.g.,achaotic c c
tained as the fraction of all surviving trajectories that
saddle[56]: the set of orbits that never escape either for-
escape in the next time step or, equivalently, by fitting
ward or backwardin time [18, 20, 26]. For systems with
γ to p (n) and inverting (11).
strongchaos,thesaddleisaninvariantfractalsetofzero e e
measure. Long-time transients correspond to trajecto-
ries that approach it closely, along its stable manifold,
andleaveitalongitsunstablemanifold. Beinganinvari- B. Systems with leaks
ant object, the chaotic saddle is, of course, independent
of the initial density ρ . Therefore, assuming that the Open systems M˜ canbe obtained fromthe closedsys-
0
distribution ρ is non-vanishing in at least some region temM defined onΓ,describedin Sec.II, by introducing
0
aroundthe stable manifold of the saddle (and that there a leak in a region I ⊂Γ
areno disjointsaddles)the escaperate γ is independent
e
oftheinitialdensity. Itcanberelatedtothegeometrical M(~x ) if ~x ∈/ I
~x =M˜(~x )= n n (12)
and dynamical properties of the saddle by the Kantz- n+1 n escape if x ∈I.
n
(cid:26)
Grassbergerformulas [34], whichfor two-dimensionalin-
vertible maps are Notice that, since the escape happens one step after en-
teringI,themapM˜ isdefinedinI andinitialconditions
γ =λ(1−d ), λd =−λ′d , (9)
e u u s can be in I. This procedure defines an open system as
where λ (λ′) is the positive (negative) Lyapunov expo- that of Sec. IIIA, with A≡Γ.
nentonthe saddle andd (d )isthe informationdimen- An estimate γ∗ of the escape rate γ in systems with
u s e e
sion of the saddle along its unstable (stable) manifold leaks is usually given in terms of the Frobenius-Perron
4
relation [5, 20] which expresses that the number of par- Natural density ρ = ρ : coherent with the idea of
0 µ
ticles not escaping within one time step is proportional systems with leaks, the initial density is chosen accord-
to the natural measure outside of the leak: exp(−γ∗)= ing to the natural measure of the closed system over the
e
1−µ(I), i.e., fullΓ. Itcorrespondstoleakingthe systemafterequilib-
rium has been achieved in the closed system. We show
γ∗ =−ln(1−µ(I)) [≈µ(I) for µ(I)→0], (13)
e in Sec. V that
where µ(I) is the natural measure of leak I. This re- 1 1
hni ≈ = with, ρ =ρ . (16)
lation is equivalent to the binomial estimate for recur- µ µc(I) 1−e−γe 0 µ
rence times (6) and we call it hereafter as the naive es-
Similar results are obtained for densities proportional
timate of γ . Deviations of γ from γ∗ were reported in
e e e to ρ (equal to ρ apart from normalization in a sub-
µ µ
Refs. [5, 7] and associated to the existence of short-time
space of Γ).
periodic orbits of the closed system inside I.
Smooth density ρ = ρ : in some systems, and for
0 s
TheresultsofSec.IIIAonopensystemscanbeapplied
numerical simulations, it is natural to define the initial
to systems with leaks. Since M˜−1(A)≡A\I, it follows
density to be a smooth function over Γ or a part of Γ.
that µ (M˜−1(A)) = µ (A)−µ (I) = 1−µ (I), and the
c c c c A particular case is that of a uniform density. Estimate
following simple relation is obtained from (11)
(16) remains valid for such initial densities as well, as
shown in Sec. V.
γ =−ln(1−µ (I)). (14)
e c
This exact expression, which was previously obtained in
Ref. [5], corresponds to the naive estimate (13) replac- IV. RELATION BETWEEN RECURRENCE
AND ESCAPE
ing µ(I) by µ (I). They become equivalent in the limit
c
of small leak region µ(I) → 0 because µ tends to µ
c
[37, 38]. It is the c-measureof the leak which determines In this section we show that there is an initial density
the escape rate. Even though numerically such a mea- ρ0 =ρr for which pr(T)=pe(n) for T =n. We use this
sure is easily calculated. Proofing the existence of such special initial condition to establish a relation between
conditionally invariant measure may be a very involving recurrence and escape.
mathematical task [1, 35] (see, e.g., Ref. [36] for a class
~
of maps with leaks). Here we employ a pragmatic ap- M(I)=escape
proachand implicitly assume that such a measure exists
in the large class of systems with leaks where numerical 1 iteration .I. 1 iteration~
(closed map M ) (open map M )
simulations indicate exponential decay of p (n).
e
Phase space
.
M(I) Γ
C. Dependence of the mean escape time on the
initial density
T−1 iterations
While the escape rate γ is independent of the initial
e
density ρ , as expressed by Eq. (14), the mean escape
0
FIG.1: (Coloronline)Illustrationoftheequivalencebetween
time hni and the full distribution p (n) change consid-
e e recurrence and escape times.
erably with ρ . It is thus instructive to discuss the fol-
0
lowing special initial conditions:
Densityequivalenttorecurrenceρ =ρ : considerthat
Conditionally invariant density ρ = ρ : from a for- 0 r
0 c trajectories are injected from outside through the leak.
mal mathematical perspective the natural choice of ini-
More precisely, consider the distribution ρ obtained as
tial conditions is according to the density of the c- r
the first iterate through map M(~x) of the points ~x ∈ I
measure ρ (~x), which can be considered as the natural
c distributedaccordingtothenaturaldensityρ . Thiscan
measureofthe open system. This correspondstosetting µ
be obtainedapplying the Perron-Frobeniusoperator[20]
the initial density in agreement with the escape process.
as
Therefore, no oscillations are observed in Eq. (8), i.e.,
n∗ =1. Since γe is given by (14) and the distribution is ρ (~x)= ρµ(M−1(~x)∩I) for ~x∈M(I), (17)
normalizable, pe(n) follows a binomial distribution r J(M−1(~x)∩I)µ(I)
p (n)=µ (1−µ )n−1 = µc eln(1−µc)n, whereM−1(x)∩I denotesthepointsthatcomefromI,J
e c c
1−µ is the Jacobian of the map, and µ(I) ensures normaliza-
c
tion [57]. In the case of invertible maps, M−1 is unique
with µ = µ (I), in analogy with Eq. (5). The mean
c c and the density (17) is equivalent to
escape time is in this case
1 1 ρ (~x)= J(ρMµ(−M1(−~x1)()~xµ)()I) if ~x∈M(I), (18)
hni = = , with ρ =ρ . (15) r
c µc(I) 1−e−γe 0 c (0 else.
5
Large leaks Small leaks
Measures Finite µ(I)6=µc(I) µ(I)=µc(I)→0
∗
Escape rate γ =−ln(1−µc(I))6=−ln(1−µ(I))=γ γ =µc(I)=µ(I)
1 1 1 1 1 1 1 1
Mean time hnir = µ(I) 6= 1−e−γ hnic = µc(I) = 1−e−γ hniµ,s ≈ µc(I) = 1−e−γ hni= µ(I) = γ
ρ0 Recurrence: ρr c-measure: ρc natural, smooth: ρµ,s ρr,c,µ,s
TABLE I: (Color Online) The escape/recurrence rate γ = γr = γe and the mean escape time (recurrence time for initial
density ρr) hni expressed in terms of the natural invariant measure µ(I) and the conditionally invariant measure µc(I) of the
leak/recurrence region I, for large and small leaks.
Consider now aninfinitely long trajectoryused to calcu- to choosinginitial densities ρ andρ or ρ , respectively
r µ s
late the recurrence times T described in Sec. II. In view (seeTableI). Sincetheescaperateisindependentofthis
oftheergodictheorem(timeaverageequalsensembleav- choice,weseethatbothreferencesaregivingintuitivein-
erage), we see that one iteration after returning to I the terpretations for the deviation of the actual γ from the
points of this trajectory are distributed precisely as ρ . naive estimate γ∗. Analogous observations in Hamilto-
r
As illustrated in Fig. 1, due to the one step time shift niansystemshavebeenreportedin[22](recurrence)and
insertedin definition(12), allescape times n ofρ corre- in [7] (escape). In a different approach, γ has been re-
r
spond to a recurrence time T[58]. Since this is valid for lated to the diffusion coefficient in both recurrence [23]
all n, p (n) ≡ p (T) for ρ = ρ . In particular, γ = γ , and escape [39] problems.
e r 0 r r e
andsinceγ isindependentoftheinitialdensity,thisim- Formallythe problemcanbe describedintermsofthe
e
pliesthatforafixedrecurrence/escaperegionI thedecay c-measure and the convergence to it. The difference be-
rate of the Poincar´e recurrences and the escape rate of tweenγ andγ∗correspondtothedifferencebetweenµ(I)
the corresponding leaked system coincide: and µ (I), as given by Eqs. (13) and (14). As empha-
c
sized in our previous publication [31], this is surprising
γ =γ ≡γ. (19)
r e since a measure of open systems is employed to describe
properties of closed systems (recurrence). In practice,
The mean recurrence time is given, however, according
initial densities are most often taken proportionally to
to Kac’s lemma (4) as
thenaturaldensityρ ortoasmoothdensityρ . Insuch
µ s
1 caseswe expect a fast convergenceto ρ , andthis is why
c
hni =hTi= with ρ =ρ . (20)
r µ(I) 0 r 1/µc(I)[Eq.(15)]isabetterestimateofhni[inEq.(16)]
than 1/µ(I) [Eq. (4), which is obtained by the atypi-
Themeanrecurrencetimeisthusdeterminedbythenat- cal ρ = ρ ]. The escape rate is related to µ (I) which
0 r c
ural measure µ(I) and deviates considerably from the is itself specially sensitive to the choice of the location,
typical mean escape time given by (16). The reason is shape,andsizeofthefiniteleakI. Itisnaturaltoexpect
that ρr in (17) is very atypical from the point of view of thatµc(I)<µ(I)sincethesaddleexistsonlyinA\I and
the c-measure. In fact, ρc is concentrated along the sad- thereforeµc isconcentratedonlyalongitsunstableman-
dle’s unstable manifold, i.e., points ~x such that M˜−i(~x) ifold. This implies that the mean recurrence time hTi,
never escape. On the other hand, all points on ρr came i.e., hni with ρ0 = ρr should be shorter than the mean
from the leak and therefore the support of ρr does not escape time hni for a typicalρ0. This statement is rigor-
overlap with the unstable manifold of the saddle (but it ously proved in Ref. [40] for ρ =ρ .
0 µ
does with the stable manifold).
In view of the results above, we consider hereafter
the recurrence problem as an escape problem with ρ0 = V. NUMERICAL RESULTS
ρ [59]. Coherently, we omit the indices r and e, and T
r
willalsobedenotedbyn. Thevaluesoftheescaperateγ
We illustrate the previous results through numerical
and of the mean escape time hni for the different initial
simulations of the dissipative H´enon map
densities ρ are summarized in Table I.
It is now possible to compare the results of Ref. [5] for x =1−1.4x2 +y , y =0.3x . (21)
n+1 n n n+1 n
escapes with those of Ref. [27] for recurrence. In both
cases the fully chaotic logistic map was carefully investi- Initialconditionsinsidealargebasinofattractioncloseto
gatedanddeviationsofγfromthenaiveestimateγ∗were the origin converge to the well studied H´enon attractor,
reported. The deviations were associated with the pres- shown in Fig. 2(a). We let trajectories leak the system
ence of periodic orbits inside I and interpreted in terms through the leak region I: circle of radius δ centered at
of the overlap of the pre-images of I (in Ref. [5]) and some (x ,y ) on the attractor. The Sinai-Ruelle-Bowen
c c
of the short time oscillations (in Ref. [27]). In our uni- measure is the natural measure µ describing the closed
fied perspective, results of Refs. [27] and [5] correspond system. All the initial densities discussed in Sec. IIIC
6
10-1 (a)
10-2
10-3 Estimation γ∗=0.0385
T)
p(r10-4
p(n), e10-5 n*r
Actual γ = 0.0545
10-6 0.01 n* (i)
e (ii)
10-7 0 10 20 30 40 50 (iii)
γ n)p(0.1 (b) n*r n*e ((iii))
x
e
n) (iii)
p( 0 n* 50 100 150 200 250
e n, T
FIG. 3: (Color online) (a) Escape time distributions for the
FIG. 2: (Color online) H´enon map (21) with a leak
H´enon map (21) with a leak of radius δ = 0.05 at (xc,yc) =
at (xc,yc) = (1.0804,0.0916) of radius δ = 0.1, depicted (1.0804,0.0916). Results for initial densities ρr,ρµ, and ρs
as a red circle. (a) Different initial densities: ρµ of the (shown in Fig. 2) are presented from top to bottom, respec-
natural measure on the H´enon attractor, ρs = const. on tively. The thin solid line corresponds to γ∗ = 0.0385 given
0< x<1,−0.2< y <0.2, and ρr, equivalent to recurrence. by (6) and strongly differs from the actual exponent γ =
The inset shows the nontrivial dependence of the projection
0.545. The inset shows the short time behavior. (b) The
ρr(y) of ρr on the y axis. (b) Invariant saddle composed of same as in (a) multiplied by exp(γn). Three different pro-
thepointsthatdonotenterI forforward andbackwarditer- ∗
cedures were employed to fit γ using p(n) for n < n (see
ations. (c)Unstablemanifoldoftheinvariantsaddle(ρc)and Sec.VII)andareindicatedbythedashedlines(i)-(iii). Inall
(d) stable manifold in grey (orange) of the invariant saddle. ∗
cases n =35 and (i) corresponds tothe useof relation (33),
(ii) follows from relations (31) and (32) with the numerical
valueofhniforρ0 =ρr,and(iii)fromrelations(31)and(32)
are marked in Fig. 2. Panels (b)-(d) show the invariant with thenumerical value of hni for ρ0 =ρs.
saddle and its manifolds. A new feature in comparison
withsaddlesofnaturallyopensystemsisthatthe unsta-
these two curves is a measure of the error of the naive
ble manifold is not a single fractal curve but consists of
estimate (13). The general dependence, for δ → 0, is
disjoint pieces cut by the leak and its images.
in agreementwith the expected δ−D1 relation,by taking
In Fig. 3 we show the escape time distribution for the
into account that D / D , where D is the fractal di-
initialdensities depicted inFig.2. It is apparentthat all 1 0 0
mension of the attractor. From the inset it is apparent
initial densities lead to different short time behavior but
thatinitialconditionsρ andρ accordingtothenatural
to the sameescaperateγ,whichissignificantlydifferent µ s
measure and to the homogeneous distribution, respec-
from the naive estimate (13).
tively are closer to the results obtained with the density
ρ of the c-measure than that obtained with initial den-
c
We investigate now the dependence of γ and µ(I) on
sity ρ of the recurrence problem. In other words, the
r
the radius δ of the leak. Since the natural and the c-
mean escape time for typical smooth densities is better
measure have different fractal properties, for small (but
approximated by Eq. (16) than by 1/µ(I) [Eq. (4)].
not infinitesimally small) δ one expects the scaling rela-
A systematic investigation of the dependence of γ on
tion
the position of the leak in the H´enonmap would be very
involved. Apart fromthe difficulty of having two dimen-
1 =µ(I)∼δD1,
hTi (22) sions of the phase space, the natural (SRB) measure of
γ ≈µc(I)∼δ1+ds =δ1+(λ−γ)/|λ′|, the system is not known a priori. Therefore, in order
to fix the natural measure of the leak µ(I), a different
where D1 is the informationdimensionofthe chaoticat- value ofδ should be chosenfor eachnew (xc,yc). We in-
tractor, ds is the partial information dimension of the vestigate therefore this issue in the fully chaotic logistic
saddle (system with leak) along the stable direction (the map
c measure is smooth along the unstable one), and (9)
x =4x (1−x ). (23)
has been used. The numerical comparison can be seen n+1 n n
in Fig. 4 where the inverse mean escape time is plot-
In this case the density ρ of the natural measure of the
µ
ted for different initial densities. For the c-measure and
system is given by [18]
δ ≪1, 1/hni correspondsto µ (I) throughrelation (14),
c
while 1/hni = µ(I) with ρ = ρ . The deviation between ρ (x)=π−1[x(1−x)]−1/2. (24)
r µ
7
1 3
(a)
0.1
0.8
2
0.01 0.6
n> +1 x)
1/< xn ρ(
0.4
0.001 1/<n>=δ−1.27 (D=1.27) M(M(I)) 1
0
ρ - smooth square }
ρs - c-measure 1/<t>=µ(Ι)=1-e-γ 0.2
c c
0.0001 ρµ - invariant measure
ρ - recurrence 1/<T>= µ(Ι)
r 0 0
0.001 0.δ01 0.1 0 0.2 }I 0.4 xn 0.6 M}(I) 1
(b) 1.1
FIG. 4: (Color online) Dependence of the inverse mean es-
capetime1/hniontheradiusδoftheleakforfixed(xc,yc)= >]e1.0
(0.015714,0.268892). Initial densities equivalent to those <n
shown in Fig. 2 are taken. The straight line corresponds to I) 0.9
δtfoo−rrDl[0a5,r3gw]e.hδTe.rheeDin0s=et1s.h2o7wisstahemfarganctifialcadtiimonenosfiotnheofmthaienagttrraapch- µ(n>=1 / [e0.8 123... ρρρ000===ρρρrcµ::: cirne-vmc.u emrareesaunsrceue,r, e 11//<<nn>> == µµc((II))=1-e-γ
> / <r0.7 4. ρ0=ρs: uniform in [0,1]
n 0.6
TheresultsareshowninFig.5(b)andconfirmthestrong <
dependence of γ on I, as well as the conclusion drawn
0.5
from Fig. 4 that for general smooth initial densities γ is 0 0.2 0.4 0.6 0.8 1
x
better approximated by (16) than by 1/µ(I), reflecting c
once more the fast convergence of ρ to ρ . Moreover,
0 c FIG. 5: (Color online) (a) Graphical representation of the
it is clear that for most x one has hni > hni , i.e.,
c e r logistic map (23) with a leak I at xc =0.29427 of half-width
µ(I)>µ (I),inagreementwiththediscussionattheend
c δ=0.00716sothatµ(I)=0.01. Thedashed-linecorresponds
of Sec. IV. The remarkable exceptions are related to the
to ρµ(x), Eq. (24), and the (blue) histogram to a numerical
existence of short time periodic orbits inside the leak, as distribution of thec-measure of thechaotic repeller obtained
previously described in Refs. [5, 17, 27]. An explanation by leaking the system at I. The dotted (red) histogram cor-
for these cases can be given as follows (see Sec. VII for responds to ρr(x) given by (17) multiplied by µ(M(I)). (b)
details): if a periodic orbit of period n is inside I we Dependence of the inverse mean escape time, multiplied by
p
expect a high recurrence probability P(T = np); since hnir,ontheleakpositionxc inthelogisticmap(23)atanin-
hTi is fixed by (4), we predict that γ <γ∗ and therefore tervallengthsuchthatµ(I)=0.01. Theinitialdensitiesused
hn i/hn i<1, as observed in Fig. 5(b). are (see Table I): ρr: equivalent to recurrence, ρµ: natural
r e density Eq.(24), and ρs: chosen uniform in 0<x<1.
VI. NONHYPERBOLIC HAMILTONIAN
(KAM)islandscoexistinginachaoticsea[21,25,42,43],
SYSTEMS WITH POWER-LAW TAILS
andalsoinhigherdimensions[49]. Thepower-lawbehav-
ioris due to the nonhyperbolicityofthe dynamics andit
A. Theory
is present both for recurrences and escapes. The power-
law exponent is independent of the choice of I and ρ ,
0
Our discussion has covered so far the case of strongly provided the latter is concentrated away from the non-
chaotic systems where asymptotically in time, the dis- hyperbolic regions. Below we show that the same holds
tribution of escape and recurrence times is exponential forthe intermediatetime exponentialdecayoftherecur-
(for certain classes of systems this has been formally rence and escape time distributions, for a given I.
proved [44]). We concentrate in this section on weakly The distribution of escape and recurrence times in
chaotic systems where deviations from exponential de- Hamiltonian systems with mixed phase space can thus
cay are observed for long times. The simplest exam- be given as
ples of weakly chaotic systems are one-dimensional in-
termittent maps with a marginal fixed point, where an
asymptoticpower-lawdecayisproved[45,46]. Themost irregular for n<n∗,
prominent examples are, however, Hamiltonian systems p(n)≈γae−γn for n∗ <n<nα, (25)
with mixed phase space, where asymptotically a power- γ[ae−γn+b(γn)−α] for n>nα,
law decayis observed. This occursin systems possessing
isolated marginally unstable orbits [47], sharply divided where ae−γnα ≫ b(γn)−nα. The factor γ is written out
phase space [48], hierarchicalKolmogorov-Arnold-Moser on the right hand side in order to facilitate the connec-
8
tion to the continuous time limit. Contrary to what has
previously been claimed [50], γ(a+b)6=1 in Eq. (25).
Thequestionwhethertheasymptoticregimehasawell
defined (universal) power-law is still under investigation
for the case of area-preserving maps (see Ref. [43] for
the latest result that indicates that α≈2.57). Here, for
simplicity we write the asymptotic decay as n−α, but it
is meant to describe the power-law like behavior usually
observed.
We define the cross-over time n between exponential
c
and power-law decay as:
ae−γnc =b(γn )−α ⇒p(n )=2γae−γnc. (26)
c c FIG. 6: (Color online) (a) Phase space of the standard
map(28)withI =[0.25<x<0.25+ε,−0.5<y<−0.5+ε]
In the framework of recurrences, the different times in
of size ε=0.15, indicated by a checkered box. The KAM is-
Eqs. (25) and (26) can be interpreted as:
land(nonhyperbolicpartofthesaddle)isinthecenterofthe
figure. The support of ρr at M(I) is marked as a dark (red)
n∗: istheshort-timememory,withinwhichfluctuations region. (b) The unstable manifold of the hyperbolic part of
duetoshorttimeperiodicorbitsappear. Afterthis thesaddle(blackdots),obtainedbyusingthemethodof[26]
time the recurrences loose correlation and a ran- choosinganescapetimens =80<nc =307. Thesupportof
domapproximationfor the returnis valid(Poisson ρr is fully outside theunstable manifold.
process).
n : is the minimum time a trajectory inside I takes to trajectoriesthat reachthe nonhyperbolic part is propor-
α
approachthenonhyperbolicregionandreturntoI. tional to those that reach the hyperbolic one. When the
size of the leak I decreases, the hyperbolic component
nc: is the time when returning without touching the increaseswhile the nonhyperbolic one remains the same,
nonhyperbolic region becomes as probable as re- meaningthattheabovepictureisevenmoreaccurate. In
turning after touching. this case we can expect that the fraction of trajectories
that arrive at the nonhyperbolic component is propor-
If I is smalland awayfromany nonhyperbolic regionwe tional to the fraction of trajectories that approach the
expect n∗ <nα <nc. hyperbolic component and that therefore the ratio b/a
In terms of escapes, the times in Eq. (25) are inter- [see Eq. (25)] has a weak dependence on γ. Introducing
preted as: thisassumptioninthedefinitionofthecross-overtimen
c
[Eq. (26)] we obtain
n∗: is a convergence time which is proportional
to 1/|λ′|, where λ′ is the negative Lyapunov ex- n ∼1/γ [≈1/µ(I) for small µ(I)], (27)
c
ponent of the saddle (the time to relax to the hy-
perbolic component of the saddle along its stable meaning that nc is proportional to the reciprocal of the
manifold). escape rate γ. The proportionality constant depends on
the size of the nonhyperbolic regions. We have verified
n : is the time needed to approach the nonhyperbolic relation (27) for different maps, what emphasizes once
α
part of the saddle. more that for small γ a well defined exponential decay
exists for intermediate times n∗ <n<n .
c
n : isthetimewhenthenonhyperbolicpartofthesad-
c
dle becomes as important as the hyperbolic one.
B. Numerical results for the standard map
In this case, the key assumption n∗ < n is related to
α
the initial density ρ and leak I being away from the
0 We illustrate the previous results in the standard
stickyregions. Beinginvariant,thedivisionofthesaddle
map [24, 25]:
is of course independent of the initial density. However,
for initial conditions touching the KAM island the im-
y =y −0.52sin(2πx ), x =x +y . (28)
n+1 n n n+1 n n+1
portance of the exponential decay can be made arbitrar-
ily small (see Fig. 8 below) and the power-law exponent The phase space of this map is shown in Fig. 6. Chaotic
in (25) is modified to α′ = α−1, see Ref. [48, 51] (see trajectories stick for an algebraic long time to the hier-
also Ref. [44]). archical border of the KAM island, shown in the center
The above interpretation of p(n) given by (25) ex- of Fig. 6(a), that constitutes the nonhyperbolic compo-
presses the view, first suggested in [32], that the nonhy- nent of the saddle. The exponential decay is governed
perbolic component of the saddle is approached through bythehyperboliccomponentofthesaddle,whoseunsta-
the hyperbolic part [32, 50, 52, 54], i.e., the fraction of ble manifold is shown in Fig. 6(b) for a specific leak I.
9
(a)
10-2 n) 1100--64 α = 2.5
p( 10-8
p(n)10-3 1124 1100--1120102 n 10α3 = 2.5104 105
10-4γnc10 c n
8
10-5 6 0.001 0.01 γ 0.1
(b)
n)0.07
γp(0.06 FIG. 8: The saddle characterizing the dynamics at different
ex0.05 times,ns,givenontopofthepanels,obtainedbythemethod
p(n)00..0043 ofRef.[26]. For ns <nc =307 in(a),thehyperboliccompo-
0 50 100 150 200 250 300 350 400 nent obtained is clearly disjoint from he region of the KAM
n nc island. By increasing ns as in (b) and (c) the nonhyperbolic
component appears with an increasing weight.
FIG. 7: (Color online) (a) Escape time distributions in the
standard map with ε = 0.15, as shown in Fig. 6. Results
are shown for ρ0 = ρr (squares/below) and for ρ0 = ρµ in However,thisrelationisoflittlepracticaluseifonewants
|x| > 0.25. Lower inset: scaling of ncγ with γ obtained by to determine γ. The reason is that usually only the
changingεin0.03≤ε≤0.4. Upperinset: log-log plotof the Lebesgue measure is known a priori and the c-measure
main graphoveralongperiodoftime. (b)pe,r multipliedby µ may change dramatically depending on the position
c
exp(γn), where γ =0.027403.
of the leak. Its numerical calculation is typically much
more involved than the determination of γ itself. In this
section we search for an efficient method to determine γ
Thisunstablemanifoldcorrespondstothesupportofthe
thatgoesbeyondtheroughnaiveestimate−ln(1−µ(I)),
conditionallyinvariantdensityρ . Noticeitsapparentfil-
c given by relations (13) and (6).
amentary character and that it is present inside I, but
In general open hyperbolic systems a dense set of
does not enter M(I).
periodic orbits exists embedded into the nonattracting
The distribution ofescape andrecurrencetime for the chaotic set. This can be used to effectively obtain γ
system presented in Fig. 6 is shown in Fig. 7. As in as [18, 20]
the case of the H´enon map (Fig. 3), short time oscilla-
tions and an exponential decay are clearly visible. How- e−nγ = |Λ(Γ )|, (29)
i,n
ever, in this case a slower decay is observed for times
ΓXi,nY
n > n ≈ 307. In the upper inset of Fig. 7(a), a log-log
c
plot of the main graph shows that the long-time decay wheretheproductisovertheexpandingeigenvaluesΛof
is roughly a power-law with α ≈ 2.5. We have also ob- each periodic point Γi,n belonging to the same periodic
tained numerically the value of the cross-over time n orbit of length n. The sum runs over all periodic orbits
c
for leak/recurrence regions I of different sizes by chang- of period n that remain inside the system, i.e., outside
ing ε defined in Fig. 6. The results are shown in the the leak region I.
lowerinsetofFig. 7(a), andsupportthe theoreticalscal- We want to take further advantage here of the possi-
ing (27). Remarkably, the value of n is almost identical bility of closing the leak of the system. Note that in the
c
forrecurrenceandescapetimes,thesamebeingvalidalso closed system the decay exponent is γ = 0 and the sum
for other initial distributions ρ away form the KAM is- in Eq. (29) runs over all periodic orbits (inside and out-
0
land. This provides additional support to the picture side the leak region). We can therefore give γ in terms
that ρ quickly converges to the hyperbolic component of the periodic orbits inside the leak region as
0
of the saddle, and that the nonhyperbolic component is
approached in an universal way through the hyperbolic 1−e−nγ = |Λ(Γi,n)|, (30)
component. The transition between the hyperbolic and Γi,nXinsideY
nonhyperbolic components of the saddle is illustrated in
From this expression, it is clear that short time periodic
Fig. 8.
orbits have a strong influence on γ since not only the
primitive orbits, but also all their multiples appear in
the sum. This dependence is apparent in Fig. 5(b) and
VII. FITTING PROCEDURE TO OBTAIN THE was previously reported in Refs. [5, 17, 27]. The appli-
ESCAPE RATE
cation of (30) requires, however, again the identification
ofthe unstable periodic orbits of highperiods. Since the
In the previous sections we have seen that the escape deviations of γ from the estimate γ∗ = −ln(1−µ(I))
rate γ can be formally written as a function of the c- has been relatedto short time periodic orbits we can ex-
measureµ (I) of the leak regionI throughrelation(14). pect to determine γ from these orbits. A method for
c
10
this purpose was described in Ref. [27] in the context of VIII. CONCLUSIONS
recurrences. We adapt it here in the context of escape.
The distribution of escape times is written as in
Experimental and observational measurements often
Eq. (8). The oscillatory part for times n < n∗ will be
occur through holes or leaks that naturally exist or are
denoted by p (n) and the remaining exponential part
0 deliberately introduced in an otherwise closed dynami-
by p (n) ≡ gexp(−γn). The two parameters g and
exp cal system [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. The relevant
γ of the asymptotic exponential decay can be obtained
observable quantity in such systems with leaks is the
then by imposing the following two conditions: normal-
distribution of escape times from inside the system. In
ization
strongly chaotic systems, this distributions decays expo-
n∗−1 ∞ nentially. Applying the ergodictheoryoftransientchaos
p (n)+ p (n)=1, (31) [1] to chaotic systems with leaks an expression for the
0 exp
n=1 n=n∗ escape rate γe [Eq. (14)] is obtained [5] in terms of the
X X
c-measure (associated to the invariant chaotic saddle) of
and the value of the mean
the leak I. We have provided a theoretical framework
n∗−1 ∞ 1 to understand the strong dependence of γ and hnei on
hni= np (n)+ np (n)= . (32) the leak size (Fig. 4) and position (Fig. 5). In the (un-
0 exp
n=1 n=n∗ µi(I) realistic)limit ofsmallleak,the expressionofthe escape
X X
rate converges to a previously known relation [Eqs. (2)],
For recurrences, or ρ0 =ρr, we can use Kac’s lemma (4) basedonthe invariantmeasureofthe closedsystem. Al-
and thus µi(I) = µ(I), the natural measure of the leak. together, our results help to understand previous obser-
On the other hand, when initial densities are propor- vations [5, 7, 27, 29] and set a theoretical framework for
tional to the naturalmeasure in the escape problem, the future applications.
best procedure is to replace (4) by (16) and therefore
We have shown that the classical problem of Poincar´e
use µ (I) = µ (I) = 1−e−γ on the right hand side of
i c recurrencesinclosedsystemscanbedescribedasaprob-
Eq.(32). Inthelastcase,wecansolveEqs.(31)and(32)
lem of escape from a system with a leak, once the re-
in terms of γ, which is expressed as the solution of
currence region is identified with the leak I and the ini-
n∗eγ +1−n∗ 1 tial density is chosen properly. This allows us to treat
(1−S1) eγ −1 = 1−e−γ −S2, (33) bothproblemsinanunifiedframeworkandcomparepre-
vious similar results that were reported independently
where S = n∗−1p (n) and S = n∗−1np (n). Both in both fields (compare Refs. [5, 7] and [27]), and to
1 n=1 0 2 n=1 0
fitting procedures, the one given by Eqs. (31)-(32) and adapt (in Sec. VII) a previous method to efficiently ob-
the one usinPg Eq. (33), were emplPoyed for the case of tain γe. More surprisingly, we show that the relaxation
rateofthedistributionofPoincar´erecurrencesγ isequal
the H´enonmapandaremarkedas dottedlines in Fig.3. r
Notice that the fitting obtained using Eq. (33) alone is to γe [Eq. (19)] but that the mean recurrence time hnri
given by Kac’s lemma (4) differs from the typical mean
worse than the one obtained using as additional infor-
mation the numerical value of hn i in Eqs (31)-(32), but escape time [the analog of Kac’s lemma for leaked sys-
e
tems is (15)]. These results are summarized in Table I
better than the random estimate (6).
The main advantage of this procedure is that, even if and provide a more detailed account of the results pub-
theoreticallylargen∗ inEqs.(31)-(33)wouldrenderbet- lished in Ref. [31].
ter results, in practice the value n∗ is fairly small as nu- In weakly chaotic systems the asymptotic exponential
merical experience shows (n∗ / hni) The values of p(n) decay is replacedby a power-lawdecay. We have shown,
for n < n∗ are thus obtained without much computa- however, that if the leak region and the initial distribu-
tional effort already with a high precision but allow the tionareawayfromnonhyperbolicregions,anexponential
computation of the asymptotic decay γ. Moreover, this decayisstillwelldefinedforintermediatetimes. Wehave
shows that important information about the long-time obtainedthescalingofthecross-overtimencbetweenthe
dynamics is already contained in the short time dynam- two regimes [Eq. (27)] which confirms the importance of
ics. the exponential decay for small leaks. Altogether, these
Inthe nonhyperboliccaseonehastotakeintoaccount results justify an effective splitting of the chaotic saddle
also the contribution of the power-law tail. In the con- inanonhyperboliccomponent,relatedtotheasymptotic
tinuous limit, the following contributions: power-law decay, and an hyperbolic component, related
to the intermediate exponentialdecayfor which our pre-
S = ∞ γb(γn)−α =b(γn )1−α/(α−1), vious results apply. This splitting is particularly impor-
3 n=nα α (34)
S = ∞ γnb(γn)−α =bγ1−αn2−α/(α−2), tant in the case of Hamiltonian systems (found, e.g., in
4 Pn=nα α fluiddynamics oropticalapplications)where the generic
are to bPe added as new terms to the left hand side of phase space shows a mixture of regions of regular and
Eqs.(31)and(32),respectively. Sincen ∼n ∼1/µ(I), chaotic motion.
α c
and α > 2, one sees that these terms become negligible Finally,itisworthconsideringbrieflytwoextensionsof
for small recurrence/leak regions. the previous results that have potentially interesting ap-