Diffusion in randomly perturbed dissipative dynamics
Christian S. Rodrigues, Aleksei V. Chechkin, Alessandro P. S. de Moura, Celso Grebogi, Rainer Klages
DDiffusion in randomly perturbed dissipative dynamicsEPL 108 40002 (2014) - doi:10.1209/0295-5075/108/40002
Christian S. Rodrigues, ∗ Aleksei V. Chechkin, Alessandro P. S. de Moura, Celso Grebogi, and Rainer Klages Max Planck Institute for Mathematics in the Sciences, Inselstr., 22, 04103 Leipzig, Germany Institute for Theoretical Physics, NSC KIPT, ul. Akademicheskaya 1, UA-61108 Kharkov, Ukraine Department of Physics and Institute for Complex Systems and Mathematical Biology,King’s College, University of Aberdeen - Aberdeen AB24 3UE, UK Queen Mary University of London, School of Mathematical Sciences, Mile End Road - London E1 4NS, UK (Dated: October 2, 2018)Dynamical systems having many coexisting attractors present interesting properties from both fun-damental theoretical and modelling points of view. When such dynamics is under bounded randomperturbations, the basins of attraction are no longer invariant and there is the possibility of trans-port among them. Here we introduce a basic theoretical setting which enables us to study thishopping process from the perspective of anomalous transport using the concept of a random dy-namical system with holes. We apply it to a simple model by investigating the role of hyperbolicityfor the transport among basins. We show numerically that our system exhibits non-Gaussian posi-tion distributions, power-law escape times, and subdiffusion. Our simulation results are reproducedconsistently from stochastic Continuous Time Random Walk theory.
I. INTRODUCTION
Understanding the dynamics of systems exhibiting co-existing attractors is fundamental for modelling processeshaving many possible asymptotic states. Although notrestricted to, this multi-stable dynamics is particularlyimportant in systems experiencing very weak dissipa-tion [1, 2]. In contrast to strongly dissipative ones, theseare typically not dominated by one or few attractors.There are many areas from which we could pick up suchexamples. For instance, if one considers finite-size parti-cle in advection dynamics, the low dissipative interactionbetween the advected particles and the fluid can be char-acterised by the presence of multiple attractors trappingadvected particles even in open flows [3]. Another ex-ample is found in the dynamics of space dust and itsrole in the formation of planetesimals [4], among others.Even when most of the attracting sets are periodic, achaotic component may be present in the form of a frac-tal boundary separating the basins of attraction [1, 2].If the dynamics is fully deterministic, the attractors areinvariant structures. Hence, once a particle or trajectoryis trapped in one of the basins of attraction, it remainsthere indefinitely. However, since most natural processesare not realistically isolated from external random per-turbations, it is natural to study their impact.The presence of random noise dramatically changes thedynamics. In contrast to deterministic systems, for ran-domly perturbed dynamics the invariance of attractorsmay not be true. If the considered perturbation is set tobe unbounded Gaussian noise, the whole phase space maybe the support of a unique invariant measure [5]. When ∗ Electronic address: [email protected] bounded perturbations are used, on the other hand, theremight be many coexisting invariant measures. In partic-ular, depending on the amplitude of the noise orbits canescape from the attracting domains [6] creating the pos-sibility of transport across their basins. This sort of hop-ping process has been reported before [1, 7–9], yet thereis a lack of understanding of its statistical properties,in particular from the anomalous transport perspective[10, 11].In this paper we analyse the statistical properties ofsystems lying on the border between dissipative and con-servative dynamics which evolve under random perturba-tions and their similarities to Hamiltonian dynamics. Westart by introducing what we call effective attractors . Be-low a certain level of dissipation the dynamics naturallygives rise to these attracting sets, which defined under fi-nite resolution are indistinguishable from topological at-tractors. We then extend the description of escape interms of a closed systems with a hole [6] to the case ofcoexisting attractors and establish the conditions allow-ing a hopping dynamics among them. We show that it ispossible to characterise the hopping process by a distribu-tion of first recurrence times to an appropriately chosennon-zero measure set. We find that such a recurrence (orescape time) distribution approaches the one expected fornon-hyperbolic dynamics as the dissipation is decreasedand the dynamics approaches the non-hyperbolic limit.This effect is similar to stickiness in Hamiltonian non-hyperbolic dynamics [10, 12]. We verify our argumentsby computer simulations for the single rotor or dissipa-tive standard map [1, 13]. The results match well toanalytical predictions from stochastic Continuous TimeRandom Walk theory [11, 14, 15]. Our discussion is basedon general arguments and not restricted to this particularmodel. a r X i v : . [ n li n . C D ] N ov II. DYNAMICS AND EFFECTIVEATTRACTORS
We are interested in adding bounded random noiseto our deterministic dynamics. More precisely, supposeour deterministic dynamics is given by the iteration of asmooth function f : M → M with differentiable inversein our phase space M , for example,[28] M ⊂ R n . Anorbit ( x n ) n ≥ is the sequence generated by the dynami-cal system x n +1 = f ( x n ) from a given initial condition x ∈ M . We will be concerned with subsets of M towhich most orbits in their neighbourhood converge forsufficiently long but finite time , what we shall call effec-tive attractors or attracting sets. In other words, thoseare f -invariant subsets of M contained in basins of at-traction, which are open sets of initial conditions withpositive Lebesgue (volume) measure converging to the at-tracting sets. Note that our requirements on convergencedemand this to happen within finite time, which is veryimportant for numerical/experimental investigations. Inthese cases, contrary to a rigorous mathematical frame-work and due to physical limitations one cannot ask fortime going to infinity or infinitely small length intervals.By making such finite-size assumptions on the dynamicsone may include among the detected invariant sets homo-clinic tangencies and Newhouse attractors which supportsome invariant measure at least within finite scales, thusbeing indistinguishable under finite resolution from moregeneral “real” attractors [2, 16].We will focus on the case where there is only a finitenumber of coexisting attractors. This is not a restriction,because for compact spaces the finiteness of the numberof effective attractors follows. Indeed, it is only possibleto fit a finite number of non overlapping balls of radiibounded from below in a compact space. Furthermore,for the case of randomly perturbed dynamics we shalldeal with it can be proven that the system has only afinite number of invariant physical measures [17]. There-fore, we represent the set of coexisting effective attractorsby { Λ i } Ni =1 , a family of pairwise disjoint compact sets, i.e.Λ i ∩ Λ j = ∅ , for i (cid:54) = j . Another important fact is thatwe also assume that the union of the basins of attractioncovers every point of the whole phase space, up to a zeroLebesgue measure set. So we write m (cid:32) M \ N (cid:91) i =1 W s (Λ i ) (cid:33) = 0 , (1)where m denotes Lebesgue measure and W s (Λ i ) thebasin of attraction of Λ i . This plays a very important rolein the definition of the hopping process between differentattractors, because the trajectories are always expectedto converge to some attractor. The boundary betweenbasins of attraction is a zero Lebesgue measure compo-nent, the so-called basin boundary , which we denote by ∂ . The basin boundary plays a fundamental role in thehopping process, as we shall see in what follows. III. RANDOM PERTURBATIONS
We now perturb the dynamics exhibiting multipleattractors by assuming physical random perturbation ;see [17] and Appendix D of [18] for a formal definition.Roughly speaking we add bounded random uniformlydistributed noise to the dynamics. That is, given thedeterministic system f defined as before, we consider thedynamical system F ( x j ) = f ( x j ) + ε j , (2)with || ε j || < ξ , where ε j is the random vector of noiseadded to the deterministic dynamics at the iteration j ,and ξ is its maximum amplitude. We require the noise toasymptotically cover uniformly a ball around the unper-turbed dynamics, representing the idea that the pertur-bation has no preferential direction and amplitude. Theorbit thus jumps from x to f ( x ) but misses the point atrandom with the conditional probability of finding theperturbed orbit in an ξ -neighbourhood of f ( x ) given x ,see [19] for a comprehensive treatment of this topic. IV. ESCAPE
If the amplitude of the perturbations is small enough,an orbit in the domain of attraction approaches the at-tracting set, wanders around without escaping and is ex-pected to be trapped there forever. Although the tra-jectory may seem very intricate, it is actually well de-scribed from a statistical perspective. In these cases, onehas a unique invariant ergodic probability distributionrepresenting a given attracting set [17]. If the system is stochastically stable, such distributions for the randomlyperturbed system approach those of the deterministic oneas the amplitude of the perturbations decreases to zero.The dynamics inside the basin can be described as thatof a closed system if the amplitude of the perturbationsis small enough [6]. When the amplitude of the noiseincreases beyond a threshold ξ the attracting sets losetheir stability. This effect can be seen as the introductionof a hole I ∂ = I ∂ ( ξ ) in the basin by which the orbits canescape from the domain of attraction; see [6] and furtherreferences therein for the general setting. Under someassumptions it is possible to estimate the size of such ahole, or its measure µ ( I ∂ ) > V. HOPPING PROCESS
Now we are ready to translate the problem of noiseinduced escape from pseudo attractors into that of aclosed system with a hole I ∂ , or a recurrence problem.We call pseudo attractors the sets where the orbits re-main trapped for some amount of time before escapingdue to noise. Rigourously speaking they are not attrac-tors or attracting sets, since the invariance condition isnot fulfilled. In our context, a pseudo attractor A is a quasi-invariant set when the amplitude of the randomperturbations is increased beyond ξ . With the assump-tion above we can describe our dynamics and the escapefrom a single attractor as x j +1 = F ( x j ) if x j ∈ A or escape if x j ∈ I ∂ . We do not define the dynamics in I ∂ as it is irrelevant to our discussion, hence when theorbit falls into I ∂ we stop considering it. However, weallow the trajectory to come back from the hole to A . Ifso, we restart the process of counting the time in A byneglecting the number of iterations that it had spent in I ∂ .Similar arguments apply to systems with many coex-isting pseudo attractors A i for which Eq. (1) holds. Insuch dynamics, when a trajectory falls into the i th hole I ∂ i there is the possibility of swapping basins. Usinga Markov assumption we argue this to be equivalent torestarting the process. Although for the i th hole there isa distinct measure µ i ( I ∂ i ) >
0, according to our assump-tion we treat all holes qualitatively in the same way. Ig-noring the dependence on i we simplify the recurrence inprobability space to the i th interval by dropping the in-dex i . We are thus characterising the dynamics in termsof a representative hole I ∂ with average measure µ ( I ∂ ).Correspondingly we reduce the sojourn time distributionof the hopping process to the statistics of the time in-tervals that a random orbit takes to access the repre-sentative hole I ∂ . Furthermore, we assume the general basin property to hold, which tells us that up to a setof zero Lebesgue measure the time averages of orbits inthe basins of attraction converge to the space averagewith respect to the invariant measures supported on theattractors; see Chap 1.6 in [18]. VI. PSEUDO STICKINESS
Let us now look further at the microscopic dynamicsin order to understand the overall statistical behaviour ofthe noise induced hopping process between different at-tractors. In particular we shall explore its analogy withnon-hyperbolic Hamiltonian dynamics where stickiness plays a fundamental role for explaining the statistical dy-namics.To set the scene let us forget about the noise for themoment. Recall that Hamiltonian non-hyperbolic dy-namics is characterised by elliptic orbits, whose eigenval-ues are purely imaginary. These orbits are surroundedby complex structures formed by marginally stable peri-odic orbits, known as Kolmogorov-Arnold-Moser (KAM)invariant tori or islands, as well as regions of chaoticmotion. Large islands are surrounded by smaller oneswhich, on the other hand, are surrounded by even smallerones, repeating this pattern on smaller scales ad infini-tum . Trajectories starting in the chaotic region exhibit intermittent dynamics : they spend long sporadic peri- ods of time performing almost regular motion near theborders of the islands before escaping to the chaotic seaagain. Even small islands can have a great impact onthe dynamics of an orbit. Given the hierarchical struc-ture of the phase space, when an orbit eventually escapesfrom the neighbourhood of an island it may spend sometime wandering in the chaotic sea before it gets trapped once more by the same or another island. This effect,generally known as stickiness [12], slows down the dy-namics. Among its statistical signatures one typicallyobserves power-law decay of correlations and anomalousdiffusion [10].
Uniformly hyperbolic dynamics, on the other hand, ischaracterised by exponential-like laws. Roughly speak-ing a system is called hyperbolic if at each point on theattracting set distances are contracted or expanded withexponential rate. If the rate of convergence does not de-pend on the point, the system is called uniformly hyper-bolic [18]. In what follows we argue that, from a statisti-cal point of view, in our case the presence of random per-turbations destroys uniformly hyperbolic behaviour.
Thatis, the perturbations destroy uniform contraction andexpansion rates, therefore exponential statistical signa-tures are lost. Furthermore, when the noise amplitudeis set above a threshold, the orbits can escape from theattracting sets as explained in the previous section
Es-cape . The general statistical effect is similar to that ob-served in non-hyperbolic Hamiltonian systems. Namely,the pseudo attractors behave in a manner similar to theKAM islands, where the orbits perform an almost regularmotion for a limited time interval. The presence of noisefurthermore washes out fine scale structures of the phasespace. Thus, the trapping regions of small attractorshave less but non-negligible importance, since the orbitsmight stay inside them only for a short time by perform-ing almost regular motion before escaping again. Oncean orbit escapes from a pseudo attractor, it undergoesan erratic motion until it falls again into the same or an-other trapping region. Although some of the trapping re-gions may be very small, yet they have great influence onthe statistical characterisation of the dynamics because,just like small KAM islands in the case of non-hyperbolicdynamics, every pseudo-attractor has a stickiness-like ef-fect. An important difference nevertheless is that for thedissipative case, the attractiveness to a nearly invariantsets determines the type of diffusion. The mean squaredisplacement is thus expected to show a slower diffusivedynamics compared to Hamiltonian systems.
VII. SOJOURN TIME DISTRIBUTION ANDHYPERBOLICITY
In the previous sections we focused on the connectionbetween a hopping process and escape in a dynamicalsystem with holes. As a consequence, the sojourn timedistribution for the hopping process given by the distri-bution of escape times P ( t ) for a system with holes de-pends on the dynamics in the pseudo attractors ( i.e. thesets A i ) governed by their hyperbolic properties. We con-sider two “extreme types” of dynamics: on the one side,the escape of orbits from sets in uniformly hyperbolic dy-namical systems has been shown to follow an exponentialtime distribution. On the other side, escape in Hamilto-nian systems with mixed phase space yields power-lawtails [12, 20–22].Now suppose that in a given dynamical system wecould somehow control “how hyperbolic” it is. Wemight then switch the escape time distribution between P ( t ) ≈ ae − αt and P ( t ) ≈ bt − β , where the parameters a and b depend on the hyperbolicity of the dynamics.They are determined by the dynamics in the pseudo at-tractors, or more generally, in the set with a hole fromwhere the trajectories escape. For uniformly hyperbolicsystems the parameter a is large and the dynamics in thepseudo attractor has hyperbolic characteristics. There-fore, we have a hyperbolic recurrence time distributionto I ∂ , and the asymptotic decay of the corresponding es-cape times is exponential. On the other hand, when thenon-hyperbolic component of the dynamics is increased,the parameter b gains importance and the diffusion of therandom orbit in the support of the conditionally invari-ant measure[29] experiences a stickiness effect, resultingin a slower distribution of recurrence times to I ∂ with apower-law tail. Such an increase of non-hyperbolic char-acteristics under parameter change may be the result ofhomoclinic tangencies with highly non-uniformly hyper-bolic properties [16, 18]. Since we deal with dynamicsunder finite resolution, we cannot distinguish them fromthe other attractors. Note that this behaviour should beindependent of the noise amplitude within some rangeof it, because its amplitude will control the number ofpseudo attractors, but the type of escape should be con-trolled by the hyperbolicity of the system. In the nextsection we present numerical evidence supporting ourarguments, showing that for systems close to the non-hyperbolic regime the escape time distribution indeed hasthe power law signature of non-hyperbolicity rather thanbeing exponential as expected for uniformly hyperbolicdynamics. VIII. NUMERICAL RESULTS
We illustrate our results by simulations of the per-turbed system defined by F ( x j , y j ) = f ( x j , y j ) +( ε x,j , ε y,j ) with uniformly distributed i.i.d random noise.For f we choose the single rotor map [13] f (cid:18) x j y j (cid:19) = (cid:18) x j + y j mod 2 π (1 − ν ) y j + f sin( x j + y j ) (cid:19) , (3)with x ∈ [0 , π ], y ∈ R and damping parameter ν ∈ [0 , ν (cid:54) = 0 the dynamics is dissipative. In the stronglydissipative limit ν → ν → ν = 0 we recover the area preserving standard map withHamiltonian dynamics [23]. Therefore, we can think of ν as a control parameter measuring how far the dynamics isaway from the non-hyperbolic regime. We use f = 4 . ν (cid:54) = 0 [1]. Atthis parameter value and ν = 0 the standard map dis-plays superdiffusion, due to the existence of acceleratormodes [24].If we evolve our system under the presence of randomnoise beyond a certain amplitude ξ ≥ ξ the attractingsets lose their stability, as discussed in the section Es-cape . Note that each attractor may have a differentvalue of minimum noise amplitude such that escape takesplace, which is proportional to the size of their basinsof attraction. We choose as a global ξ the minimumvalue for the escape from the largest trapping region.For ξ ≥ ξ escape from the attracting sets consequentlygives rise to diffusion of trajectories through the phasespace.Fig. 1(a) shows the time dependence of the y -positionprobability density function of such a process. It confirmsour hypothesis that diffusion of trajectories induced byrandom perturbations indeed takes place. While at firstview the included fits to Gaussian distributions seem tomatch well to the simulation data, the inset shows devia-tions in the tails especially for long times. This deviationwill be explained later on by matching the data with astochastic theory. Note also the existence of a periodicfine structure, which reflects the spatial distribution ofthe attracting sets along the y axis [1]. Analogous resultshave been obtained for simulations under different levelsof random noise, for different dissipation parameters ν ,and also for different values of f .For general systems a rigourous investigation of the so-journ time distribution and the identification of pseudoattractors can be a very difficult task [9, 17]. Even fromthe numerical point of view the fact that, a priori , neitherthe physical nor the conditionally invariant measures areknown can represent an obstacle to the identification ofpseudo attractors. A way to detect whether an orbit istrapped in the trapping region of some pseudo attractorfor a period of time is given in terms of finite-time Lya-punov exponents. Equivalently, one can calculate theeigenvalues of the Jacobian matrix of F along the or-bit. As a consequence of meta-stability of the pseudoattractors, while an orbit remains trapped the maximumeigenvalue of the Jacobian has, on average, magnitudeless than one; see Theorem V1.1 in [25] for a rigourousdiscussion on characteristic exponents in the case of ran-dom transformations. Fig. 1(b) illustrates our criterionfor the random dynamical system Eq. (3) where we have,without loss of generality, plotted y = 30 when a pseudoattractor is identified and y = 20 otherwise. Also with-out loss of generality we only consider trajectories thatremain trapped for more than 20 iterations. -100 -50 0 50 100y0.00010.0010.010.1 P ( n , y ) n=10 iterationsn=10 iterationsn=10 iterationsn=10 iterationsGaussianstretched exponential 70 90 110y6.1e-050.000244 P ( n , y ) (a) y time seriescriteriaPseudo attractor (b) FIG. 1: (a) Probability density function P ( n, y ) at position y for different iteration numbers n . An ensemble of 10 randominitial conditions uniformly distributed around x = y = 0was iterated by the map Eq. (3) randomly perturbed by noiseof level ξ = 0 .
06 and dissipation ν = 0 . n , the upper (dark green) lines are stretchedexponential fits with Eq. (4). The inset shows a blowup oftwo tails. (b) The black graph depicts a representative timeseries of the noisy system for ν = 0 .
02 and ξ = 0 .
2. The corre-sponding result by our eigenvalue criterion to identify pseudoattractors (see text) is given by the red line. The plateausat y = 30 reveal pseudo attractors, which coincide with thevisual identification of localisation in the time series. Once a proper identification of the different dynamicalregimes, i.e. trapped or wandering, is obtained, we areready to statistically analyse these different behaviours.We start by computing the probability distributions forthe times an orbit stays trapped for n < t iterations in apseudo attractor. For a range of larger values of ν in oursimulations we observe a predominantly exponential es-cape, as was to be expected [12, 20–22]. However, whenthe damping is decreased below ν = 0 .
02 the probabilitydistribution is roughly described by a power law, similarto the case of non-hyperbolic Hamiltonian dynamics [21].In Fig. 2(a) we show the probability distributions of es-cape times from pseudo attractors, or equivalently, thefirst recurrence time distributions to I ∂ , for fixed smalldissipation ν but different noise amplitudes ξ . Approx- t10 -5 -4 -3 -2 -1 P ( t ) ξ = 0.04ξ = 0.06ξ = 0.08ξ = 0.10β = 1.95
500 1000 1500 2000 2500t 10 -5 -4 -3 -2 -1 P ( t ) (a)
10 100 1000 10000n101001000 〈 y ( n ) 〉 ξ = 0.02ξ = 0.04ξ = 0.06ξ = 0.08ξ = 0.1ξ = 0.2γ = 0.85γ = 0.95 (b) FIG. 2: (a) Double-logarithmic plot of the probability distri-bution P ( t ) of escape times t for an orbit to stay trapped ina pseudo attractor for n < t . The map Eq. (3) was iterated10 times for dissipation ν = 0 .
002 and different values ofthe noise amplitude ξ . The dashed line represents a powerlaw decay with exponent β = 1 .
95. The inset shows the cor-responding semi-logarithmic plot. (b) Mean square displace-ment (cid:104) y ( n ) (cid:105) for the coordinate y as a function of time n .An ensemble of 10 initial conditions was iterated by the mapEq. (3) for different amplitudes ξ of random noise and fixedsmall dissipation ν = 0 . γ = 0 .
95, the upper dashed lineto an exponent γ = 0 . imately up to times t <
300 the escape time distribu-tions match reasonably well to power laws with expo-nents around β = 1 .
95 as shown in the figure. This willbe justified later by matching all data consistently with atheoretical prediction. The value is in agreement with therange of exponents 1 . ≤ β ≤ ξ depends on the parameters f and ν , for ampli-tudes ξ ≥ ξ the existence of a power law decay is in-dependent from the amplitude of the noise. This is notshown here but observed in further simulations. Whenwe decrease ξ the orbit typically takes longer to escape,consequently the probability distributions are stretchedto longer times. In Fig. 2(a) we observe a cross-over toexponential laws which changes with ξ , as is highlightedby the inset. The most important result of this analy-sis is that when the dynamics is near the non-hyperbolicHamiltonian limit, i.e. for small dissipation parameters ν , the behaviour of diffusive trajectories indeed has, fromthe statistical point of view, non-hyperbolic characteris-tics. This is what we shall address next.In order to understand the type of diffusion process weare dealing with, we computed the mean square displace-ment (cid:104) y ( n ) (cid:105) for the coordinate y , the relevant one fordiffusion, as a function of time n . The two lines shownin Fig. 2(b) represent power laws (cid:104) y ( n ) (cid:105) ∼ n γ with ex-ponents γ <
1. They reveal power law behaviour forthe data up to approximately t <
300 by providing up-per and lower bounds for the exponents. For the cor-responding subdiffusive hopping process among the dif-ferent basins the power laws persist independently of ξ but with a slightly varying exponent. Changing otherparameters such as ν typically generates the same be-haviour. This finding is in agreement with our analogy tonon-hyperbolic Hamiltonian dynamics generating sticki-ness to pseudo attractors as discussed in Section PseudoStickiness . Note that for t > (cid:104) y ( n ) (cid:105) are close to zero. This is due to thefact that the fastest particles have reached the region inphase space where the pseudo attractors of the map ceaseto exist [1] meaning they cannot move any further, andtrivial localization sets in.In the area preserving standard map superdiffusionhas successfully been modeled by stochastic ContinuousTime Random Walk (CTRW) theory [27]. As our ran-domly perturbed dissipative model displays subdiffusion,here we test the subdiffusive CTRW version put forwardin Refs. [11, 14, 15] to explain our simulation results.This theory predicts that if the mean square displace-ment exhibits a power law with exponent (cid:104) y ( n ) (cid:105) ∼ n γ ,the respective escape (or waiting) time distribution mustbe P ( t ) ∼ t − ( γ +1) on corresponding time scales. It fur-thermore predicts that the position distribution functionof the subdiffusive process must approximately be of thestretched exponential form.[30] P ( n, y ) ∼ exp (cid:16) − c ( n ) y / (2 − γ ) (cid:17) . (4)The lower straight line in Fig. 2(b) representing a powerlaw with exponent γ = 0 .
95 matches well to the meansquare displacement of ξ = 0 .
06. The dashed line inFig. 2(a) yields the corresponding power law with expo-nent γ + 1 = 1 .
95 as predicted by CTRW theory, whichmatches well to the numerical result for the escape timedistribution for the same ξ = 0 .
06 in the regime of t < ξ = 0 .
06 in Fig. 1(a) have all been per- formed with Eq. (4) by using the very same value of γ .Evidently, these fits match much better to the numericalresults in the tails than the corresponding Gaussian dis-tributions, at least for long enough times. We thus con-clude that the subdiffusive CTRW of Refs. [11, 14, 15]consistently explains our numerical findings, thus con-firming theoretically that our randomly perturbed dissi-pative dynamics generates a subdiffusive process that iswell-known in stochastic theory. This is quite surprising,as we did not take the strongly non-uniform distributionof pseudo attractors along the y axis into account butjust averaged over all of them by performing a kind ofmean field approximation. IX. CONCLUSION
We have investigated the hopping process of pointsgenerated by randomly perturbed dissipative dynamics.We have set up a theoretical framework that describesescape in terms of a closed system with a hole. Es-cape occurs when the support of the conditional invariantmeasure of one pseudo attractor overlaps with the neigh-bourhood of another basin boundary. In this setting thesojourn time distribution becomes the recurrence timedistribution of the orbit wandering to a hole. We thenshowed by simulations that for the randomly perturbedweakly dissipative single rotor map the distribution ofsojourn times is described by a power law up to relevanttime scales, in contrast to an exponential distribution forstrong dissipation. We found that the hopping processamong different basins is subdiffusive for a wide rangeof perturbation strengths. Using only the subdiffusivepower law exponent as a fit parameter, we showed thatstochastic CTRW theory consistently explains all of oursimulation data by revealing stretched exponential tailsin the position distribution function. We conclude thatbounded random perturbations generate a kind of non-hyperbolic stickiness in the diffusion process for the con-sidered dissipative dynamics which leads to non-Gaussianposition distributions, power laws in the escape time dis-tributions, and subdiffusion. It would be interesting toinvestigate whether similar phenomena occur in other dif-fusive randomly perturbed deterministic dynamical sys-tems.
Acknowledgments
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