Recurrence time statistics for finite size intervals

Abstract

We investigate the statistics of recurrences to finite size intervals for chaotic dynamical systems. We find that the typical distribution presents an exponential decay for almost all recurrence times except for a few short times affected by a kind of memory effect. We interpret this effect as being related to the unstable periodic orbits inside the interval. Although it is restricted to a few short times it changes the whole distribution of recurrences. We show that for systems with strong mixing properties the exponential decay converges to the Poissonian statistics when the width of the interval goes to zero. However, we alert that special attention to the size of the interval is required in order to guarantee that the short time memory effect is negligible when one is interested in numerically or experimentally calculated Poincaré recurrence time statistics.

Poincaré recurrences and transient chaos in systems with leaks

“…Despite its simplicity, relation (2) is of little practical relevance since the limit µ(I) → 0 is never achieved in either numerical or experimental applications [27,28,29]. Here we do not restrict ourselves to this limit and find that, apart from being unrealistic, it masks different interesting relations.…”

“…We define the recurrence region as a subset I ⊂ Γ with µ(I) > 0. As stated above, we do not restrict ourselves to the unrealistic limit µ(I) → 0 [27,28,29]. The Poincaré recurrence theorem ensures that for almost all initial conditions x 0 ∈ I, there are infinitely many time instants n = n 1 , n 2 , .…”

“…The recurrence time distribution of an uncorrelated random process with fixed return probability µ follows a binomial distribution [27] …”

“…(1) can be estimated from the probability of escaping or returning. For small leaks, this probability is obtained by computing the closed systems natural measure µ of a typical leak region I, and it follows that [5,24,27] γ r = γ e = µ(I) = 1 t for 1 ≫ µ(I) > 0,…”

“…We also use this connection to apply and adapt a method to determine γ [27] based only on the escape probabilities for short times. This is a main advantage over genuine open systems where long-term periodic orbits are needed to determine γ e .…”