Poincaré recurrences and transient chaos in systems with leaks

Abstract

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 the closed 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 Poincaré recurrences in closed systems, which is treated in the same framework. Finally, we show how our results apply to weakly chaotic systems and justify a split of the invariant saddle in hyperbolic and nonhyperbolic components, related, respectively, to the intermediate exponential and asymptotic power-law decays of the survival probability.

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

“…This universal output directionality of single modes was proven without ambiguity in experiments on a liquid-jet microcavity (Lee et al, 2007c). Using this concept provided numerical evidence that all long-lived modes in the limaçon cavity (23) In the case of mixed phase space the chaotic saddle is divided into hyperbolic and nonhyperbolic components (Altmann, 2009). The mechanism of escape of electromagnetic radiation along the unstable manifold works also in this case as demonstrated by experiments on microwave cavities of quadrupolar shape .…”

“…A systematic and clear discussion of this extended version of the chaotic saddle and its relation to the ergodic theory of transient chaos can be found in (Altmann, 2009;Altmann et al, 2013). Altmann (2009) pointed out that the unstable manifolds of short periodic orbits (which are part of the chaotic saddle) close to the critical line as discussed by Schwefel et al (2004) are parallel to the unstable manifold of the chaotic saddle and therefore lead to nearly the same far-field emission.…”

“…However, as emphasized by Altmann (2009), the term "chaotic saddle" is more appropriate as the dynamics is time reversible. A "chaotic repeller" appears in noninvertible dynamical systems and possess only unstable manifolds (Lai and Tél, 2010).…”

In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

“…[23][24][25][26][27] From the mathematical point of view, a billiard is defined by a connected region Q & R D , with boundary @Q & R DÀ1 which separates Q from its complement. Basically, they are settled in three classes, namely (i) integrable, (ii) ergodic, and (iii) mixed.…”

Recurrence-time statistics in non-Hamiltonian volume-preserving maps and flows

“…It was observed that the RTS is capable of describing the relevant aspects of the dynamics in complex systems. In this context, we mention that the RTS is able to describe universal algebraic decays in Hamiltonian systems [2][3][4], including random walk penetration of the Kolmogorov-Arnold-Moser (KAM) islands [5,6], biased random walk to escape from KAM island [7], DNA sequence [8], synchronization of oscillator [9], generalized bifurcation diagram of conservative systems [10], fine structure of resonance islands [11], transient chaos in systems with leaks [12], among others.…”