Number of first-passage times as a measurement of information for weakly chaotic systems

Abstract

We consider a general class of maps of the interval having Lyapunov subexponential instability |δxt|∼|δx0|exp[Λt(x0)ζ(t)], where ζ(t) grows sublinearly as t→∞. We outline here a scheme [J. Stat. Phys. 154, 988 (2014)] whereby the choice of a characteristic function automatically defines the map equation and corresponding growth rate ζ(t). This matching approach is based on the infinite measure property of such systems. We show that the average information that is necessary to record without ambiguity a trajectory of the system tends to 〈Λ〉ζ(t), suitably extending the Kolmogorov-Sinai entropy and Pesin's identity. For such systems, information behaves like a random variable for random initial conditions, its statistics obeying a universal Mittag-Leffler law. We show that, for individual trajectories, information can be accurately inferred by the number of first-passage times through a given turbulent phase-space cell. This enables us to calculate far more efficiently Lyapunov exponents for such systems. Lastly, we also show that the usual renewal description of jumps to the turbulent cell, usually employed in the literature, does not provide the real number of entrances there. Our results are supported by exhaustive numerical simulations.

From Weakly Chaotic Dynamics to Deterministic Subdiffusion via Copula Modeling

“…In studies of Pomeau-Manneville maps, such fixed points are singularities in the invariant measure of the map, which produces small intervals around itself, called laminar regions, where the particle spend most part of the time of its trajectory. In this context, plenty of works has studied that the number of first-passage times N t to the particle to return to laminar region obeys a Mittag-Leffler distribution [21][22][23][24]. Thus, as the particle has two senses to return to some laminar region, and the displacements R t and L t are proportional respectively 1 The correspondence between the initial conditions from each map is: x 0 =x 0 , for x 0 ≥ 0 and x 0 = x 0 − 0.5, for x 0 < 0.…”

“…We perform then a copula modeling, where these new random variables will work as the marginal distributions. Thus, as R t and L t are connected to the number of first-passages of Pomeau-Manneville maps, whose distribution characterize the weakly chaotic dynamics [21][22][23][24], they will obey the same statistical quantity. Therefore, the jumps distribution of deterministic subdiffusion is connected to the distribution of weakly chaotic dynamics.…”

Exact invariant measures: How the strength of measure settles the intensity of chaos

“…We intend to explore here an idea that was recently introduced in Ref. [29]: the precise knowledge of the measure also enables us to estimate information by observing the transition dynamics along the phase space.…”

“…The approach, initially developed in Ref. [29], is extended to more general shapes of maps, also enabling us to identify geometric characteristics of them. Based on this approach we also provide a simple proof that infinite measure implies universal Mittag-Leffler statistics of observables, also the so-called Aaronson-Darling-Kac theorem, a central result of infinite ergodic theory [15].…”

“…For maps having infinite invariant measure the conven-tional Lyapunov exponent vanishes, resulting in weakly chaotic behavior. Such kinds of maps are the motivation of so-called infinite ergodic theory [15], an issue that is gaining increasing interest in the literature [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. Despite its sound development, this theory still suffers from a paucity of models whose invariant measures are exactly known, and this paper aims to bring further alternatives.…”