Spectral analysis and an area-preserving extension of a piecewise linear intermittent map

Abstract

We investigate spectral properties of a 1-dimensional piecewise linear intermittent map, which has not only a marginal fixed point but also a singular structure suppressing injections of the orbits into neighborhoods of the marginal fixed point. We explicitly derive generalized eigenvalues and eigenfunctions of the Frobenius-Perron operator of the map for classes of observables and piecewise constant initial densities, and it is found that the Frobenius-Perron operator has two simple real eigenvalues 1 and λ d ∈ (−1, 0), and a continuous spectrum on the real line [0,1]. From these spectral properties, we also found that this system exhibits power law decay of correlations. This analytical result is found to be in a good agreement with numerical simulations. Moreover, the system can be extended to an area-preserving invertible map defined on the unit square. This extended system is similar to the baker transformation, but does not satisfy hyperbolicity. A relation between this area-preserving map and a billiard system is also discussed.

Effect of noise and detector sensitivity on a dynamical process: Inverse power law and Mittag-Leffler interevent time survival probabilities

“…In this article we focus on power laws connected to temporal complexity and intermittent dynamics of critical fluctuations. As shown by Contoyiannis et al [1], these fluctuations have a close connection with the intermittency generating maps [2][3][4][5][6]. The Pomeau-Manneville (PM) map generates sporadic randomness and the laminar regions between two consecutive randomness bursts have a distribution density that is an inverse power law.…”

Ergodic Problems for Real Complex Systems in Chemical Physics

“…Second, we discuss an extension of the Birkhoff's individual ergodicity theorem [28,[38][39][40][41][42][43][44]47]. and its relation to nonstationary processes in reactions [45].…”

“…Moreover, its distribution reveals a certain universal behavior. For example, Aizawa and his group have shown these universal characteristics for a class of one-dimensional maps and certain billiard systems [39][40][41][42][43][44]. The existence of universal fluctuation suggests that the statistical reaction theory can be extended to those reactions in which the traditional concept of ergodicity does not hold.…”

A piecewise linear model of self-organized hierarchy formation

“…Piecewise linear approximations are often used in the studies of nonlinear dynamical systems. In fact, even for systems in which rigorous approaches are difficult, more detailed analysis is possible for the piecewise linear versions [23][24][25][26]. Here, we derive the stable ranges of two-level and three-level solutions for the piecewise-linear model.…”