Transition to anomalous dynamics in a simple random map

Abstract

The famous Bernoulli shift (or dyadic transformation) is perhaps the simplest deterministic dynamical system exhibiting chaotic dynamics. It is a piecewise linear time-discrete map on the unit interval with a uniform slope larger than one, hence expanding, with a positive Lyapunov exponent and a uniform invariant density. If the slope is less than one the map becomes contracting, the Lyapunov exponent is negative, and the density trivially collapses onto a fixed point. Sampling from these two different types of maps at each time step by randomly selecting the expanding one with probability $p$, and the contracting one with probability $1-p$, gives a prototype of a random dynamical system. Here we calculate the invariant density of this simple random map, as well as its position autocorrelation function, analytically and numerically under variation of $p$. We find that the map exhibits a non-trivial transition from fully chaotic to completely regular dynamics by generating a long-time anomalous dynamics at a critical sampling probability $p_c$, defined by a zero Lyapunov exponent. This anomalous dynamics is characterised by an infinite invariant density, weak ergodicity breaking and power law correlation decay.

22 pages, 6 figures

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Cited by in corpus (1)

• Explicit dynamical properties of the Pelikan random map in the chaotic region and at the intermittent critical point towards the non-chaotic region