Steady State Of Random Dynamical Systems

TL;DR

This paper investigates the behavior of bounded random dynamical systems when the usual conditions for reaching a unique steady state fail, revealing attraction to common fixed points and their probabilistic weights.

Contribution

It characterizes the steady state of RDS beyond Pelikan's criterion, especially when multiple fixed points attract the system, and maps the problem to random walk hitting probabilities.

Findings

System is attracted to common fixed points when Pelikan's criterion breaks down.

Initial density distribution influences the probability weights at fixed points.

Mapping to hitting problem of random walks provides a way to compute these weights.

Abstract

Random dynamical systems (RDS) evolve by a dynamical rule chosen independently with a certain probability, from a given set of deterministic rules. These dynamical systems in an interval reach a steady state with a unique well-defined probability density only under certain conditions, namely Pelikan's criterion. We investigate and characterize the steady state of a bounded RDS when Pelikan's criterion breaks down. In this regime, the system is attracted to a common fixed point (CFP) of all the maps, which is attractive for at least one of the constituent mapping functions. If there are many such fixed points, the initial density is shared among the CFPs; we provide a mapping of this problem with the well known hitting problem of random walks and find the relative weights at different CFPs. The weights depend upon the initial distribution.