Effect of randomness in logistic maps

TL;DR

This paper investigates how randomness in the parameters of logistic maps affects their dynamical behavior, revealing conditions under which the system becomes ergodic, nonchaotic, or chaotic, including chaos at parameter values below the classical threshold.

Contribution

It introduces analysis of a random logistic map with bounded parameters, showing how randomness influences chaos and ergodicity, and identifies novel chaotic regimes at lower thresholds.

Findings

System exhibits ergodic behavior at maximum parameter value 4.

Chaos can occur even when parameters are below the classical threshold 3.5699.

Different random variable sets can induce chaos for all parameter differences.

Abstract

We study a random logistic map $x_{t+1} = a_{t} x_{t}[1-x_{t}]$ where $a_t$ are bounded ($q_1 \leq a_t \leq q_2$), random variables independently drawn from a distribution. $x_t$ does not show any regular behaviour in time. We find that $x_t$ shows fully ergodic behaviour when the maximum allowed value of $a_t$ is $4$. However $< x_{t \to \infty}>$, averaged over different realisations reaches a fixed point. For $1\leq a_t \leq 4$ the system shows nonchaotic behaviour and the Lyapunov exponent is strongly dependent on the asymmetry of the distribution from which $a_t$ is drawn. Chaotic behaviour is seen to occur beyond a threshold value of $q_1$ ($q_2$) when $q_2$ ($q_1$) is varied. The most striking result is that the random map is chaotic even when $q_2$ is less than the threshold value $3.5699......$ at which chaos occurs in the non random map. We also employ a different method in which a different set of random variables are used for the evolution of two initially identical $x$ values, here the chaotic regime exists for all $q_1 \neq q_2 $ values.