Effect of randomness in logistic maps

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.

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