Dynamical characterization of stochastic bifurcations in a random logistic map

Abstract

The emergence of noise-induced chaos in a random logistic map with bounded noise is understood as a two-step process consisting of a topological bifurcation flagged by a zero-crossing point of the supremum of the dichotomy spectrum and a subsequent dynamical bifurcation to a random strange attractor flagged by a zero crossing point of the Lyapunov exponent. The associated three consecutive dynamical phases are characterized as a random periodic attractor, a random point attractor, and a random strange attractor, respectively. The first phase has a negative dichotomy spectrum reflecting uniform attraction to the random periodic attractor. The second phase no longer has a negative dichotomy spectrum - and the random point attractor is not uniformly attractive - but it retains a negative Lyapunov exponent reflecting the aggregate asymptotic contractive behaviour. For practical purposes, the extrema of the dichotomy spectrum equal that of the support of the spectrum of the finite-time Lyapunov exponents. We present detailed numerical results from various dynamical viewpoints, illustrating the dynamical characterisation of the three different phases.