Dynamical characterization of stochastic bifurcations in a random logistic map

Yuzuru Sato; Thai Son Doan; Jeroen S. W. Lamb; Martin Rasmussen

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arXiv · 1811.03994

Dynamical characterization of stochastic bifurcations in a random logistic map

Yuzuru Sato·Thai Son Doan·Jeroen S. W. Lamb·Martin Rasmussen

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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.

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Yuzuru Sato, Thai Son Doan, Jeroen S. W. Lamb, Martin Rasmussen. 2018-11-09. Dynamical characterization of stochastic bifurcations in a random logistic map. https://arxiv.org/abs/1811.03994

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