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The emergence of noise-induced chaos in a random logistic map with boundednoise is understood as a two-step process consisting of a topologicalbifurcation flagged by a zero-crossing point of the supremum of the dichotomyspectrum and a subsequent dynamical bifurcation to a random strange attractorflagged by a zero crossing point of the Lyapunov exponent. The associated threeconsecutive dynamical phases are characterized as a random periodic attractor,a random point attractor, and a random strange attractor, respectively. Thefirst phase has a negative dichotomy spectrum reflecting uniform attraction tothe random periodic attractor. The second phase no longer has a negativedichotomy spectrum - and the random point attractor is not uniformly attractive- but it retains a negative Lyapunov exponent reflecting the aggregateasymptotic contractive behaviour. For practical purposes, the extrema of thedichotomy spectrum equal that of the support of the spectrum of the finite-timeLyapunov exponents. We present detailed numerical results from variousdynamical viewpoints, illustrating the dynamical characterisation of the threedifferent phases.

Dynamical characterization of stochastic bifurcations in a random logistic map