  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.
