Effective and randomly perturbed dynamical systems

We start by analysing the effect of random perturbations on non-hyperbolic scattering dynamics.  We show that, under small random fluctuations of the field, trajectories starting inside KAM islands, not only escape but also gain hyperbolic-like time decay.  We show a random walk modelling this distribution to the amplitude of noise, what is expected to be a universal quadratic power law.  Due to the effect of the random perturbations, we show that the typical dimension of invariant sets is that which is obtained for hyperbolic dynamics. For dissipative systems, we show that when the dissipation is decreased, the number of periodic attractors increase.  Furthermore, we show that the dynamics is then better described by invariants that are time and length scale dependent. Employing random maps we described the escape of a random orbit from an attractor in terms of a dynamical system with a chosen “hole”. We show that the mean escape time depends on the measure of this hole, and we analytically obtain one universal power law describing such dependence.  Once we have described the escape from one attractor, we tackle dynamics of a system having finitely many coexisting attractors in terms of conditional invariant measures.  We study conditions for the hopping process to happen and the sojourn time distribution, according to hyperbolic properties. In the second part of this thesis, we study random dynamical systems from a mathematical perspective. Using set dynamic, we prove the existence of a finite number of random attractors.