Effects of bounded random perturbations on discrete dynamical systems
Christian S. Rodrigues, Alessandro P. S. de Moura, and Celso Grebogi
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Submission date: 22. Jun. 2012
PACS-Numbers: 05.45.Ac, 61.43.Hv, 02.50.Ey
Keywords and phrases: Dynamical Systems, random perturbation, stochastic stability
In this chapter we discuss random perturbations and their effect on dynamical systems. We focus on discrete time dynamics and present different ways of implementing the random dynamics, namely the dynamics of random uncorrelated noise and the dynamics of random maps. We discuss some applications in scattering and in escaping from attracting sets. As we shall see, the perturbations may dramatically change the asymptotic behaviour of these systems. In particular, in randomly perturbed non-hyperbolic scattering trajectories may escape from regions where otherwise they are expected to be trapped forever. The dynamics also gains hyperbolic-like characteristics. These are observed in the decay of survival probability as well as in the fractal dimension of singular sets. In addition, we show that random perturbations also trigger escape from attracting sets, giving rise to transport among basins. Along the chapter, we motivate the application of such processes. We finish by suggesting some possible further applications.