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Effects of Bounded Random Perturbations on Discrete Dynamical Systems

  • Christian S. Rodrigues
  • Alessandro P. S. de Moura
  • Celso Grebogi
Chapter
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Abstract

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.

Keywords

Bounded noises Discrete-time dinamical systems Random perturbations Escape from attracting sets Fractal dimension 

Notes

Acknowledgements

C.S.R. is grateful to J. Jost, R. Klages, M. Kell, J. Lamb, M. Rasmussen, and P. Ruffino for inspiring discussions along these subprojects and acknowledges the financial support from the University of Aberdeen and from the Max-Planck Society.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Christian S. Rodrigues
    • 1
  • Alessandro P. S. de Moura
    • 2
  • Celso Grebogi
    • 2
  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Department of Physics and Institute for Complex Systems and Mathematical Biology, King’s CollegeUniversity of AberdeenAberdeenUK

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