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Effects of Bounded Random Perturbations on Discrete Dynamical Systems

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Bounded Noises in Physics, Biology, and Engineering

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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.

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Notes

  1. 1.

    More generally one should consider M to be a manifold.

  2. 2.

    Within the mathematical literature the random perturbations are defined in terms of spaces of maps. In this setting, we have a family of maps and the iteration is obtained by randomly selecting them. Thus it is said to be a family of random maps even when different sequences are applied to different orbits.

  3. 3.

    Because scattering dynamics are so closely related to scattering of particles in physical systems, we shall refer to the dynamics of initial conditions in a region of the phase space as dynamics of particles started in such region.

  4. 4.

    The term subdynamics is used here as a simplification of the splitting of the tangent bundle. See, for example, [4, 6].

  5. 5.

    In the case of discrete time dynamical systems, t ≡ n, the number of iterations.

  6. 6.

    We say artificially placed hole when the hole is defined as a region [35–37]. Our intention is just to contrast to the case discussed in [20] and presented in Sect. 10.4, where the hole is given by the random perturbations.

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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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Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, Inselstr., 22, 04103, Leipzig, Germany

    Christian S. Rodrigues

  2. Department of Physics and Institute for Complex Systems and Mathematical Biology, King’s College, University of Aberdeen, Aberdeen, AB24 3UE, UK

    Alessandro P. S. de Moura & Celso Grebogi

Authors
  1. Christian S. Rodrigues
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  2. Alessandro P. S. de Moura
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  3. Celso Grebogi
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Correspondence to Christian S. Rodrigues .

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  1. Department of Experimental Oncology, European Institute of Oncology, Milan, Italy

    Alberto d'Onofrio

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Rodrigues, C.S., de Moura, A.P.S., Grebogi, C. (2013). Effects of Bounded Random Perturbations on Discrete Dynamical Systems. In: d'Onofrio, A. (eds) Bounded Noises in Physics, Biology, and Engineering. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-7385-5_10

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