Noise-induced phenomena in slow-fast dynamical systems. A sample-paths approach.

*(English)*Zbl 1098.37049
Probability and its Applications. London: Springer (ISBN 1-84628-038-9/hbk; 1-84628-186-5/ebook). xiv, 276 p. (2006).

The book concerns the particular class of Markov processes which are solutions to systems of stochastic differential equations. These systems are derived by adding noise to slow-fast system of ordinary differential equations. In the book, an approach to slow-fast stochastic differential equations based on a characterization of typical sample paths is developed. It is proposed that the basic determinate dynamics is sufficiently well known. The main results on deterministic slow-fast systems concerning stable slow manifolds, dynamical bifurcations and associated systems admitting stable periodic orbits are presented. In the study of the effect of noise, the one-dimensional slow-fast systems with slowly varying parameters are considered firstly. The dynamical saddle-node, pitchwork and transcritical bifurcations with noise are discussed in detail. Further, such systems which display stochastic resonance, are examined. The results obtained are generalized to the case of multidimensional, fully coupled slow-fast systems with noise. There are proved results on the concentration of sample paths in an explicitly constructed neighborhood of the manifold.

Reviewer: Elena Ya. Gorelova (Samara)

##### MSC:

37H10 | Generation, random and stochastic difference and differential equations |

37H20 | Bifurcation theory for random and stochastic dynamical systems |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

34F05 | Ordinary differential equations and systems with randomness |

70K70 | Systems with slow and fast motions for nonlinear problems in mechanics |

70K30 | Nonlinear resonances for nonlinear problems in mechanics |

37N05 | Dynamical systems in classical and celestial mechanics |

37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |