A random dynamical systems perspective on stochastic resonance.

*(English)*Zbl 1379.37097A random dynamical systems approach is used to prove that the nonautonomous stochastic differential equation (NSDE)
\[
dx= (\alpha x-\beta x^3)\,dt+ A\cos\nu\,dt+ \sigma dW_t,
\]
where \(\alpha, \beta,\sigma>0\) and \(W_t\) is a Wiener process, has a unique globally attracting random periodic orbit. Stochastic resonances are discussed. General theorems that provide existence of global nonautonomous random attractors and random periodic orbits of continuous-time nonautonomous random dynamical systems generated by the NSDE \(dx= f(t,x)\,dt+\sigma dW_t\), \(\sigma>0\), are proved.

Reviewer: Melvin D. Lax (Long Beach)

##### MSC:

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

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

34F05 | Ordinary differential equations and systems with randomness |

34F15 | Resonance phenomena for ordinary differential equations involving randomness |

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