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A random dynamical systems perspective on stochastic resonance. (English) Zbl 1379.37097
A 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.

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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