Starting from the classical Van der Pol equation, after suitable changes of variables, we derive a reaction-diffusion type forcedVan der Pol equation with values in a suitable infinite dimensionalHilbert space. In particular we will perturb previous equation with a small additive Brownian noise. After some preliminary results concerning the non trivial existence and uniqueness of a solution, due to the presence of anon-Lipschitz non-linearity, we provide a rigorous asymptoticexpansions in term of the small parameter ε of the related solution up to order 3. We will then explicitly write the first threeorder of the rigorous expansion and provide as well an upperbound for the remainder.

Small Noise Asymptotic Expansion for a Infinite Dimensional Stochastic Reaction-Diffusion Forced Van Der Pol Equation

Francesco Cordoni;DI PERSIO, Luca
2015-01-01

Abstract

Starting from the classical Van der Pol equation, after suitable changes of variables, we derive a reaction-diffusion type forcedVan der Pol equation with values in a suitable infinite dimensionalHilbert space. In particular we will perturb previous equation with a small additive Brownian noise. After some preliminary results concerning the non trivial existence and uniqueness of a solution, due to the presence of anon-Lipschitz non-linearity, we provide a rigorous asymptoticexpansions in term of the small parameter ε of the related solution up to order 3. We will then explicitly write the first threeorder of the rigorous expansion and provide as well an upperbound for the remainder.
2015
asymptotic expansions; Van der Pol oscillator; Stochastic reaction- diffusion systems
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11562/909983
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