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The Van der Pol equation is a paradigmatic model of relaxation oscillations.This remarkable nonlinear phenomenon of self-sustained oscillatory motionunderlies important rhythmic processes in nature and electrical engineering.Relaxation oscillations in a real system are usually coupled to environmentalnoise, which further enriches their dynamics, but makes theoretical analysis ofsuch systems and determination of the equation's parameter values a difficulttask. In a companion paper we have proposed an analytic approach to a similarproblem for another classical nonlinear model, the bistable Duffing oscillator.Here we extend our techniques to the case of the Van der Pol equation driven bywhite noise. We analyze the statistics of solutions and propose a method toestimate parameter values from the oscillator's time series. We useexperimental data of active oscillations in a biological system to demonstratehow our method applies to real observations and how it can be generalized formore complex models.

Volterra-series approach to stochastic nonlinear dynamics: linear response of the Van der Pol oscillator driven by white noise