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

## TL;DR

This paper extends a Volterra-series analytical method to study the linear response of the Van der Pol oscillator driven by white noise, analyzing its statistics and proposing parameter estimation techniques using experimental biological data.

## Contribution

It introduces a novel application of Volterra-series techniques to the stochastic Van der Pol oscillator, enabling parameter estimation from noisy biological oscillation data.

## Key findings

- Analytic characterization of the stochastic Van der Pol oscillator's response.
- A method for estimating model parameters from noisy time series.
- Application demonstrated on biological oscillation data.

## Abstract

The Van der Pol equation is a paradigmatic model of relaxation oscillations. This remarkable nonlinear phenomenon of self-sustained oscillatory motion underlies important rhythmic processes in nature and electrical engineering. Relaxation oscillations in a real system are usually coupled to environmental noise, which further enriches their dynamics, but makes theoretical analysis of such systems and determination of the equation's parameter values a difficult task. In a companion paper we have proposed an analytic approach to a similar problem for another classical nonlinear model, the bistable Duffing oscillator. Here we extend our techniques to the case of the Van der Pol equation driven by white noise. We analyze the statistics of solutions and propose a method to estimate parameter values from the oscillator's time series. We use experimental data of active oscillations in a biological system to demonstrate how our method applies to real observations and how it can be generalized for more complex models.