# A Volterra-series approach to stochastic nonlinear dynamics: The Duffing oscillator driven by white noise

## TL;DR

This paper develops a Volterra-series based method to analyze and infer parameters of the stochastic Duffing oscillator driven by white noise, enabling accurate reconstruction of its dynamics from observed time series.

## Contribution

It introduces a novel analytical and statistical inference framework for the stochastic Duffing oscillator, extending techniques used for linear models to nonlinear bistable systems.

## Key findings

- Accurately reconstructs Duffing oscillator parameters from simulated data.
- Provides theoretical formulas for the statistics of stochastic nonlinear oscillations.
- Demonstrates applicability to experimental time series of bistable systems.

## Abstract

The Duffing oscillator is a paradigm of bistable oscillatory motion in physics, engineering, and biology. Time series of such oscillations are often observed experimentally in a nonlinear system excited by a spontaneously fluctuating force. One is then interested in estimating effective parameter values of the stochastic Duffing model from these observations--a task that has not yielded to simple means of analysis. To this end we derive theoretical formulas for the statistics of the Duffing oscillator's time series. Expanding on our analytical results, we introduce methods of statistical inference for the parameter values of the stochastic Duffing model. By applying our method to time series from stochastic simulations, we accurately reconstruct the underlying Duffing oscillator. This approach is quite straightforward--similar techniques are used with linear Langevin models--and can be applied to time series of bistable oscillations that are frequently observed in experiments.