Reconstructing bifurcation behavior of a nonlinear dynamical system by introducing weak noise

TL;DR

This paper demonstrates how introducing weak noise into a nonlinear dynamical system allows for the reconstruction of its bifurcation behavior by analyzing the conditional probability of configurations, using Fokker-Planck and path-integral methods.

Contribution

It introduces a novel approach to identify bifurcation behavior in nonlinear systems through weak noise analysis and provides exact solutions using path-integral methods.

Findings

Conditional probability captures bifurcation behavior.

Path-integral approach yields exact expressions.

Predictions match numerical simulations.

Abstract

For a model nonlinear dynamical system, we show how one may obtain its bifurcation behavior by introducing noise into the dynamics and then studying the resulting Langevin dynamics in the weak-noise limit. A suitable quantity to capture the bifurcation behavior in the noisy dynamics is the conditional probability to observe a microscopic configuration at one time, conditioned on the observation of a given configuration at an earlier time. For our model system, this conditional probability is studied by using two complementary approaches, the Fokker-Planck and the path-integral approach. The latter has the advantage of yielding exact closed-form expressions for the conditional probability. All our predictions are in excellent agreement with direct numerical integration of the dynamical equations of motion.