Bifurcation Behavior of a Nonlinear System by Introducing Noise

Abstract

In this chapter, we unveil a useful facet of noise: we show how one may obtain the bifurcation behavior of a nonlinear dynamical system by introducing noise into its dynamics. This is done by adding a Gaussian, white noise term to the dynamics and then studying the resulting Langevin dynamics in the weak-noise limit. We show that the behavior of the noisy dynamics can be effectively captured via a conditional probability of observing microscopic configurations at a given instant, conditioned on having observed a given configuration at an earlier instant. We study this conditional probability for our model system by using two distinct approaches, namely, the Fokker-Planck and the path integral approach. An exact closed-form expression for the conditional probability can be obtained within the latter approach. By studying the long-time behavior of the conditional probability in the weak-noise limit, we show how to reconstruct the bifurcation diagram of the noiseless dynamics. All our predictions are validated with direct numerical integration of the dynamical equations of motion. This chapter is based on our published work, Ref. [1].