Probabilistic analysis of bifurcations in stochastic nonlinear dynamical systems

[+] Author and Article Information
Ehsan Mirzakhalili

Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA

Bogdan I. Epureanu

Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA

1Corresponding author.

ASME doi:10.1115/1.4043669 History: Received December 20, 2018; Revised April 22, 2019


Bifurcation diagrams are limited most often to deterministic dynamical systems. However, stochastic dynamics can substantially affect the interpretation of such diagrams because the deterministic diagram often is not simply the mean of the probabilistic diagram. We present an approach based on the Fokker-Planck equation to obtain probabilistic bifurcation diagrams for stochastic nonlinear dynamical systems. We propose a systematic approach to expand the analysis of nonlinear and linear dynamical systems from deterministic to stochastic when the states or the parameters of the system are noisy. We find stationary solutions of the Fokker-Planck equation numerically. Then, marginal probability density function (MPDF) is used to track changes in the shape of probability distributions as well as determining the probability of finding the system at each point on the bifurcation diagram. Hence, we explore how the deterministic bifurcation diagrams of dynamical systems of different dimensions are affected by noise. For example, we show that additive noise can lead to an earlier bifurcation in one-dimensional subcritical pitchfork bifurcation. We further show that multiplicative noise can have dramatic changes such as changing one-dimensional subcritical pitchfork bifurcations into supercritical pitchfork bifurcations or annihilating the bifurcation altogether. We demonstrate how the joint probability density function can show the presence of limit cycles in the FitzHugh-Nagumo neuron model or chaotic behavior in the Lorenz system. We study coupled Brusselators to show how our approach can be used to construct bifurcation diagrams for higher dimensional systems.

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