Probabilistic Analysis of Bifurcations in Stochastic Nonlinear Dynamical Systems

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 (FPE) 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 FPE 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. Using MPDFs is necessary for multidimensional dynamical systems and allows direct visual comparison of deterministic bifurcation diagrams with the proposed probabilistic bifurcation diagrams. Hence, we explore how the deterministic bifurcation diagrams of different 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 (1D) subcritical pitchfork bifurcation. We further show that multiplicative noise can have dramatic changes such as changing 1D subcritical pitchfork bifurcations into supercritical pitchfork bifurcations or annihilating the bifurcation altogether. We demonstrate how the joint probability density function (PDF) can show the presence of limit cycles in the FitzHugh–Nagumo (FHN) neuron model or chaotic behavior in the Lorenz system. Moreover, we reveal that the Lorenz system has chaotic behavior earlier in the presence of noise. We study coupled Brusselators to show how our approach can be used to construct bifurcation diagrams for higher dimensional systems.