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Research Papers

Probabilistic Analysis of Bifurcations in Stochastic Nonlinear Dynamical Systems

[+] Author and Article Information
Ehsan Mirzakhalili

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: mirzakh@umich.edu

Bogdan I. Epureanu

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: epureanu@umich.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 20, 2018; final manuscript received April 22, 2019; published online June 10, 2019. Assoc. Editor: Zaihua Wang.

J. Comput. Nonlinear Dynam 14(8), 081009 (Jun 10, 2019) (14 pages) Paper No: CND-18-1563; 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 (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.

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Figures

Grahic Jump Location
Fig. 1

(a) Time evolution of stochastic system governed by Eq. (23). The system oscillates between two extrema located atx=±1 and lingers near each one of them. The system also lingers near x=0, between the two extrema. (b) The PDF for system governed by Eq. (23) obtained from all three methods matches well. The PDF shows that the likelihood of finding the system is equal and largest at the two extrema. The Monte Carlo simulation has been computed using 106 samples with initial conditions evenly distributed between values of −2 and 2. A time-step of 0.05 was used, and the final state of each sample after 1000 time steps was stored and used for plotting.

Grahic Jump Location
Fig. 2

Analytic and numerical solution of the FPE for the dynamical system of Eq. (23) in the presence of additive noise (σ=0.5) matches well. Black lines show the stable branches and white lines correspond to unstable branches of the deterministic bifurcation diagram.

Grahic Jump Location
Fig. 3

Probabilistic bifurcation diagram of the dynamical system of Eq. (23) in the presence of additive noise ((a) and (b)) and multiplicative noise. G(x)=σ(1+x2) for ((c) and (d)) and G(x)=σ(1+x−2) for ((e) and (f)). Black lines correspond to stable branches and white lines to unstable branches in the deterministic bifurcation diagram. The colorbars are removed because the values of the normalized PDF vary between 0 and 1 for all cases.

Grahic Jump Location
Fig. 4

Probabilistic phase planes of FHN model when the external current has white noise I=I0+N(0,σ2) with σ2=0.0025

Grahic Jump Location
Fig. 5

Probabilistic bifurcation digrams of FHN model in the presence of additive noise ((a)–(d)) and multiplicative noise ((e)–(h)). The additive noise is associated with white noise in the injected current in Eq. (29). The multiplicative noise is due to white noise in the membrane voltage that leads to Eqs. (37) and (38). Black lines correspond to stable branches, white lines to unstable branches, and gray lines to extrema of limit cycles in the deterministic bifurcation diagram. The colorbars are omitted because the values of the normalized PDF vary between 0 and 1 for all cases.

Grahic Jump Location
Fig. 6

Probabilistic phase planes of Lorenz system in the presence of additive noise when ρ=15. The two-dimensional projections show MPDFs.

Grahic Jump Location
Fig. 7

Probabilistic bifurcation digrams of Lorenz equations in the presence of additive noise ((a)–(c)) and multiplicative noise ((d)–(f)). The multiplicative noise is due to white noise in the parameters. Black lines correspond to stable branches, white lines to unstable branches, and gray lines to extrema of limit cycles in the deterministic bifurcation diagram. The colorbars are omitted because the values of the normalized PDF vary between 0 and 1 for all cases.

Grahic Jump Location
Fig. 8

Probabilistic phase planes of coupled Brusselators in the presence of noise when B=2.5. The two-dimensional projections show MPDFs.

Grahic Jump Location
Fig. 9

Probabilistic bifurcation digrams of coupled Brusselators in the presence of noise. The noise is due to white noise in A and B. Black lines correspond to stable branches, white lines to unstable branches, and gray lines to extrema of limit cycles in the deterministic bifurcation diagram. The colorbars are removed because the values of the normalized PDF vary between 0 and 1 for all cases.

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