Toward an understanding of stochastic Hopf bifurcation: A case study

In this paper asymptotic and numerical methods are used to study the phenomenon of stochastic Hopf bifurcation. The analysis is carried out through studying a noisy Duffing-van der Pol oscillator which exhibits a Hopf bifurcation in the absence of noise as one of the parameters is varied. In the first part of this paper, we present briefly various concepts that are essential to describe stochastic bifurcations. We also present the definitions of P-Bifurcation and D-Bifurcation and illustrate these concepts through a one-dimensional example. In the second part of this paper, we construct an asymptotic expansion for the maximal Lyapunov exponent, the exponential growth rate of solutions to a linear stochastic system, and the moment Lyapunov exponents for a two-dimensional dynamical system driven by a small intensity real noise process. The nonlinear analysis is performed using both the method of stochastic averaging and stochastic normal forms. In this study, stochastic bifurcation implies either qualitative changes to the invariant measures which can be observed by examining the Fokker-Planck equation, or the appearance of a new invariant measure which is, at present, generated numerically through the forward and backward solutions of the stochastic differential equations. For a detailed study of this topic the reader is referred to Arnold et al.