Largest Lyapunov Exponents and Bifurcations of Stochastic Nonlinear Systems

Two commonly adopted expressions for the largest Lyapunov exponents of linearized stochastic systems are reviewed. Their features are discussed in light of bifurcation analysis and one expression is selected for evaluating the largest Lyapunov exponent of a linearized system. An independent method, developed earlier by the authors, is also applied to determine the bifurcation points of a van der Pol oscillator under parametric random excitation. It is shown that the bifurcation points obtained by the independent technique agree qualitatively and quantitatively with those evaluated by using the largest Lyapunov exponent of the linearized oscillator.