Stochastic bifurcation

This investigation is concerned with the dynamic response, stability, and bifurcation behavior of nonlinear dynamical systems under stochastic excitation. The excitation may be either multiplicative or additive or a combination of both. The effect of small stochastic perturbations on systems that exhibit codimension-one and -two bifurcations is examined. The asymptotic behavior of nonlinear dynamical systems in the presence of noise is studied using both the methods of stochastic averaging and stochastic normal forms. It is shown that for systems with rapidly oscillating and decaying components, these techniques yield a set of equations of considerably smaller dimension. The Markov diffusion approximation is used to obtain analytical results relating to the statistical properties of the stochastic response. Both moment and sample stability conditions along with the stationary moments are obtained. For the reduced nonlinear systems, stationary and transient probability densities are found. It is shown that in nonlinear systems a shift of the bifurcation point takes place due to the presence of noise.