Stochastic bifurcations and transient dynamics of probability responses with radial basis function neural networks

摘要

Stochastic noise is common in many engineering systems. Stochastic noise and nonlinearity of dynamical systems will lead to the occurrence of complex dynamic phenomena, such as stochastic bifurcations and transitions. In this paper, a radial basis function neural networks (RBF-NN) method is applied to study stochastic bifurcations and transient dynamics of probability response. A Duffing oscillator under additive and multiplicative random noise is presented to show the effectiveness of the proposed solution method. A new type of stochastic bifurcations is found with an increase of system control parameter, which is called a stochastic double P-bifurcation. It occurs when a stationary probability density function (PDF) changes from a single peak to another single one through two double peaks with their maximum exchange. It is revealed that such a double P-bifurcation is linked to two catastrophic bifurcations of its deterministic counterpart. Moreover, these different types of transient evolutions of the PDFs are observed with transitions of probability maximum peak to one, and to the other, as well as to the both of stable invariant sets from different initial PDFs. The evolutionary orientation of transient PDFs aligns with unstable invariant manifolds leading to stable invariant sets. The previous conjecture is confirmed that the noise-induced transient dynamics and bifurcations are dominated by the global topology and its sudden changes of corresponding deterministic systems without noise.