On Asymptotic Measure of Quasi-Potential Systems with Applications to Stochastic Bifurcations

Abstract. The asymptotic measure of stationary measures for a gradient system undersmall additive noise perturbation is concentrated on the global minima set of the po-tential function. Hwang (1980) and Huang et al.(2016) determined the weights of theasymptotic measure supported on equilibria if the global minima set consists of only fi-nite points at which the first nonzero homogeneous polynomials of Taylor expansions areeven order. This talk will focus on the asymptotic measure problem of a quasi-potentialsystem which is the orthogonal sum of a gradient system and a divergence-free system. Itis given that stationary measures of the gradient system and the quasi-potential systemunder the same additive noise perturbation admit the same density. We also provideorthogonal group invariance criterion of the density, which helps us to determine thesupporting components' weights. Combining these with the global dynamics of quasi-potential system, we give exact asymptotic measure and its support, including stableequilibria, stable periodic orbits, saddles, and chaotic motions et al., of a large numberof quasi-potential systems, which are used to study stochastic bifurcations of two or threedimensional quasi-potential systems. Finally, we present Freidlin and Wentzell's methodto compute the transitive difficulty matrix and apply it to determine the asymptoticmeasure and asymptotic limits of solutions of the Cauchy problem for reaction-diffusionequation with small diffusion coefficient.

This is a joint work with Chen Lifeng.