Abstract
In this paper, the stability and bifurcation of a class of two-dimensional stochastic differential equations with multiplicative excitations are investigated. Firstly, we employ Taylor expansions, polar coordinate transformation and stochastic averaging method to transform the original system into an Itô averaging diffusion system. Secondly, we apply the maximum Lyapunov exponent and the singular boundary theory to analyze the local and global stability of the fixed point. Thirdly, we explore the stochastic dynamical bifurcation of the Itô averaging amplitude equation by studying the qualitative changes of invariant measures, and investigate the phenomenological bifurcation by utilizing Fokker–Planck equation. Finally, an example is given to illustrate the effectiveness of our analyzing procedure.
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Acknowledgments
The research was supported by the NSFC (Grant No. 11271115), by the Doctoral Fund of Ministry of Education of China, and by the Hunan Provincial Natural Science Foundation (Grant No. 14JJ1025).
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Communicated by Rosihan M. Ali.
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Luo, C., Guo, S. Stability and Bifurcation of Two-dimensional Stochastic Differential Equations with Multiplicative Excitations. Bull. Malays. Math. Sci. Soc. 40, 795–817 (2017). https://doi.org/10.1007/s40840-016-0313-7
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DOI: https://doi.org/10.1007/s40840-016-0313-7
Keywords
- Stochastic averaging
- Lyapunov exponent
- Singular boundary theory
- Stochastic stability
- Stochastic bifurcation