1. Stochastic generalized Kolmogorov systems with small diffusion: II. Explicit approximations for periodic solutions in distribution
- Author
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Zhou, Baoquan, Wang, Hao, Wang, Tianxu, and Jiang, Daqing
- Subjects
Mathematics - Dynamical Systems ,37H05, 37H30, 45M15, 60H10 - Abstract
This paper is Part II of a two-part series on coexistence states study in stochastic generalized Kolmogorov systems under small diffusion. Part I provided a complete characterization for approximating invariant probability measures and density functions, while here, we focus on explicit approximations for periodic solutions in distribution. Two easily implementable methods are introduced: periodic normal approximation (PNOA) and periodic log-normal approximation (PLNA). These methods offer unified algorithms to calculate the mean and covariance matrix, and verify positive definiteness, without additional constraints like non-degenerate diffusion. Furthermore, we explore essential properties of the covariance matrix, particularly its connection under periodic and non-periodic drift coefficients. Our new approximation methods significantly relax the minimal criteria for positive definiteness of the solution of the discrete-type Lyapunov equation. Some numerical experiments are provided to support our theoretical results., Comment: 39 pages, 5 figures
- Published
- 2024