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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.09901 |
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| _version_ | 1866910526846533632 |
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| author | Zhou, Baoquan Wang, Hao Wang, Tianxu Jiang, Daqing |
| author_facet | Zhou, Baoquan Wang, Hao Wang, Tianxu Jiang, Daqing |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09901 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stochastic generalized Kolmogorov systems with small diffusion: II. Explicit approximations for periodic solutions in distribution Zhou, Baoquan Wang, Hao Wang, Tianxu Jiang, Daqing Dynamical Systems 37H05, 37H30, 45M15, 60H10 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. |
| title | Stochastic generalized Kolmogorov systems with small diffusion: II. Explicit approximations for periodic solutions in distribution |
| topic | Dynamical Systems 37H05, 37H30, 45M15, 60H10 |
| url | https://arxiv.org/abs/2407.09901 |