Stochastic bifurcation of a three-dimensional stochastic Kolmogorov system
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| Format: | Preprint |
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2024
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| _version_ | 1866914899413696512 |
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| author | Xiao, Dongmei Zhang, Deng Zhou, Chenwan |
| author_facet | Xiao, Dongmei Zhang, Deng Zhou, Chenwan |
| contents | In this paper we systematically investigate the stochastic bifurcations of both ergodic stationary measures and global dynamics for stochastic Kolmogorov differential systems, which relate closely to the change of the sign of Lyapunov exponents. It is derived that there exists a threshold $σ_0$ such that, if the noise intensity $σ\geqσ_0$, the noise destroys all bifurcations of the deterministic system and the corresponding stochastic Kolmogorov system is uniquely ergodic. On the other hand, when the noise intensity $σ<σ_0$, the stochastic system undergoes bifurcations from the unique ergodic stationary measure to three different types of ergodic stationary measures: (I) finitely many ergodic measures supported on rays, (II) infinitely many ergodic measures supported on rays, (III) infinitely many ergodic measures supported on invariant cones. Correspondingly, the global dynamics undergo similar bifurcation phenomena, which even displays infinitely many Crauel random periodic solutions in the sense of \cite{ELR21}. Furthermore, we prove that as $σ$ tends to zero, the ergodic stationary measures converge to either Dirac measures supported on equilibria, or to Haar measures supported on non-trivial deterministic periodic orbits. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_01560 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stochastic bifurcation of a three-dimensional stochastic Kolmogorov system Xiao, Dongmei Zhang, Deng Zhou, Chenwan Dynamical Systems Probability 60H10, 37G35, 37H15, 34F05 In this paper we systematically investigate the stochastic bifurcations of both ergodic stationary measures and global dynamics for stochastic Kolmogorov differential systems, which relate closely to the change of the sign of Lyapunov exponents. It is derived that there exists a threshold $σ_0$ such that, if the noise intensity $σ\geqσ_0$, the noise destroys all bifurcations of the deterministic system and the corresponding stochastic Kolmogorov system is uniquely ergodic. On the other hand, when the noise intensity $σ<σ_0$, the stochastic system undergoes bifurcations from the unique ergodic stationary measure to three different types of ergodic stationary measures: (I) finitely many ergodic measures supported on rays, (II) infinitely many ergodic measures supported on rays, (III) infinitely many ergodic measures supported on invariant cones. Correspondingly, the global dynamics undergo similar bifurcation phenomena, which even displays infinitely many Crauel random periodic solutions in the sense of \cite{ELR21}. Furthermore, we prove that as $σ$ tends to zero, the ergodic stationary measures converge to either Dirac measures supported on equilibria, or to Haar measures supported on non-trivial deterministic periodic orbits. |
| title | Stochastic bifurcation of a three-dimensional stochastic Kolmogorov system |
| topic | Dynamical Systems Probability 60H10, 37G35, 37H15, 34F05 |
| url | https://arxiv.org/abs/2408.01560 |