Instability of small-amplitude periodic waves from fold-Hopf bifurcation
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866911922574589952 |
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| author | Chen, Shuang Duan, Jinqiao |
| author_facet | Chen, Shuang Duan, Jinqiao |
| contents | We study the existence and stability of small-amplitude periodic waves emerging from fold-Hopf equilibria in a system of one reaction-diffusion equation coupled with one ordinary differential equation. This coupled system includes the FitzHugh-Nagumo system, caricature calcium models, consumer-resource models and other models in the real-world applications. Based on the recent results on the averaging theory, we solve periodic solutions in related three-dimensional systems and then prove the existence of periodic waves arising from fold-Hopf bifurcations. Numerical computation in [J. Tsai, W. Zhang, V. Kirk, and J. Sneyd, SIAM J. Appl. Dyn. Syst. 11 (2012), 1149--1199] once suggested that the periodic waves from fold-Hopf bifurcations in a caricature calcium model are spectrally unstable, yet without a proof. After analyzing the linearization about periodic waves by the relatively bounded perturbation, we prove the instability of small-amplitude periodic waves through a perturbation of the unstable spectra for the linearizations about the fold-Hopf equilibria. As an application, we prove the existence and stability of small-amplitude periodic waves from fold-Hopf bifurcations in the FitzHugh-Nagumo system with an applied current. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_07484 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Instability of small-amplitude periodic waves from fold-Hopf bifurcation Chen, Shuang Duan, Jinqiao Dynamical Systems We study the existence and stability of small-amplitude periodic waves emerging from fold-Hopf equilibria in a system of one reaction-diffusion equation coupled with one ordinary differential equation. This coupled system includes the FitzHugh-Nagumo system, caricature calcium models, consumer-resource models and other models in the real-world applications. Based on the recent results on the averaging theory, we solve periodic solutions in related three-dimensional systems and then prove the existence of periodic waves arising from fold-Hopf bifurcations. Numerical computation in [J. Tsai, W. Zhang, V. Kirk, and J. Sneyd, SIAM J. Appl. Dyn. Syst. 11 (2012), 1149--1199] once suggested that the periodic waves from fold-Hopf bifurcations in a caricature calcium model are spectrally unstable, yet without a proof. After analyzing the linearization about periodic waves by the relatively bounded perturbation, we prove the instability of small-amplitude periodic waves through a perturbation of the unstable spectra for the linearizations about the fold-Hopf equilibria. As an application, we prove the existence and stability of small-amplitude periodic waves from fold-Hopf bifurcations in the FitzHugh-Nagumo system with an applied current. |
| title | Instability of small-amplitude periodic waves from fold-Hopf bifurcation |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2012.07484 |