Relaxation Oscillations and the Entry-Exit Function in Multi-Dimensional Slow-Fast Systems
For a slow-fast system of the form $\dot{p}=\epsilon f(p,z,\epsilon)+h(p,z,\epsilon)$, $\dot{z}=g(p,z,\epsilon)$ for $(p,z)\in \mathbb R^n\times \mathbb R^m$, we consider the scenario that the system has invariant sets $M_i=\{(p,z): z=z_i\}$, $1\le i\le N$, linked by a singular closed orbit formed b...
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| Main Authors | , |
|---|---|
| Format | Journal Article |
| Language | English |
| Published |
14.10.2019
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| Subjects | |
| Online Access | Get full text |
| DOI | 10.48550/arxiv.1910.06318 |
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| Abstract | For a slow-fast system of the form $\dot{p}=\epsilon
f(p,z,\epsilon)+h(p,z,\epsilon)$, $\dot{z}=g(p,z,\epsilon)$ for $(p,z)\in
\mathbb R^n\times \mathbb R^m$, we consider the scenario that the system has
invariant sets $M_i=\{(p,z): z=z_i\}$, $1\le i\le N$, linked by a singular
closed orbit formed by trajectories of the limiting slow and fast systems.
Assuming that the stability of $M_i$ changes along the slow trajectories at
certain turning points, we derive criteria for the existence and stability of
relaxation oscillations for the slow-fast system. Our approach is based on a
generalization of the entry-exit relation to systems with multi-dimensional
fast variables. We then apply our criteria to several predator-prey systems
with rapid ecological evolutionary dynamics to show the existence of relaxation
oscillations in these models. |
|---|---|
| AbstractList | For a slow-fast system of the form $\dot{p}=\epsilon
f(p,z,\epsilon)+h(p,z,\epsilon)$, $\dot{z}=g(p,z,\epsilon)$ for $(p,z)\in
\mathbb R^n\times \mathbb R^m$, we consider the scenario that the system has
invariant sets $M_i=\{(p,z): z=z_i\}$, $1\le i\le N$, linked by a singular
closed orbit formed by trajectories of the limiting slow and fast systems.
Assuming that the stability of $M_i$ changes along the slow trajectories at
certain turning points, we derive criteria for the existence and stability of
relaxation oscillations for the slow-fast system. Our approach is based on a
generalization of the entry-exit relation to systems with multi-dimensional
fast variables. We then apply our criteria to several predator-prey systems
with rapid ecological evolutionary dynamics to show the existence of relaxation
oscillations in these models. |
| Author | Hsu, Ting-Hao Ruan, Shigui |
| Author_xml | – sequence: 1 givenname: Ting-Hao surname: Hsu fullname: Hsu, Ting-Hao – sequence: 2 givenname: Shigui surname: Ruan fullname: Ruan, Shigui |
| BackLink | https://doi.org/10.48550/arXiv.1910.06318$$DView paper in arXiv |
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| Snippet | For a slow-fast system of the form $\dot{p}=\epsilon
f(p,z,\epsilon)+h(p,z,\epsilon)$, $\dot{z}=g(p,z,\epsilon)$ for $(p,z)\in
\mathbb R^n\times \mathbb R^m$,... |
| SourceID | arxiv |
| SourceType | Open Access Repository |
| SubjectTerms | Mathematics - Dynamical Systems |
| Title | Relaxation Oscillations and the Entry-Exit Function in Multi-Dimensional Slow-Fast Systems |
| URI | https://arxiv.org/abs/1910.06318 |
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