Applications of the Poincaré--Hopf Theorem: Epidemic Models and Lotka--Volterra Systems
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2019
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| Acceso en línea: | |
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| _version_ | 1866910661390368768 |
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| author | Ye, Mengbin Liu, Ji Anderson, Brian D. O. Cao, Ming |
| author_facet | Ye, Mengbin Liu, Ji Anderson, Brian D. O. Cao, Ming |
| contents | This paper focuses on the equilibria and their regions of attraction for continuous-time nonlinear dynamical systems. The classical Poincaré--Hopf Theorem from differential topology is used to derive a result on a sufficient condition on the Jacobian for the existence of a unique equilibrium for the system, which is in fact locally exponentially stable. We show how to apply this result to the deterministic SIS networked model, and a nonlinear Lotka--Volterra system. We apply the result further to establish an impossibility conclusion for a class of distributed feedback controllers whose goal is to drive the SIS network to the zero equilibrium. Specifically, we show that if the uncontrolled system has a unique nonzero equilibrium (a diseased steady-state), then the controlled system also has a unique nonzero equilibrium. Applying results from monotone dynamical systems theory, we further show that both the uncontrolled and controlled system will converge to their nonzero equilibrium from all nonzero initial conditions exponentially fast. A counterpart sufficient condition for the existence of a unique equilibrium for a nonlinear discrete-time dynamical system is also presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1911_12985 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Applications of the Poincaré--Hopf Theorem: Epidemic Models and Lotka--Volterra Systems Ye, Mengbin Liu, Ji Anderson, Brian D. O. Cao, Ming Dynamical Systems Systems and Control This paper focuses on the equilibria and their regions of attraction for continuous-time nonlinear dynamical systems. The classical Poincaré--Hopf Theorem from differential topology is used to derive a result on a sufficient condition on the Jacobian for the existence of a unique equilibrium for the system, which is in fact locally exponentially stable. We show how to apply this result to the deterministic SIS networked model, and a nonlinear Lotka--Volterra system. We apply the result further to establish an impossibility conclusion for a class of distributed feedback controllers whose goal is to drive the SIS network to the zero equilibrium. Specifically, we show that if the uncontrolled system has a unique nonzero equilibrium (a diseased steady-state), then the controlled system also has a unique nonzero equilibrium. Applying results from monotone dynamical systems theory, we further show that both the uncontrolled and controlled system will converge to their nonzero equilibrium from all nonzero initial conditions exponentially fast. A counterpart sufficient condition for the existence of a unique equilibrium for a nonlinear discrete-time dynamical system is also presented. |
| title | Applications of the Poincaré--Hopf Theorem: Epidemic Models and Lotka--Volterra Systems |
| topic | Dynamical Systems Systems and Control |
| url | https://arxiv.org/abs/1911.12985 |