Applications of the Poincare-Hopf Theorem: Epidemic Models and Lotka-Volterra Systems
This paper focuses on properties of equilibria and their associated regions of attraction for continuous-time nonlinear dynamical systems. The classical Poincar\'e--Hopf Theorem is used to derive a general result providing a sufficient condition for the system to have a unique equilibrium. The...
| Main Authors: | , , , |
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| Format: | Journal Article |
| Published: |
2021
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| Online Access: | http://purl.org/au-research/grants/arc/DP160104500 http://hdl.handle.net/20.500.11937/83245 |
| _version_ | 1848764566767403008 |
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| author | Ye, Mengbin Liu, J. Anderson, B.D.O. Cao, M. |
| author_facet | Ye, Mengbin Liu, J. Anderson, B.D.O. Cao, M. |
| author_sort | Ye, Mengbin |
| building | Curtin Institutional Repository |
| collection | Online Access |
| description | This paper focuses on properties of equilibria and their associated regions of attraction for continuous-time nonlinear dynamical systems. The classical Poincar\'e--Hopf Theorem is used to derive a general result providing a sufficient condition for the system to have a unique equilibrium. The condition involves the Jacobian of the system at possible equilibria, and ensures the system is in fact locally exponentially stable. We apply this result to the susceptible-infected-susceptible (SIS) networked model, and a generalised Lotka--Volterra system. We use the result further to extend the SIS model via the introduction of decentralised feedback controllers, which significantly change the system dynamics, rendering existing Lyapunov-based approaches invalid. Using the Poincar\'e--Hopf approach, we identify a necessary and sufficient condition under which the controlled SIS system has a unique nonzero equilibrium (a diseased steady-state), and monotone systems theory is used to show this nonzero equilibrium is attractive for all nonzero initial conditions. A counterpart condition for the existence of a unique equilibrium for a nonlinear discrete-time dynamical system is also presented |
| first_indexed | 2025-11-14T11:21:24Z |
| format | Journal Article |
| id | curtin-20.500.11937-83245 |
| institution | Curtin University Malaysia |
| institution_category | Local University |
| last_indexed | 2025-11-14T11:21:24Z |
| publishDate | 2021 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | curtin-20.500.11937-832452021-07-12T05:59:48Z Applications of the Poincare-Hopf Theorem: Epidemic Models and Lotka-Volterra Systems Ye, Mengbin Liu, J. Anderson, B.D.O. Cao, M. This paper focuses on properties of equilibria and their associated regions of attraction for continuous-time nonlinear dynamical systems. The classical Poincar\'e--Hopf Theorem is used to derive a general result providing a sufficient condition for the system to have a unique equilibrium. The condition involves the Jacobian of the system at possible equilibria, and ensures the system is in fact locally exponentially stable. We apply this result to the susceptible-infected-susceptible (SIS) networked model, and a generalised Lotka--Volterra system. We use the result further to extend the SIS model via the introduction of decentralised feedback controllers, which significantly change the system dynamics, rendering existing Lyapunov-based approaches invalid. Using the Poincar\'e--Hopf approach, we identify a necessary and sufficient condition under which the controlled SIS system has a unique nonzero equilibrium (a diseased steady-state), and monotone systems theory is used to show this nonzero equilibrium is attractive for all nonzero initial conditions. A counterpart condition for the existence of a unique equilibrium for a nonlinear discrete-time dynamical system is also presented 2021 Journal Article http://hdl.handle.net/20.500.11937/83245 10.1109/TAC.2021.3064519 http://purl.org/au-research/grants/arc/DP160104500 http://purl.org/au-research/grants/arc/DP190100887 fulltext |
| spellingShingle | Ye, Mengbin Liu, J. Anderson, B.D.O. Cao, M. Applications of the Poincare-Hopf Theorem: Epidemic Models and Lotka-Volterra Systems |
| title | Applications of the Poincare-Hopf Theorem: Epidemic Models and Lotka-Volterra Systems |
| title_full | Applications of the Poincare-Hopf Theorem: Epidemic Models and Lotka-Volterra Systems |
| title_fullStr | Applications of the Poincare-Hopf Theorem: Epidemic Models and Lotka-Volterra Systems |
| title_full_unstemmed | Applications of the Poincare-Hopf Theorem: Epidemic Models and Lotka-Volterra Systems |
| title_short | Applications of the Poincare-Hopf Theorem: Epidemic Models and Lotka-Volterra Systems |
| title_sort | applications of the poincare-hopf theorem: epidemic models and lotka-volterra systems |
| url | http://purl.org/au-research/grants/arc/DP160104500 http://purl.org/au-research/grants/arc/DP160104500 http://hdl.handle.net/20.500.11937/83245 |