Instability of equilibrium and convergence to periodic orbits in strongly 2-cooperative systems
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| Format: | Preprint |
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2025
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| author | Katz, Rami Giordano, Giulia Margaliot, Michael |
| author_facet | Katz, Rami Giordano, Giulia Margaliot, Michael |
| contents | We consider time-invariant nonlinear $n$-dimensional strongly $2$-cooperative systems, that is, systems that map the set of vectors with up to weak sign variation to its interior. Strongly $2$-cooperative systems enjoy a strong Poincare-Bendixson property: bounded solutions that maintain a positive distance from the set of equilibria converge to a periodic solution. For strongly $2$-cooperative systems whose trajectories evolve in a bounded and invariant set that contains a single unstable equilibrium, we provide a simple criterion for the existence of periodic trajectories. Moreover, we explicitly characterize a positive-measure set of initial conditions which yield solutions that asymptotically converge to a periodic trajectory. We demonstrate our theoretical results using two models from systems biology, the $n$-dimensional Goodwin oscillator and a $4$-dimensional biomolecular oscillator with RNA-mediated regulation, and provide numerical simulations that verify the theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09155 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Instability of equilibrium and convergence to periodic orbits in strongly 2-cooperative systems Katz, Rami Giordano, Giulia Margaliot, Michael Dynamical Systems We consider time-invariant nonlinear $n$-dimensional strongly $2$-cooperative systems, that is, systems that map the set of vectors with up to weak sign variation to its interior. Strongly $2$-cooperative systems enjoy a strong Poincare-Bendixson property: bounded solutions that maintain a positive distance from the set of equilibria converge to a periodic solution. For strongly $2$-cooperative systems whose trajectories evolve in a bounded and invariant set that contains a single unstable equilibrium, we provide a simple criterion for the existence of periodic trajectories. Moreover, we explicitly characterize a positive-measure set of initial conditions which yield solutions that asymptotically converge to a periodic trajectory. We demonstrate our theoretical results using two models from systems biology, the $n$-dimensional Goodwin oscillator and a $4$-dimensional biomolecular oscillator with RNA-mediated regulation, and provide numerical simulations that verify the theoretical results. |
| title | Instability of equilibrium and convergence to periodic orbits in strongly 2-cooperative systems |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2503.09155 |