Global Dynamics for Steep Nonlinearities in Two Dimensions
- Gedeon, Tomas [ Montana State University: Mathematical Sciences ]
- Harker, Shaun [ Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, Piscataway, NJ 08854-8019, United States ]
- Kokubu, Hiroshi [ Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan ]
- Mischaikow, Konstantin [ Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, Piscataway, NJ 08854-8019, United States ]
- Oka, Hiroe [ Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu 520-2194, Japan ]
This paper discusses a novel approach to obtaining mathematically rigorous results on the global dynamics of ordinary differential equations. We study switching models of regulatory networks. To each switching network we associate a Morse graph, a computable object that describes a Morse decomposition of the dynamics. In this paper we show that all smooth perturbations of the switching system share the same Morse graph and we compute explicit bounds on the size of the allowable perturbation. This shows that computationally tractable switching systems can be used to characterize dynamics of smooth systems with steep nonlinearities.