Planar Bistable Structures Detection via the Conley Index and Applications to Biological Systems
Junbo Jia,
Pan Yang,
Huaiping Zhu,
Zhen Jin,
Jinqiao Duan,
Xinchu Fu
Affiliations
Junbo Jia
Key Laboratory of Systems Health Science of Zhejiang Province, School of Life Science, Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Hangzhou 310024, China
Pan Yang
School of Mathematical Sciences, Changsha Normal University, Changsha 410100, China
Huaiping Zhu
Laboratory of Mathematical Parallel Systems (LAMPS), Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada
Zhen Jin
Complex Systems Research Center, Shanxi University, Taiyuan 030051, China
Jinqiao Duan
Department of Mathematics and Department of Physics, Great Bay University, Dongguan 523000, China
Xinchu Fu
Department of Mathematics, Shanghai University, Shanghai 200444, China
Bistability is a ubiquitous phenomenon in life sciences. In this paper, two kinds of bistable structures in two-dimensional dynamical systems are studied: one is two one-point attractors, another is a one-point attractor accompanied by a cycle attractor. By the Conley index theory, we prove that there exist other isolated invariant sets besides the two attractors, and also obtain the possible components and their configuration. Moreover, we find that there is always a separatrix or cycle separatrix, which separates the two attractors. Finally, the biological meanings and implications of these structures are given and discussed.
