Detection of bistable structures via the Conley index and applications to biological systems

TL;DR

This paper uses Conley index theory to identify and analyze bistable structures in dynamical systems, revealing invariant sets, separatrices, and biological implications of these phenomena.

Contribution

It introduces a novel application of Conley index theory to classify and understand bistable structures in biological and dynamical systems.

Findings

Existence of additional isolated invariant sets beyond attractors.

Identification of separatrices and cycle separatrices.

Biological implications of bistable structures discussed.

Abstract

Bistability is a ubiquitous phenomenon in life sciences. In this paper, two kinds of bistable structures in 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.