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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.
Conley index theory is a very powerful tool in the study of dynamical systems, differential equations and bifurcation theory. In this paper, we make an attempt to generalize the Conley index to discrete random dynamical systems. And we mainly follow
The topological method for the reconstruction of dynamics from time series [K. Mischaikow, M. Mrozek, J. Reiss, A. Szymczak. Construction of Symbolic Dynamics from Experimental Time Series, Physical Review Letters, 82 (1999), 1144-1147] is reshaped t
Multivector fields provide an avenue for studying continuous dynamical systems in a combinatorial framework. There are currently two approaches in the literature which use persistent homology to capture changes in combinatorial dynamical systems. The
In the context of Mathers theory of Lagrangian systems, we study the decomposition in chain-transitive classes of the Mather invariant sets. As an application, we prove, under appropriate hypotheses, the semi-continuity of the so-called Aubry set as a function of the Lagrangian.
The Conley theory has a tool to guarantee the existence of periodic trajectories in isolating neighborhoods of semi-dynamical systems. We prove that the positive trajectories generated by a piecewise-smooth vector field $Z=(X, Y)$ defined in a closed