  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.
