Linking Combinatorial and Classical Dynamics: Conley Index and Morse Decompositions

We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system F on the geometric realization of the simplicial complex. Moreover, F may be chosen in such a way that the isolated invariant sets, Conley indices, Morse decompositions and Conley–Morse graphs of the combinatorial vector field give rise to isomorphic objects in the multivalued map case.

Strong Morse–Lyapunov functions for Morse decompositions of attractors of random dynamical systems

In this paper, we first construct strong Lyapunov functions for random attractors. On this basis, we then establish strong Morse–Lyapunov functions for Morse decompositions of the random attractors. Finally, the stability of the Morse decompositions is studied through the Morse–Lyapunov functions.