A User’s Guide to the Conley-Morse Database

Functoriality and duality in Morse-Conley-Floer homology

In [13] a homology theory –Morse-Conley-Floer homology– for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functoriality for Morse homology of closed manifolds is known [1, 2, 3, 8, 14], but the proofs use isom...

Morse-Conley-Floer Homology

For Morse-Smale pairs on a smooth, closed manifold the MorseSmale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for isolated invariant sets of flows on a smooth manifold, which gives an analogue ...

A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems

A generally applicable, automatic method for the efficient computation of a database of global dynamics of a multiparameter dynamical system is introduced. An outer approximation of the dynamics for each subset of the parameter range is computed using rigorous numerical methods and is represented by means of a directed graph. The dynamics is then decomposed into the recurrent and gradient-like ...

CATALOGING THE GLOBAL BEHAVIOR OF DYNAMICAL SYSTEMS: ADAPTIVELY SEARCHING PARAMETER SPACE USING THE CONLEY-MORSE DATABASE by

The random case of Conley’s theorem: III. Random semiflow case and Morse decomposition

In the first part of this paper, we generalize the results of the author [25, 26] from the random flow case to the random semiflow case, i.e. we obtain Conley decomposition theorem for infinite dimensional random dynamical systems. In the second part, by introducing the backward orbit for random semiflow, we are able to decompose invariant random compact set (e.g. global random attractor) into ...