Discretization strategies for computing Conley indices and Morse decompositions of flows

Conley indices and Morse decompositions of flows can be found by using algorithms which rigorously analyze discrete dynamical systems. This usually involves integrating a time discretization of the flow using interval arithmetic. We compare the old idea of fixing a time step as a parameters to a time step continuously varying in phase space. We present an example where this second strategy necessarily yields better numerical outputs and prove that our outputs yield a valid Morse decomposition of the given flow

Weak index pairs and the Conley index for discrete multivalued dynamical systems

Motivated by the problem of reconstructing dynamics from samples we revisit the Conley index theory for discrete multivalued dynamical systems. We introduce a new, less restrictive definition of the isolating neighbourhood. It turns out that then the main tool for the construction of the index, i.e. the index pair, is no longer useful. In order to overcome this obstacle we use the concept of weak index pairs

Weak index pairs and the Conley index for discrete multivalued dynamical systems. Part II: properties of the index

Motivation to revisit the Conley index theory for discrete multivalued dynamical systems stems from the needs of broader real applications, in particular in sampled dynamics or in combinatorial dynamics. The new construction of the index in [B. Batko and M. Mrozek, {\em SIAM J. Applied Dynamical Systems}, 15(2016), pp. 1143-1162] based on weak index pairs, under the circumstances of the absence of index pairs caused by relaxing the isolation property, seems to be a promising step towards this direction. The present paper is a direct continuation of [B. Batko and M. Mrozek, {\em SIAM J. Applied Dynamical Systems}, 15(2016), pp. 1143-1162] and concerns properties of the index defined therin, namely Wa\.zewski property, the additivity property, the homotopy (continuation) property and the commutativity property. We also present the construction of weak index pairs in an isolating block

CAPD::RedHom - Reduction heuristics for homology algorithms

We present an efficient software package for computing homology of sets, maps and filtrations represented as cubical, simplicial and regular CW complexes. The core homology computation is based on classical Smith diagonalization, but the efficiency of our approach comes from applying several geometric and algebraic reduction techniques combined with smart implementation

Index Pairs Algorithms

Abstract. We introduce some modifications and extensions of the concept od index pair in the Conley index theory. We then show how these concepts may be used to overcome some difficulties in obtaining efficient algorithms computing the Conley index. We also present examples of applications to computer assisted proofs in dynamics

An Algorithmic Approach To The Conley Index Theory

. We introduce a class of representable sets which is closed under the operations of set theoretical union, intersection, difference and topological interior and closure. We use this class to construct an algorithm which verifies if for a given dynamical system a given set is an isolating neighborhood. In case of a positive answer the algorithm constructs an index pair. Keywords: Conley index, isolated invariant set, index pairs, algorithms, grids, discretisation 1. Introduction The computer assisted, Conley index based proof of chaos in the Lorenz equations [4] showed the importance of the Conley index theory in rigorous numerics of dynamical systems. The proof utilized a chaos criterion developed in [3]. The role of the computer was to verify whether the assumptions of that criterion are satisfied for some region in the phase space of the Lorenz system. Since the assumptions are based on the Conley index, it was necessary to verify that the Conley indexes are defined in the appropr..