Transition Matrix without Continuation in the Conley Index Theory

TL;DR

This paper extends the concept of singular transition matrices in Conley index theory to cases without Morse decomposition continuation, aiding the study of bifurcations in topological dynamics.

Contribution

It introduces a new definition of transition matrices applicable when Morse decompositions do not continue over parameter intervals.

Findings

Provides a generalized framework for transition matrices without Morse continuation

Enables analysis of bifurcations in topological dynamics

Facilitates understanding of orbit connections without Morse decomposition continuity

Abstract

Given a one-parameter family of flows over a parameter interval $\Lambda$, assuming there is a continuation of Morse decompositions over $\Lambda$, Reineck defined a singular transition matrix to show the existence of a connection orbit between some Morse sets at some parameter points in $\Lambda$. This paper aims to extend the definition of a singular transition matrix in cases where there is no continuation of Morse decompositions over the parameter interval. This extension will help study the bifurcation associated with the change of Morse decomposition from a topological dynamics viewpoint.