Discretization strategies for computing Conley indices and Morse decompositions of flows

TL;DR

This paper compares fixed and variable time step strategies for computing Conley indices and Morse decompositions of flows, demonstrating that variable steps can improve numerical results and ensuring the validity of the Morse decompositions obtained.

Contribution

It introduces and analyzes a variable time step approach for computing Morse decompositions, showing its advantages over fixed time steps in dynamical systems analysis.

Findings

Variable time steps improve numerical accuracy.

The method guarantees valid Morse decompositions.

Comparison shows benefits over traditional fixed step methods.

Abstract

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.