Epsilon-Chains for Continuous-Time Semiflows

TL;DR

This paper introduces epsilon-chains for continuous-time semiflows, showing they align with existing chain concepts in strong compact dynamics, and are well-suited for differential equation-based systems.

Contribution

It defines epsilon-chains for semiflows and proves their equivalence to Conley's chains in certain dynamical settings, enhancing the understanding of recurrence.

Findings

Epsilon-chains generate the same chain-recurrent structures as Conley's chains in strong compact dynamics.

The new definition is naturally compatible with semiflows from differential equations.

Epsilon-chains provide a useful tool for analyzing continuous-time dynamical systems.

Abstract

We introduce a notion of $\varepsilon$-chains for continuous-time semiflows inspired by the shadow-orbit property. Although this definition differs from the $(\varepsilon,T)$-chains introduced by Conley, we prove that, for semiflows with strong compact dynamics, the two notions generate the same chain-recurrent structure. In particular, they yield the same recurrent sets, nodes and graphs. The new definition fits naturally with semiflows arising from differential equations.