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Epsilon-Chains for Continuous-Time Semiflows

Roberto De Leo, James A. Yorke

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.

Epsilon-Chains for Continuous-Time Semiflows

Abstract

We introduce a notion of -chains for continuous-time semiflows inspired by the shadow-orbit property. Although this definition differs from the -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.
Paper Structure (3 sections, 7 theorems, 47 equations)

This paper contains 3 sections, 7 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Results

Key Result

Theorem A

Let $F$ be a semiflow with strong compact dynamics. Denote by $F_\mathcal{G}$ be the restriction of $F$ to its global attractor and by $\mathcal{R}_\mathcal{C}$ its Conley chain-recurrent set and by $\mathcal{R}_{\mathcal{C}_\mathcal{G}}$ the Conley chain-recurrent set of $F_\mathcal{G}$. Then:

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Example 2.1
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem A
  • Proposition 6: $\mathcal{G}$ is invariant under $\succcurlyeq_\mathcal{S}$
  • proof
  • Proposition 7
  • ...and 9 more