Relative Chaos for $C_0$-Semigroups Beyond Topological Notions

TL;DR

This paper introduces a new, trajectory-based notion of chaos for $C_0$-semigroups that is independent of topology and more physically meaningful, demonstrated on reaction-diffusion-transport systems.

Contribution

It proposes a novel concept of relative chaos that overcomes limitations of topological chaos notions in infinite-dimensional dynamics.

Findings

Relative chaos is strictly weaker than classical Devaney chaos.

The new chaos criterion is independent of topological refinements.

Application to boundary-driven reaction-diffusion-transport semigroups shows its practical relevance.

Abstract

We investigate instability phenomena for linear evolution equations within the framework of $C_0$--semigroups on infinite--dimensional spaces. We show that Devaney chaos, being formulated in purely topological terms, may depend on the choice of topology and therefore fail to capture intrinsic dynamical behavior. To address this issue, we introduce a trajectory--based notion of relative chaos, defined with respect to a reference solution and measured in a fixed, physically meaningful norm. This criterion is independent of topological refinements and is shown to be strictly weaker than classical Devaney chaos. Its relevance is illustrated on boundary--driven reaction--diffusion--transport semigroups.