Relative Chaos for $C_0$-Semigroups Beyond Topological Notions

TL;DR

This work shows that Devaney chaos for linear $C_0$-semigroups in infinite dimensions can depend on the underlying topology, which challenges its relevance for physical instability. To address this, the authors introduce relative chaos, a trajectory-based notion defined via a fixed, physically meaningful norm and a reference trajectory, characterized by $\liminf_{t\to\infty}D_{u_0,u_{\text{ref}}}(t)=0$ and $\limsup_{t\to\infty}D_{u_0,u_{\text{ref}}}(t)=\infty$, where $D_{u_0,u_{\text{ref}}}(t)=\|T(t)u_0-T(t)u_{\text{ref}}\|$. They establish that Devaney chaos implies relative chaos but not vice versa, and show that relative chaos is robust to topology changes and bounded perturbations. Abstract criteria based on stable/unstable invariant splitting and spectral coexistence are provided, along with a boundary-driven PDE example illustrating relative chaos in parameter regimes where DSW chaos is absent. Numerics on a boundary-driven diffusion-transport model corroborate the energy-based instability pattern and highlight the practical relevance of a norm-based instability notion for infinite-dimensional dynamics. The results advocate using the physically meaningful norm to define and detect instability in PDE semigroups, offering a topology-insensitive framework for understanding complex trajectory behavior in applications.

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.

Figures (3)

• Figure 1: Time evolution of the energy $E(t)=\|u(t,\cdot)\|_{L^2(0,L)}$ for three representative parameter regimes. The curves illustrate, respectively, a dissipative behavior, a strongly amplifying dynamics, and an intermediate transition scenario.
• Figure 2: Snapshots of the spatial profile $x\mapsto u(t,x)$ at selected times in the transition regime. The profiles illustrate the deformation of the solution while the energy alternates between decay--dominated and growth--dominated phases.
• Figure 3: Energy trace $E(t)$ in the transition regime (semilogarithmic scale). The plot highlights alternating phases of attenuation and amplification along a single numerical trajectory, in agreement with a trajectory--based intermittent instability pattern.

Theorems & Definitions (152)

• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Proposition 2.6
• Definition 2.7
• Lemma 2.8
• Lemma 2.9
• Corollary 2.10