Spatially Periodic Reaction-Diffusion Patterns and Homoclinic Entropy

Definitions and mathematical statement

Let $m\in{\mathbb{N}}$ with $m>1$, and let $x\in{\mathbb{R}}^m$ denote the spatial variable. A solution $u$ is stationary in time if there is a spatial profile $v$ such that $u(t,x)=v(x)$ for every time and spatial point in its domain.

1. Can the one-dimensional complexity result referred to in the supplied report passage be extended to stationary, asymptotically spatially periodic solutions of reaction--diffusion systems on ${\mathbb{R}}^m$, with localized defects as specified in the supplied clean statement?
2. Do symmetric homoclinic tangles involving the multi-parameter families of periodic orbits referred to in the supplied clean statement always yield positive topological entropy?

These are qualified problem statements rather than fully specified mathematical propositions. The supplied excerpt calls the first target a "similar result" but does not state the one-dimensional theorem, its hypotheses, or its complexity conclusion. It does not specify the reaction--diffusion equations, the solution class, the meaning of asymptotic spatial periodicity, or the localization condition for defects. For the second question, the excerpt refers to "such families" without supplying their defining context. The relevant dynamical system, symmetry, homoclinic-tangle hypotheses, parameter space, and entropy convention are also unspecified. Consequently, no particular periodicity lattice, asymptotic condition, transversality assumption, or entropy-bearing invariant set is imposed here.

Short English statement

The questions ask whether a one-dimensional complexity result extends to stationary spatially periodic reaction--diffusion patterns with localized defects in higher dimensions, and whether the associated symmetric homoclinic tangles always have positive topological entropy. The referenced theorem and several essential hypotheses are absent from the supplied excerpt, so a complete formal statement requires the surrounding report context.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005588$, titled Spatially Periodic Reaction-Diffusion Patterns and Homoclinic Entropy; source problem code catalogue entry. The collection attributes the problem to Oberwolfach workshop open-problem discussions. No individual proposer is supplied; the recorded proposed year is 2023.
• Supplied report citation: Dynamische Systeme (2024), Oberwolfach Reports, pp. 1671--1730, DOI 10.4171/owr/2023/30. Supplied source URL and literature reference labeled "original OWR report": https://doi.org/10.4171/owr/2023/30.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels both questions open and reports that no later resolution was identified. This is supplied dated curation, not an independently verified literature assessment; the record requests independent verification of time-sensitive status and cited claims.
• The record labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report and preserves the report's hypotheses, notation, and open target. This is supplied source-verification information, not verification performed for this draft. The record also describes its original-statement passage as retained verbatim.