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arXiv · 2610.33519

Ambient Chain-Relation Reduction in Contracting Controlled Skew-Product Semiflows

Robert Vrabel
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Abstract

The qualitative dynamics of nonautonomous differential equations is often studied through cocycles and skew-product semiflows over a driving base. We develop a chain-recurrence reduction principle for systems admitting an equivariant attracting section and uniform asymptotic tracking. The abstract theorem determines the complete chain-attainability relation in the ambient phase space: chains may start off the attracting section, but every attainable target lies on it and is selected solely by the factor semiflow. Applied to contracting controlled skew-product semiflows, this gives exact descriptions of the chain recurrent set, chain components, Conley order, compact strictly invariant sets, attractors, basins, and dual repellers. A weighted-supremum transverse gauge yields a complete Lyapunov function for the full semiflow while preserving the recurrent values of the base. The differential hypotheses are verified through subadditive fiber-derivative growth and averaged moving-metric logarithmic norms, allowing transient expansion and non-unique ergodicity. Quantitative persistence is established for finite-window maps and smooth vector-field perturbations. The results provide a unified framework for chain recurrence, Lyapunov structure, and attractor--repeller geometry in contracting nonautonomous dynamics.

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Robert Vrabel. 2026-10-24. Ambient Chain-Relation Reduction in Contracting Controlled Skew-Product Semiflows. https://arxiv.org/abs/2610.33519

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