Ambient Chain-Relation Reduction in Contracting Controlled Skew-Product Semiflows
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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BibTeX
@misc{vrabel2026ambient,
title = {{Ambient Chain-Relation Reduction in Contracting Controlled Skew-Product Semiflows}},
author = {Robert Vrabel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33519},
eprint = {2610.33519},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33519},
}
RIS
TY - UNPB
TI - Ambient Chain-Relation Reduction in Contracting Controlled Skew-Product Semiflows
AU - Robert Vrabel
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33519
UR - https://arxiv.org/abs/2610.33519
AB - 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.
ER -