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arXiv · 2508.00165

$C^1$ invariant, stable and inertial manifolds for non-autonomous dynamical systems

Radosław Czaja·Piotr Kalita·Alexandre N. Oliveira-Sousa
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Abstract

We use the version of the Lyapunov--Perron method operating on individual solutions to investigate the existence of invariant manifolds for non-autonomous dynamical systems, focusing in particular on inertial and stable manifolds. We establish a characterization of both types of manifolds in terms of solutions exhibiting a common growth behavior, analogous to the classical characterization involving hyperbolicity. Furthermore, we introduce a unified formulation of the gap condition, from which known sharp versions are derived. Finally, we show that the constructed inertial manifolds have $C^1$ regularity.

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Radosław Czaja, Piotr Kalita, Alexandre N. Oliveira-Sousa. 2025-07-31. $C^1$ invariant, stable and inertial manifolds for non-autonomous dynamical systems. https://arxiv.org/abs/2508.00165

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