An application of the parameterization method to evolution flows associated to nonautonomous systems
Abstract
Let M be a finite-dimensional complete Riemannian manifold of bounded geometry. We consider a time-dependent vector field X : ℝ × M → TM with Xt:=X(t,·) ∈ Xr(M) for all t ∈ ℝ that satisfies some further uniform continuity assumptions. We suppose that X admits a globally defined integral curve γ: ℝ → M. To any isolated subset Ω of the Sacker-Sell spectrum of t ↦ ∇ Xt(γ(t)), Sacker-Sell theory associates a subbundle 𝒩Ω of TγM invariant under ∇ Xt. Under additional nonresonance conditions on Ω, we prove, by an application of the parameterization method (Cabré et al., 2003), the existence of an invariant topological submanifold 𝒲Ω⊂ ℝ × M with Cr fibers WtΩ:= 𝒲Ω∩ {t} × M that is unique in a sufficiently smooth subclass of invariant manifolds uniformly tangent to 𝒩Ω (made precise later). We moreover show regularity of 𝒲Ω with respect to parameters, and characterize the attraction towards 𝒲Ω via a foliation in the case Ω is the slowest part of the spectrum.
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BibTeX
@misc{buza2026application,
title = {{An application of the parameterization method to evolution flows associated to nonautonomous systems}},
author = {Gergely Buza and George Haller},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18078},
eprint = {2610.18078},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18078},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An application of the parameterization method to evolution flows associated to nonautonomous systems
AU - Gergely Buza
AU - George Haller
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18078
UR - https://arxiv.org/abs/2610.18078
AB - Let M be a finite-dimensional complete Riemannian manifold of bounded geometry. We consider a time-dependent vector field X : ℝ × M → TM with X_t:=X(t,·) ∈ X^r(M) for all t ∈ ℝ that satisfies some further uniform continuity assumptions. We suppose that X admits a globally defined integral curve γ: ℝ → M. To any isolated subset Ω of the Sacker-Sell spectrum of t ↦ ∇ X_t(γ(t)), Sacker-Sell theory associates a subbundle 𝒩^Ω of T_γM invariant under ∇ X_t. Under additional nonresonance conditions on Ω, we prove, by an application of the parameterization method (Cabré et al., 2003), the existence of an invariant topological submanifold 𝒲^Ω⊂ ℝ × M with C^r fibers W_t^Ω:= 𝒲^Ω∩ {t} × M that is unique in a sufficiently smooth subclass of invariant manifolds uniformly tangent to 𝒩^Ω (made precise later). We moreover show regularity of 𝒲^Ω with respect to parameters, and characterize the attraction towards 𝒲^Ω via a foliation in the case Ω is the slowest part of the spectrum.
ER -