Trajectories of vector fields asymptotic to formal invariant curves

2023/11/12 by Gal, Olivier Le, Sánchez, Fernando Sanz
#34A25 #34C08 #34C20 #34E05 #37C25 #37D10 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.06821

Abstract

We prove that a formal curve Γ that is invariant by a C^∞ vector field ξ of ℝ^m has a geometrical realization, as soon as the Taylor expansion of ξ is not identically zero along Γ. This means that there is a trajectory γ of ξ which is asymptotic to Γ. This result solves a natural question proposed by Bonckaert nearly forty years ago. We also construct an invariant C⁰ manifold S in some open horn around Γ which is composed entirely of trajectories asymptotic to Γ, and contains the germ of any such trajectory. If ξ is analytic, we prove that there exists a trajectory asymptotic to Γ which is, moreover, non-oscillating with respect to subanalytic sets.

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