What question did this study set out to answer?

This research explores the conditions under which invariant curves of vector fields have geometrical realizations in a higher-dimensional space.

April 3, 2026Ergodic Theory and Dynamical Systems0 citationsOpen Access

Trajectories of vector fields asymptotic to formal invariant curves

OGOLIVIER LE GAL FSFERNANDO SANZ SÁNCHEZ

Key Points

This research explores the conditions under which invariant curves of vector fields have geometrical realizations in a higher-dimensional space.
Proved the existence of trajectories asymptotic to invariant curves within Taylor expansions of vector fields.
Constructed an invariant manifold around the curves composed of these asymptotic trajectories.
Analyzed the nature of trajectories when the vector field is analytic.
Demonstrated that non-zero Taylor expansions lead to geometrical trajectories corresponding to invariant curves.
Identified an invariant manifold that includes all trajectories asymptotic to a given formal curve.
Showed the existence of non-oscillating trajectories with respect to subanalytic sets for analytic vector fields.

Abstract

Abstract We prove that a formal curve that is invariant by a C^ vector field of R^m has a geometrical realization, as soon as the Taylor expansion of is not identically zero along. This means that there is a trajectory R^m of which is asymptotic to. This result solves a natural question proposed by Bonckaert Smooth invariant curves of singularities of vector fields in R 3. Ann. Inst. Henri Poincaré 3 (2) (1986), 111–183 nearly forty years ago. We also construct an invariant C⁰ manifold S in some open horn around which is composed entirely of trajectories asymptotic to and contains the germ of any such trajectory. If is analytic, we prove that there exists a trajectory asymptotic to which is, moreover, non-oscillating with respect to subanalytic sets.

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