A mechanism for detecting normally hyperbolic invariant tori in differential equations
Abstract
Determining the existence of compact invariant manifolds is a central quest in the qualitative theory of differential equations. Singularities, periodic solutions, and invariant tori are examples of such invariant manifolds. A classical and useful result from the averaging theory relates the existence of isolated periodic solutions of non-autonomous periodic differential equations, given in a specific standard form, with the existence of simple singularities of the so-called guiding system, which is an autonomous differential equation given in terms of the first non-vanishing higher order averaged function. In this paper, we provide an analogous result for the existence of invariant tori. Namely, we show that a non-autonomous periodic differential equation, given in the standard form, has a normally hyperbolic invariant torus in the extended phase space provided that the guiding system has a hyperbolic limit cycle. We apply this result to show the existence of normally hyperbolic invariant tori in a family of jerk differential equations.
Keywords
Cite
@article{arxiv.2208.10989,
title = {A mechanism for detecting normally hyperbolic invariant tori in differential equations},
author = {Pedro C. C. R. Pereira and Douglas D. Novaes and Murilo R. Cândido},
journal= {arXiv preprint arXiv:2208.10989},
year = {2023}
}