A mechanism for detecting normally hyperbolic invariant tori in differential equations

TL;DR

This paper establishes a new criterion linking the existence of normally hyperbolic invariant tori in periodic differential equations to hyperbolic limit cycles in an associated guiding system, extending classical averaging theory.

Contribution

It introduces a novel method for detecting invariant tori by relating their existence to hyperbolic limit cycles in the guiding system, generalizing classical results for periodic solutions.

Findings

Invariant tori exist when the guiding system has a hyperbolic limit cycle.

The method applies to jerk differential equations.

Provides a new tool for analyzing invariant structures 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.