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arXiv · 2208.10989

A mechanism for detecting normally hyperbolic invariant tori in differential equations

Pedro C. C. R. Pereira·Douglas D. Novaes·Murilo R. Cândido
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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.

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Pedro C. C. R. Pereira, Douglas D. Novaes, Murilo R. Cândido. 2022-08-23. A mechanism for detecting normally hyperbolic invariant tori in differential equations. https://doi.org/10.1016/j.matpur.2023.06.008

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