Non-Smooth Bifurcations of Uniformly Hyperbolic Invariant Manifolds in Skew Product Systems: Rigorous Results

TL;DR

This paper investigates non-smooth bifurcations in hyperbolic invariant manifolds within skew-product systems, extending results to systems with degenerate potentials and establishing the existence of Cantor-like orbit structures.

Contribution

It generalizes the anti-integrable limit analysis to systems with degenerate potentials and proves the existence of Cantor-type orbit structures and non-smooth bifurcations.

Findings

Existence of orbits with any fibered rotation number in generalized systems.

Presence of Cantor set structures under mild regularity conditions.

Confirmation of non-smooth folding bifurcation in these systems.

Abstract

In this paper we study the anti-integrable limit scenario of skew-product systems. We consider a generalization of such systems based on the Frenkel-Kontorova model, and prove the existence of orbits with any fibered rotation number in systems of both one and two degrees of freedom. In particular, our results also apply to two dimensional maps with degenerate potentials (vanishing second derivative), so extending the results of existence of Cantori for more general twist maps. We also prove that under certain mild regularity conditions on the potential the structure of the orbits is of Cantor type. From our results we deduce the existence of the non-smooth folding bifurcation (conjectured by Figueras-Haro, \textit{Different scenarios for hyperbolicity breakdown in quasiperiodic area preserving twist maps}, Chaos:25 (2015)). Lastly we present a pair of results which are useful in determining if a potential satisfies the regularity conditions required for the Cantor sets of orbits to exist and are also of independent interest.