On the Dynamics of Invariant Graphs for Dissipative Twist Maps

TL;DR

This paper studies how invariant graphs in dissipative twist maps behave under small perturbations, identifying conditions for their existence, destruction, and persistence even with limited regularity.

Contribution

It characterizes the thresholds for invariant graph breakdown and shows persistence of Lipschitz graphs under less regular perturbations.

Findings

Invariant graphs can be realized with small $C^r$ perturbations.

Sharp perturbations can destroy all invariant graphs.

Lipschitz invariant graphs may persist even when $C^1$ regularity fails.

Abstract

For two-parameter families of dissipative twist maps, we investigate the dynamics of invariant graphs as well as the thresholds for their existence and breakdown. Our main results are as follows: (1) For arbitrarily small $C^r$ perturbations with $r \geq 1$, invariant graphs with prescribed rotation numbers can be realized by adjusting the parameters; (2) We characterize sharp perturbations that lead to the complete destruction of all invariant graphs; (3) When the perturbation fails to be $C^1$, Lipschitz invariant graphs with non-differentiable points may still persist, even though the Lipschitz norm meets the conditions required by the normally hyperbolic invariant manifold theorem.