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arXiv · 2506.04938

On the Dynamics of Invariant Graphs for Dissipative Twist Maps

Qi Li·Lin Wang
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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.

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Qi Li, Lin Wang. 2025-07-14. On the Dynamics of Invariant Graphs for Dissipative Twist Maps. https://arxiv.org/abs/2506.04938

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