On the Dynamics of Invariant Graphs for Dissipative Twist Maps

For two-parameter families of dissipative twist maps, we study the dynamics of invariant graphs and the thresholds of their existence and breakdown. We obtain: (1) For arbitrary $C^1$-small perturbations, invariant graphs with prescribed rotation numbers can be realized through parameter adjustments; (2) The existence of almost sharp perturbations that trigger complete destruction of all invariant graphs; (3) When the perturbation lacks $C^1$-regularity, Lipschitz invariant graphs with non-differentiable points may persist despite the Lipschitz semi-norm satisfying the requirement of the normally hyperbolic invariant manifold theorem.