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Quantitative Destruction and Persistence of Lagrangian Torus in Hamiltonian Systems
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abstract
For an integrable Hamiltonian systems with $d$ degrees of freedom ($d\geq 2$), we consider quantitatively the existence and non-existence of the flow-invariant Lagrangian torus with given frequency under the perturbation beyond the scope of the classical KAM method in the $C^r$ topology. As applications, the non-existence result gives a partial answer to an open problem on non-existence of invariant circles by Mather from 1988. The existence result sheds a light on another open problem on the existence of invariant circles with lower regularity by Mather from 1998.
Forward citations
Cited by 2 Pith papers
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Persistence of invariant graphs for twist maps under analytic perturbations
For a new frequency set larger than Brjuno-Rüssmann, renormalized Gevrey-γ perturbations of twist maps still admit real-analytic invariant graphs, with a λ-independent threshold in the dissipative case.
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On the sharpness of the $C^1$-norm threshold for perturbations in the normally hyperbolic invariant manifold theorem---a toy model perspective
The paper claims the C^1-norm threshold (1−√λ)^2 is sharp for invariant graph regularity in a dissipative toy model, but the proof only bounds the maximum of a key derivative, not the minimum.
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