KAM Tori Reforming

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Under the influence of a small perturbation, the invariant manifolds of integrable Hamiltonian systems are slightly modified. The original manifolds are tori, according to the KAM theorem due to Kolmogorov, Arnold, and Moser. As the perturbation increases, the tori become progressively distorted and fully developed chaos sets in. But as the perturbation becomes increasingly large (infinite in the limit) chaos dissipates and tori form again, although they are not the same as those of the unperturbed system. Thus the conditions for the KAM theorem are recovered.

Contributed by: Marco Frasca (June 2009)
Based on a program by: Housam Binous and Nasri Zakia
Code improvements by: Simon Tyler
Open content licensed under CC BY-NC-SA


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Based on the author's paper: "Dual Lindstedt Series and KAM Theorem".

In this paper a dual KAM theorem is proved to hold in the limit of a very large perturbation. This Demonstration shows the sequence of integrability, chaotic behavior, and back to integrability for a harmonic oscillator perturbed by a plane wave. Tori reform from chaos.



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