A Renormalization Proof of the Kam Theorem for Non-Analytic Perturbations

Simone, Emiliano De

doi:10.1142/s0129055x07003085

Cited by 4 publications

(1 citation statement)

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“…KAM theorem has a large body of literature. Quite recent works apply renormalization group concepts to prove it [15,16,17] but our aims here are not to describe all this notable history but rather to try to see if there is another perspective to know the behavior of Hamiltonian systems in a different range of parameter space. The reason for this is that, being such systems ubiquitous, the possible applications could be a large number.…”

Section: Introductionmentioning

confidence: 99%

Dual Lindstedt series and Kolmogorov–Arnol’d–Moser theorem

Frasca¹

2009

Journal of Mathematical Physics

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We prove that exists a Lindstedt series that holds when a Hamiltonian is driven by a perturbation going to infinity. This series appears to be dual to a standard Lindstedt series as it can be obtained by interchanging the role of the perturbation and the unperturbed system. The existence of this dual series implies that a dual KAM theorem holds and, when a leading order Hamiltonian exists that is non degenerate, the effect of tori reforming can be observed with a system passing from regular motion to fully developed chaos and back to regular motion with the reappearance of invariant tori. We apply these results to a perturbed harmonic oscillator proving numerically the appearance of tori reforming. Tori reforming appears as an effect limiting chaotic behavior to a finite range of parameter space of some Hamiltonian systems. Dual KAM theorem, as proved here, applies when the perturbation, combined with a kinetic term, provides again an integrable system.

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Section: Introductionmentioning

confidence: 99%

Dual Lindstedt series and Kolmogorov–Arnol’d–Moser theorem

Frasca¹

2009

Journal of Mathematical Physics

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Resonant motions in the presence of degeneracies for quasi-periodically perturbed systems

Corsi¹,

Gentile²

2014

Ergod. Th. Dynam. Sys.

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We consider one-dimensional systems in the presence of a quasi-periodic perturbation, in the analytical setting, and study the problem of existence of quasi-periodic solutions which are resonant with the frequency vector of the perturbation. We assume that the unperturbed system is locally integrable and anisochronous, and that the frequency vector of the perturbation satisfies the Bryuno condition. Existence of resonant solutions is related to the zeroes of a suitable function, called the Melnikov function -by analogy with the periodic case. We show that, if the Melnikov function has a zero of odd order and under some further condition on the sign of the perturbation parameter, then there exists at least one resonant solution which continues an unperturbed solution. If the Melnikov function is identically zero then one can push perturbation theory up to the order where a counterpart of Melnikov function appears and does not vanish identically: if such a function has a zero of odd order and a suitable positiveness condition is met, again the same persistence result is obtained. If the system is Hamiltonian, then the procedure can be indefinitely iterated and no positiveness condition must be required: as a byproduct, the result follows that at least one resonant quasi-periodic solution always exists with no assumption on the perturbation. Such a solution can be interpreted as a (parabolic) lower-dimensional torus.

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