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Regular and Chaotic Dynamics, 2006, Volume 11, Issue 2, Pages 299–310
DOI: https://doi.org/10.1070/RD2006v011n02ABEH000353
(Mi rcd676)
 

On the 70th birthday of L.P. Shilnikov

Adiabatic deformations of integrable two-frequency Hamiltonians

V. Yu. Novokshenov

Institute of Mathematics, Ufa Science Center, RAS, 112, Chernyshevsky str., 450077 Ufa, Russia
Abstract: The paper discusses a mechanism of excitation and control of two-frequency oscillations in the integrable Hamiltonian systems. It is close to the autoresonant technique for controlling the amplitude of nonlinear modes. Autoresonance is usually associated with single frequency mode excitations due to the synchronization and phase lock of various nonlinear modes with the driving force. Despite this we propose a model of multifrequency autoresonance which occur in completely integrable systems. This phenomenon is due to a number stable invariant tori governed by integrals of motion of the integrable system. The basic autoresonant effect of phase locking appears here as Whitham deformations of the invariant tori. This provides also a possibility to transfer a certain initial n-periodic motion to a given m-periodic motion as a final state.
Keywords: integrable Hamiltonian system, Lax pairs, perturbation theory, adiabatic invariants, autoresonance, Whitham equations.
Received: 08.08.2005
Accepted: 03.11.2005
Bibliographic databases:
Document Type: Article
MSC: 34E05, 34E15
Language: English
Citation: V. Yu. Novokshenov, “Adiabatic deformations of integrable two-frequency Hamiltonians”, Regul. Chaotic Dyn., 11:2 (2006), 299–310
Citation in format AMSBIB
\Bibitem{Nov06}
\by V. Yu. Novokshenov
\paper Adiabatic deformations of integrable two-frequency Hamiltonians
\jour Regul. Chaotic Dyn.
\yr 2006
\vol 11
\issue 2
\pages 299--310
\mathnet{http://mi.mathnet.ru/rcd676}
\crossref{https://doi.org/10.1070/RD2006v011n02ABEH000353}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2245085}
\zmath{https://zbmath.org/?q=an:1164.37338}
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