01-177 Anna Litvak-Hinenzon, Vered Rom-Kedar

Resonant tori and instabilities in Hamiltonian systems (4536K, .zip) May 11, 01

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Abstract. The existence of lower dimensional resonant tori of \emph{parabolic}, hyperbolic and elliptic normal stability types is proved to be generic and persistent in a class of $n$ degrees of freedom (d.o.f.) integrable Hamiltonian systems with $n\geq 3$. In particular, in such systems the existence of normally elliptic or hyperbolic $n-1$ dimensional torus of fixed points is persistent without the use of any external parameters, and the existence of an $n-1$ dimensional normally parabolic torus of fixed points is of co-dimension one. \emph{Parabolic resonance} (respectively, hyperbolic or elliptic resonance) is created when a small Hamiltonian perturbation is added to an integrable Hamiltonian system possessing a resonant torus of the corresponding normal stability. It is numerically demonstrated that parabolic resonances cause intricate behavior and large instabilities. The place and role of lower dimensional parabolic resonant tori in the Arnold web, and the related structure of the unperturbed energy surfaces, are discussed and illustrated using models of near integrable Hamiltonian systems with three, four and five d.o.f.. Critical $n$ values for which the system first persistently possesses mechanisms for large instabilities of a certain type are found. Initial numerical studies of the rate and time of development of the most significant instabilities are presented.

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