Anna Litvak-Hinenzon, Vered Rom-Kedar
Resonant tori and instabilities in Hamiltonian systems
(4536K, .zip)
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.