Degenerate resonances in Hamiltonian systems with 3/2 degrees of freedom.
Hamiltonian systems with 3/2 degrees of freedom close to autonomous systems are considered. Special attention is focused on the case of degenerate resonances. In this case, an averaged system in the first approximation reduces to an area-preserving mapping of a cylinder whose rotation number is a nonmonotonic function of the action variable. Behavior of the trajectories of such a map is similar to that of the trajectories of a Poincare map. Three regions: B(+/-) in the upper and lower parts of the cylinder and an additional region A which contains separatrices of fixed points for the corresponding resonance are distinguished on the cylinder. It is shown that there is a nonempty set of initial points corresponding to walking trajectories in B(+/-) and, hence, there are no closed invariant curves that are homotopically nontrivial on the cylinder. Cells limited by a "stochastic network" can exist in region A. The number of cells is the greater the higher the order of degeneration of the resonance. Possible types of orbit behavior in region A are described. (c) 2002 American Institute of Physics.
Morozov
Albert D.
AD
Department of Mathematics and Mechanics, University of Nizhny Novgorod, 23 Gagarin Avenue, 603600 Nizhny Novgorod, Russia.