Dynamics of nonlinear resonances in Hamiltonian systemsM. D. Bustamante1 and E. Kartashova2
1 Mathematics Institute, University of Warwick - Coventry CV4 7AL, UK, EU
2 RISC, J. Kepler University - Linz 4040, Austria, EU
received 22 October 2008; accepted in final form 28 November 2008; published January 2009
published online 15 January 2009
It is well known that the dynamics of a Hamiltonian system depends crucially on whether or not it possesses nonlinear resonances. In the generic case, the set of nonlinear resonances consists of independent clusters of resonantly interacting modes, described by a few low-dimensional dynamical systems. We formulate and prove a new theorem on integrability which allows us to show that most frequently met clusters are described by integrable dynamical systems. We argue that construction of clusters can be used as the base for the Clipping method, substantially more effective for these systems than the Galerkin method. The results can be used directly for systems with cubic Hamiltonian.
47.10.Df - Hamiltonian formulations.
47.10.Fg - Dynamical systems methods.
02.70.Dh - Finite-element and Galerkin methods.
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