Volume 85, Number 1, January 2009
|Number of page(s)||6|
|Section||Electromagnetism, Optics, Acoustics, Heat Transfer, Classical Mechanics, and Fluid Dynamics|
|Published online||15 January 2009|
Dynamics of nonlinear resonances in Hamiltonian systems
Mathematics Institute, University of Warwick - Coventry CV4 7AL, UK, EU
2 RISC, J. Kepler University - Linz 4040, Austria, EU
Accepted: 28 November 2008
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.
PACS: 47.10.Df – Hamiltonian formulations / 47.10.Fg – Dynamical systems methods / 02.70.Dh – Finite-element and Galerkin methods
© EPLA, 2009
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