Resonances in Hamiltonian Systems
In this chapter we consider resonances in multi-degree-of-freedom Hamiltonian systems. The resonances are assumed to occur either among the eigenvalues of an elliptic equilibrium (Section 4.1) or among the frequencies of an invariant torus (Sections 4.3 and 4.5). In all cases, slow manifolds or partially slow manifolds exist in local normal forms computed near the resonant object. For resonant equilibria, fast transients among slow manifolds turn out to be responsible for resonance energy transfer, while for invariant tori, irregular motion across resonances can be shown to exist. We examine these phenomena in more detail in a model of the classical water molecule (Section 4.2) and in two concrete examples of rigid body systems (Sections 4.4 and 4.5). The first of the latter examples involves tori of elliptic stability type, while the second one deals with elliptic—hyperbolic tori.
KeywordsPeriodic Orbit Normal Form Hamiltonian System Invariant Manifold Unstable Manifold
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