The evolution of invariant manifolds in Hamiltonian-Hopf bifurcations |
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Authors: | Patrick D McSwiggen |
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Institution: | Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USA |
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Abstract: | We consider the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter, ν. The eigenvalues of the linearized system are complex for ν<0 and pure imaginary for ν>0. Thus, for ν<0 the equilibrium has a two-dimensional stable manifold and a two-dimensional unstable manifold, but for ν>0 these stable and unstable manifolds are gone. If the sign of a certain term in the normal form is positive then for small negative ν the stable and unstable manifolds of the system are either identical or must have transverse intersection. Thus, either the system is totally degenerate or the system admits a suspended Smale horseshoe as an invariant set. |
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Keywords: | Stable manifold Bifurcation Strö mgren's conjecture Restricted three-body problem |
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