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Singular perturbation theory for homoclinic orbits in a class of near- integrable dissipative systems. (English) Zbl 0835.34049

Summary: This paper presents a new unified theory of orbits homoclinic to resonance bands in a class of near-integrable dissipative systems. It describes three sets of conditions, each of which implies the existence of homoclinic or heteroclinic orbits that connect equilibria or periodic orbits in a resonance band. These homoclinic and heteroclinic orbits are born under a given small dissipative perturbation out of a family of heteroclinic orbits that connect pairs of points on a circle of equilibria in the phase space of the nearby integrable system. The result is a constructive method that may be used to ascertain the existence of orbits homoclinic to objects in a resonance band, as well as to determine their precise shape, asymptotic behavior, and bifurcations in a given example. The method is a combination of the Melnikov method and geometric singular perturbation theory for ordinary differential equations.

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
34D15 Singular perturbations of ordinary differential equations
34A26 Geometric methods in ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
37G99 Local and nonlocal bifurcation theory for dynamical systems
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