Kovačič, Gregor Singular perturbation theory for homoclinic orbits in a class of near- integrable dissipative systems. (English) Zbl 0835.34049 SIAM J. Math. Anal. 26, No. 6, 1611-1643 (1995). 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. Cited in 1 ReviewCited in 14 Documents 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 Keywords:near-integrable dissipative systems; homoclinic and heteroclinic orbits; Melnikov method; geometric singular perturbation theory PDFBibTeX XMLCite \textit{G. Kovačič}, SIAM J. Math. Anal. 26, No. 6, 1611--1643 (1995; Zbl 0835.34049) Full Text: DOI