Homoclinic orbits in reversible systems. II: Multi-bumps and saddle-centres (Q2711578)

This article reviews theory and applications of homoclinic orbits in reversible systems, either Hamiltonian or not. It continues the review [Phys. D 112, 158-186 (1998; Zbl 0986.37041)] by the same author and A. J. Rodríguez-Luis. NEWLINENEWLINENEWLINEMulti-bump homoclinic orbits to an equilibrium which is a four dimensional saddle-center with two real and two imaginary eigenvalues are considered. If the system is Hamiltonian, then it is known that a sign condition determines whether or not cascades of multi-bumps accumulate on the parameter values of a primary homoclinic solution. For non-Hamiltonian reversible systems cascades always occur, as originally explained in \textit{A. R. Champneys} and \textit{J. Härterich} [Dyn. Stab. Syst. 15, 231-252 (2000; Zbl 1003.37033)]. NEWLINENEWLINENEWLINEThe author also describes how a reversible Hopf bifurcation in four dimensional space gives rise to a heteroclinic connection between a saddle-focus equilibrium and a periodic orbit. The discussion is based on a fourth order differential equation that arises in several contexts like in a generalized Swift-Hohenberg equation. NEWLINENEWLINENEWLINETwo examples where these bifurcations occur, spatially localized buckling of cylindrical shells and a generalized massive Thirring model with dispersion, are treated numerically.