Homoclinic orbits in reversible systems II : multi-bumps and saddle-centres

This article extends a review by the author in Physica D, vol.112, pp.158-186 of the theory and application of homoclinic orbits to equilibria in even-order, time-reversible systems of autonomous ordinary differential equations, either Hamiltonian or not. Recent results in two directions are surveyed. First, a heteroclinic connection between a saddle-focus equilibrium and a periodic orbit is shown to arise from a certain codimension-two local bifurcation; a degenerate Hamiltonian-Hopf bifurcation. Under a transversality hypothesis, perturbation from normal form causes this isolated solution to break into a snaking bifurcation curve under which a primary homoclinic becomes a multi-bump with arbitrarily many bumps. Taking as a model a fourth-order equation arising in many contexts, the snaking is terminated by the existence of a heteroclinic connection to an equilibrium. Second, multi-bump homoclinic orbits are considered in the case where the equilibrium is a four-dimensional saddle-centre (having two real and two imaginary eigenvalues). 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, albeit from one sign of parameter perturbation only. Finally, aided by numerical methods, possible applications are considered to localised cylindrical shell buckling and to a generalised massive Thirring model arising in nonlinear optics.