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Bifurcation of homoclinic orbits to a saddle-center in reversible systems. (English) Zbl 1052.37043

The subject of this paper is a bifurcation study of a homoclinic loop to a nonhyperbolic equilibrium in a system of differential equations on n that is time reversible with a linear reversor. There is a pair of purely imaginary eigenvalues at the equilibrium. By the Lyapunov center theorem for reversible systems, there is a two-dimensional center manifold with periodic dynamics. We consider the case under small perturbations of the vector field.

From expressions for the center stable and center unstable manifolds of the equilibrium, intersections of these manifolds are studied. This way, the single-round homoclinic connections to the center manifold are analysed.

MSC:
37G20Hyperbolic singular points with homoclinic trajectories
34C37Homoclinic and heteroclinic solutions of ODE
37C29Homoclinic and heteroclinic orbits