Non-persistence of Homoclinic Connections for Perturbed Integrable Reversible Systems

Abstract

The dynamics of an analytic reversible vector field $V$(X,μ) is studied in $\mathbb{R}^4$ with one real parameter μ close to 0; X=0 is a fixed point. The differential D_x $V$(0,0) generates an “oscillatory” dynamics with a frequency of order 1—due to two simple, opposite eigenvalues lying on the imaginary axis—and it also generates a “slow” dynamics which changes from a hyperbolic type—eigenvalues are $\pm \sqrt { - \mu }$—to an elliptic type—eigenvalues are $\pm {\text{ }}i{\text{ }}\sqrt \mu$—as μ passes trough 0. The existence of reversible homoclinic connections to periodic orbits is known for such vector fields. In this paper we study a particular subclass of such vector fields, obtained by small reversible perturbations of the normal form. We give an explicit condition on the perturbation, generically satisfied, which prevents the existence of a homoclinic connections to 0 for the perturbed system. The normal form system of any order admits a reversible homoclinic connection to 0, which then does not survive under perturbation of higher order. It will be seen that normal form essentially decouples the hyperbolic and elliptic part of the linearization to any chosen algebraic order. However, this decoupling does not persist arbitrary reversible perturbation, which finally causes the appearance of small amplitude oscillations.