Dynamics near homoclinic orbits to a saddle in four-dimensional systems with a first integral and a discrete symmetry

Abstract

We consider a $Z_2$-equivariant 4-dimensional system of ODEs with a smooth first integral H and a saddle equilibrium state O. We assume that there exists a transverse homoclinic orbit Γ to O that approaches O along the nonleading directions. Suppose $H\left(O\right)=c$. In [3], the dynamics near Γ in the level set $H^{-1}\left(c\right)$ was described. In particular, some criteria for the existence of the stable and unstable invariant manifolds of Γ were given. In the current paper, we describe the dynamics near Γ in the level set $H^{-1}\left(h\right)$ for $h\ne c$ close to c. We prove that when $h<c$, there exists a unique saddle periodic orbit in each level set $H^{-1}\left(h\right)$, and the forward (resp. backward) orbit of any point off the stable (resp. unstable) invariant manifold of this periodic orbit leaves a small neighborhood of Γ. We further show that when $h>c$, the forward and backward orbits of any point in $H^{-1}\left(h\right)$ near Γ leave a small neighborhood of Γ. We also prove analogous results for the scenario where two transverse homoclinics to O (homoclinic figure-eight) exist. The results of this paper, together with [3], give a full description of the dynamics in a small open neighborhood of Γ (and a small open neighborhood of a homoclinic figure-eight). An application to a system of coupled Schrödinger equations with cubic nonlinearity is also considered.