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

Abstract

We consider a $\mathbb{Z}_{2}$-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 $\Gamma$ to $O$ that approaches $O$ along the nonleading directions. Suppose $H(O) = c$. In \cite{Bakrani2022JDE}, the dynamics near $\Gamma$ in the level set $H^{-1}(c)$ was described. In particular, some criteria for the existence of the stable and unstable invariant manifolds of $\Gamma$ were given. In the current paper, we describe the dynamics near $\Gamma$ in the level set $H^{-1}(h)$ for $h\neq c$ close to $c$. We prove that when $h < c$, there exists a unique saddle periodic orbit in each level set $H^{-1}(h)$, 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 $\Gamma$. We further show that when $h > c$, the forward and backward orbits of any point in $H^{-1}(h)$ near $\Gamma$ leave a small neighborhood of $\Gamma$. 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 \cite{Bakrani2022JDE}, give a full description of the dynamics in a small open neighborhood of $\Gamma$ (and a small open neighborhood of a homoclinic figure-eight).