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 $Γ$ to $O$ that approaches $O$ along the nonleading directions. Suppose $H(O) = c$. In \cite{Bakrani2022JDE}, the dynamics near $Γ$ in the level set $H^{-1}(c)$ 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 $H^{-1}(h)$ 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 $Γ$. We further show that when $h > c$, the forward and backward orbits of any point in $H^{-1}(h)$ 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 \cite{Bakrani2022JDE}, give a full description of the dynamics in a small open neighborhood of $Γ$ (and a small open neighborhood of a homoclinic figure-eight).

Figures (8)

• Figure 1: The transverse homoclinic loop $\Gamma$ in the invariant plane $\lbrace u_{1}=v_{1}=0 \rbrace$.
• Figure 2: The homoclinic orbit $\Gamma$ lies in the invariant $(u_{2}, v_{2})$-plane and the level set $\{H=0\}$. There exists a family of periodic orbits $L_h$ (shown by blue color) in the invariant plane close to $\Gamma$ in the region surrounded by the closed curve $\Gamma \cup \{O\}$. The periodic orbit $L_h$ lies in the level set $\{H=h\}$ for $-h_{0} \leq h < 0$, where $h_{0} > 0$ is sufficiently small.
• Figure 3: The three-dimensional cross-sections $\Sigma^{\mathrm{in}}$ and $\Sigma^{\mathrm{out}}$ to the homoclinic orbit $\Gamma$ are shown. Each cross-section is foliated by two-dimensional sections $\Pi^{\mathrm{in/out}}(h)$.
• Figure 4: A pair of transverse homoclinic loops in the invariant plane $\lbrace u_{1}=v_{1}=0 \rbrace$.
• Figure 5: The homoclinic orbits $\Gamma_{+}$ and $\Gamma_{-}$ lie in the invariant $(u_{2}, v_{2})$-plane. A continuum of periodic orbits $L_{h}^{+}$ (resp. $L_{h}^{-}$) shown by blue (resp. green) lies in the invariant plane inside the region surrounded by $\Gamma_{+} \cup \{O\}$ (resp. $\Gamma_{-} \cup \{O\}$). Moreover, a continuum of periodic orbits lies in the invariant plane (shown by red) outside the regions surrounded by $\Gamma_{+} \cup \{O\} \cup \Gamma_{-}$. In this figure, the homoclinic orbits are straightened near the equilibrium $O$. This does not need to be the case in general, however, this is the case when the system is brought into the normal forms (see Section \ref{['Normal_forms_section']}).

Theorems & Definitions (10)

• proof : Proof of Lemma \ref{['flowlemmaresonant']}
• proof : Proof of Lemma \ref{['flowlemmaResonant2']}
• proof
• proof : Proof of Lemma \ref{['Lem11u67tt6']}
• proof : Proof of Lemma \ref{['Lem568g67eywfg76']} for the case $\lambda_{1} = \lambda_{2}$
• proof
• proof : Proof of Lemma \ref{['Lem568g67eywfg76']} for the case $\lambda_{1} < \lambda_{2} < 2\lambda_{1}$
• proof : Proof of Lemma \ref{['Lem568g67eywfg76']} for the case $\lambda_{2} = 2\lambda_{1}$
• proof : Proof of Lemma \ref{['Lem568g67eywfg76']} for the case $\lambda_{2} > 2\lambda_{1}$
• proof