Dynamics near homoclinic orbits to a saddle in four-dimensional systems with a first integral and a discrete symmetry
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913571782262784 |
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| author | Bakrani, Sajjad |
| author_facet | Bakrani, Sajjad |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02641 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dynamics near homoclinic orbits to a saddle in four-dimensional systems with a first integral and a discrete symmetry Bakrani, Sajjad Dynamical Systems Mathematical Physics Classical Analysis and ODEs Chaotic Dynamics Pattern Formation and Solitons 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). |
| title | Dynamics near homoclinic orbits to a saddle in four-dimensional systems with a first integral and a discrete symmetry |
| topic | Dynamical Systems Mathematical Physics Classical Analysis and ODEs Chaotic Dynamics Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2411.02641 |