2024/11/04 by Sajjad Bakrani, Bakrani, Sajjad
Engineering · Physics and Astronomy · #Astro and Planetary Science #Chaotic Dynamics (nlin.CD) #Classical Analysis and ODEs (math.CA) #Control and Dynamics of Mobile Robots #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Pattern Formation and Solitons (nlin.PS) #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2411.02641
openalex publication_date 2024/11/04 · openalex created_date 2024/11/15 · openalex updated_date 2026/07/28
We consider a ℤ2-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 \citeBakrani2022JDE, 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≠ 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 \citeBakrani2022JDE, give a full description of the dynamics in a small open neighborhood of Γ (and a small open neighborhood of a homoclinic figure-eight).