| Full text | |
| Author(s): |
Bakrani, Sajjad
Total Authors: 1
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| Document type: | Journal article |
| Source: | Journal of Differential Equations; v. 446, p. 33-pg., 2025-11-25. |
| Abstract | |
We consider a Z(2)-equivariant 4-dimensional system of ODEs with a smooth first integral Hand 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 [3], 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 not equal 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 P. We further show that when h > c, the forward and backward orbits of any point in H-1(h) near P 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 [3], give a full description of the dynamics in a small open neighborhood of Gamma (and a small open neighborhood of a homoclinic figure-eight). An application to a system of coupled Schrodinger equations with cubic nonlinearity is also considered. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies. (AU) | |
| FAPESP's process: | 23/04294-0 - Normal Forms and Sparse Reconstruction for Networks |
| Grantee: | Sajjad Bakrani Balani |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |