Polynomial normal forms for ODEs near a center-saddle equilibrium point
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| Format: | Preprint |
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2024
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| author | Delshams, Amadeu Zgliczynski, Piotr |
| author_facet | Delshams, Amadeu Zgliczynski, Piotr |
| contents | In this work we consider a saddle-center equilibrium for general vector fields as well as Hamiltonian systems, and we transform it locally into a polynomial normal form in the saddle variables by a change of coordinates. This problem was first solved by Bronstein and Kopanskii in 1995, as well as by Banyaga, de la Llave and Wayne in 1996 [BLW] in the saddle case. The proof used relies on the deformation method used in [BLW], which in particular implies the preservation of the symplectic form for a Hamiltonian system, although our proof is different and, we believe, simpler. We also show that if the system has sign-symmetry, then the transformation can be chosen so that it also has sign-symmetry. This issue is important in our study of shadowing non-transverse heteroclinic chains (Delshams and Zgliczynski 2018 and 2024) for the toy model systems (TMS) of the cubic defocusing nonlinear Schrödinger equation (NLSE) on $2D$-torus or similar Hamiltonian PDE, which are used to prove energy transfer in these PDE. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_04600 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomial normal forms for ODEs near a center-saddle equilibrium point Delshams, Amadeu Zgliczynski, Piotr Dynamical Systems Classical Analysis and ODEs 34C20, 34C45 In this work we consider a saddle-center equilibrium for general vector fields as well as Hamiltonian systems, and we transform it locally into a polynomial normal form in the saddle variables by a change of coordinates. This problem was first solved by Bronstein and Kopanskii in 1995, as well as by Banyaga, de la Llave and Wayne in 1996 [BLW] in the saddle case. The proof used relies on the deformation method used in [BLW], which in particular implies the preservation of the symplectic form for a Hamiltonian system, although our proof is different and, we believe, simpler. We also show that if the system has sign-symmetry, then the transformation can be chosen so that it also has sign-symmetry. This issue is important in our study of shadowing non-transverse heteroclinic chains (Delshams and Zgliczynski 2018 and 2024) for the toy model systems (TMS) of the cubic defocusing nonlinear Schrödinger equation (NLSE) on $2D$-torus or similar Hamiltonian PDE, which are used to prove energy transfer in these PDE. |
| title | Polynomial normal forms for ODEs near a center-saddle equilibrium point |
| topic | Dynamical Systems Classical Analysis and ODEs 34C20, 34C45 |
| url | https://arxiv.org/abs/2411.04600 |