Invariant manifolds of parabolic fixed points (I). Existence and dependence on parameters
In this paper we study the existence and regularity of stable manifolds associated to fixed points of parabolic type in the differentiable and analytic cases, using the parametrization method. The parametrization method relies on a suitable approximate solution of a functional equation. In the case...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2020 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/404094 |
| Acceso en línea: | https://hdl.handle.net/2117/404094 https://dx.doi.org/10.1016/j.jde.2019.11.100 |
| Access Level: | acceso abierto |
| Palabra clave: | Differentiable dynamical systems Sistemes dinàmics diferenciables Classificació AMS::37 Dynamical systems and ergodic theory::37D Dynamical systems with hyperbolic behavior Classificació AMS::37 Dynamical systems and ergodic theory::37M Approximation methods and numerical treatment of dynamical systems Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics |
| Sumario: | In this paper we study the existence and regularity of stable manifolds associated to fixed points of parabolic type in the differentiable and analytic cases, using the parametrization method. The parametrization method relies on a suitable approximate solution of a functional equation. In the case of parabolic points, if the manifolds have dimension two or higher, in general this approximation cannot be obtained in the ring of polynomials but as a sum of homogeneous functions and it is given in [BFM]. Assuming a sufficiently good approximation is found, here we provide an “a posteriori” result which gives a true invariant manifold close to the approximated one. In the differentiable case, in some cases, there is a loss of regularity. We also consider the case of parabolic periodic orbits of periodic vector fields and the dependence of the manifolds on parameters. Examples are provided. We apply our method to prove that in several situations, namely, related to the parabolic infinity in the elliptic spatial three body problem, these invariant manifolds exist and do have polynomial approximations. |
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