Global periodicity conditions for maps and recurrences via normal forms
We face the problem of characterizing the periodic cases in parametric families of rational diffeomorphisms of K, where K is ℝ or ℂ, having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2013 |
| 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:2117/21188 |
| Acceso en línea: | https://hdl.handle.net/2117/21188 https://dx.doi.org/10.1142/S0218127413501824 |
| Access Level: | acceso abierto |
| Palabra clave: | Differential equations Differentiable dynamical systems Periodic orbits Normal forms Maps Difference Equations Equacions diferencials Sistemes dinàmics diferenciables Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory Classificació AMS::39 Difference and functional equations::39A Difference equations Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals |
| Sumario: | We face the problem of characterizing the periodic cases in parametric families of rational diffeomorphisms of K, where K is ℝ or ℂ, having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two- and three-dimensional classes of polynomial or rational maps. In particular, we find the global periodic cases for several Lyness-type recurrences. |
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