Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tort with quadratic and cubic frequencies
We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector omega = (1, Omega), where Omega is a quadratic irrat...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2014 |
| 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:2117/23508 |
| Acceso en línea: | https://hdl.handle.net/2117/23508 |
| Access Level: | acceso abierto |
| Palabra clave: | Invariant manifolds splitting of separatrices Melnikov integrals quadratic and cubic frequencies INTEGRABLE HAMILTONIAN-SYSTEMS ADIABATIC INVARIANTS CONTINUED FRACTIONS MELNIKOV METHOD MCMILLAN MAP UPPER-BOUNDS RENORMALIZATION APPROXIMATION PERTURBATION PENDULUM Sistemes dinàmics diferenciables Equacions diferencials ordinàries Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector omega = (1, Omega), where Omega is a quadratic irrational number, or a 3-dimensional torus with a frequency vector w = (1, Omega, Omega(2)), where Omega is a cubic irrational number. Applying the Poincare-Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which Q is the so-called cubic golden number (the real root of x(3) x - 1= 0), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases. |
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