Exponentially small splitting of separatrices in the perturbed McMillan map
The McMillan map is a one-parameter family of integrable symplectic maps of the plane, for which the origin is a hyperbolic xed point with a homoclinic loop, with small Lyapunov exponent when the parameter is small. We consider a perturbation of the McMillan map for which we show that the loop break...
| Autores: | , , |
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| Formato: | artículo |
| Fecha de publicación: | 2011 |
| País: | España |
| Recursos: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/13218 |
| Acesso em linha: | https://hdl.handle.net/2117/13218 https://dx.doi.org/10.3934/dcds.2011.31.301 |
| Access Level: | acceso abierto |
| Palavra-chave: | McMillan map exponentially small phenomena splitting of separatrices asymptotic formula Sistemes dinàmics diferenciables Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Classificació AMS::34 Ordinary differential equations::34C Qualitative theory Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory Classificació AMS::34 Ordinary differential equations::34E Asymptotic theory Classificació AMS::34 Ordinary differential equations::34M Differential equations in the complex domain Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Resumo: | The McMillan map is a one-parameter family of integrable symplectic maps of the plane, for which the origin is a hyperbolic xed point with a homoclinic loop, with small Lyapunov exponent when the parameter is small. We consider a perturbation of the McMillan map for which we show that the loop breaks in two invariant curves which are exponentially close one to the other and which intersect transversely along two primary homoclinic orbits. We compute the asymptotic expansion of several quantities related to the splitting, namely the Lazutkin invariant and the area of the lobe between two consecutive primary homoclinic points. Complex matching techniques are in the core of this work. The coe cients involved in the expansion have a resurgent origin, as shown in |
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