descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Jan 2011 English Publisher:American Institute of Mathematical Sciences (AIMS)Journal:Discrete & Continuous Dynamical Systems - A, volume 31, pages 301-372 (issn: 1553-5231,

Authors: Martín, Pau; Sauzin, D.; Martínez-Seara Alonso, M. Teresa;

doi: 10.3934/dcds.2011.31.301

handle: 2117/13218 , 2072/50793

Exponentially small splitting of separatrices in the perturbed McMillan map

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Abstract

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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asymptotic formula, Àrees temàtiques de la UPC::Matemàtiques i estadística, :34 Ordinary differential equations::34C Qualitative theory [Classificació AMS], Pertorbació (Matemàtica), Classificació AMS::34 Ordinary differential equations::34C Qualitative theory, Classificació AMS::34 Ordinary differential equations::34E Asymptotic theory, :37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [Classificació AMS], :Matemàtiques i estadística [Àrees temàtiques de la UPC], :34 Ordinary differential equations::34E Asymptotic theory [Classificació AMS], 517 - Anàlisi, Sistemes dinàmics diferenciables, exponentially small phenomena, Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory, Classificació AMS::34 Ordinary differential equations::34M Differential equations in the complex domain, :37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory [Classificació AMS], splitting of separatrices, and nonholonomic systems, McMillan map, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, :34 Ordinary differential equations::34M Differential equations in the complex domain [Classificació AMS]

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Resurgence of inner solutions for perturbations of the McMillan map
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