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Singular separatrix splitting and Melnikov method: An experimental study
Delshams Valdés, Amadeu; Ramírez Ros, Rafael
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
We consider families of analytic area-preserving maps depending on two pa- rameters: the perturbation strength E and the characteristic exponent h of the origin. For E=0, these maps are integrable with a separatrix to the origin, whereas they asymptote to flows with homoclinic connections as h->0+. For fixed E!=0 and small h, we show that these connections break up. The area of the lobes of the resultant turnstile is given asymptotically by E exp(-Pi^2/h)Oª(h), where Oª(h) is an even Gevrey-1 function such that Oª(0)!=0 and the radius of convergence of its Borel transform is 2Pi^2. As E->0 the function Oª tends to an entire function Oº. This function Oº agrees with the one provided by the Melnikov theory, which cannot be applied directly, due to the exponentially small size of the lobe area with respect to h. These results are supported by detailed numerical computations; we use an expensive multiple-precision arithmetic and expand the local invariant curves up to very high order.
Dynamical systems
Bifurcation theory
Ordinary Differential Equations and Operators, Symposium on
Area-preserving map
singular separatrix splitting
Melnikov method
numerical experiments
Equacions diferencials ordinàries
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::37G Local and nonlocal bifurcation theory
Classificació AMS::37 Dynamical systems and ergodic theory::37M Approximation methods and numerical treatment of dynamical systems
Classificació AMS::65 Numerical analysis::65L Ordinary differential equations
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
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