Pere Gutierrez
Av. Diagonal 647, 08028 Barcelona, Catalonia, Spain
Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya
Publications:
Delshams A., Gonchenko M. S., Gutierrez P.
Continuation of the Exponentially Small Transversality for the Splitting of Separatrices to a Whiskered Torus with Silver Ratio
2014, vol. 19, no. 6, pp. 663680
Abstract
We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast frequencies in nearly integrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega = \sqrt{2}−1$. We show that the Poincaré–Melnikov method can be applied to establish the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide asymptotic estimates for the transversality of the splitting whose dependence on the perturbation parameter ε satisfies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilon$, generalizing the results previously known for the golden number.

Delshams A., Gutierrez P., Koltsova O. Y., Pacha J. R.
Transverse intersections between invariant manifolds of doubly hyperbolic invariant tori, via the Poincaré–Mel’nikov method
2010, vol. 15, no. 23, pp. 222236
Abstract
We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptic–hyperbolic type, having a homoclinic orbit. More precisely, we consider an $(n+2)$degreeoffreedom near integrable Hamiltonian with $n$ centers and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the center manifold near the equilibrium, there is a Cantorian family of hyperbolic KAM tori, and we study the homoclinic intersections between the stable and unstable manifolds associated to such tori. We establish that, in general, the manifolds intersect along transverse homoclinic orbits. In a more concrete model, such homoclinic orbits can be detected, in a first approximation, from nondegenerate critical points of a Mel’nikov potential. We provide bounds for the number of transverse homoclinic orbits using that, in general, the potential will be a Morse function (which gives a lower bound) and can be approximated by a trigonometric polynomial (which gives an upper bound).
