Amadeu Delshams

Av. Diagonal 647, 08028 Barcelona, Catalonia
Departament de Matematica Aplicada I, Universitat Politècnica de Catalunya


Delshams A., Schaefer R. G.
Arnold Diffusion for a Complete Family of Perturbations
2017, vol. 22, no. 1, pp.  78-108
In this work we illustrate the Arnold diffusion in a concrete example — the a priori unstable Hamiltonian system  of $2+1/2$ degrees of freedom  $H(p,q,I,\varphi,s) = p^{2}/2+\cos q -1 +I^{2}/2 + h(q,\varphi,s;\varepsilon)$ — proving that for   any small periodic perturbation of the form $h(q,\varphi,s;\varepsilon) = \varepsilon\cos q\left( a_{00} + a_{10}\cos\varphi + a_{01}\cos s  \right)$  ($a_{10}a_{01} \neq 0$) there is global instability for the action.  For the proof we apply a geometrical mechanism based in the so-called Scattering map. This work has the following structure: In a first stage, for a more restricted case ($I^*\thicksim\pi/2\mu$, $\mu = a_{10}/a_{01}$), we use only one scattering map, with a special property: the existence of simple paths of diffusion called highways. Later, in the general case we combine a scattering map with the inner map (inner dynamics) to prove the more general result (the existence of the instability for any $\mu$). The bifurcations of the scattering map are also studied as a function of $\mu$. Finally, we give an estimate for the time of diffusion, and we show that this time is primarily the time spent under the scattering map.
Keywords: Arnold diffusion, normally hyperbolic invariant manifolds, scattering maps
Citation: Delshams A., Schaefer R. G.,  Arnold Diffusion for a Complete Family of Perturbations, Regular and Chaotic Dynamics, 2017, vol. 22, no. 1, pp. 78-108
Delshams A., Gonchenko M. S., Gutierrez P.
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.
Keywords: transverse homoclinic orbits, splitting of separatrices, Melnikov integrals, silver ratio
Citation: 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, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 663-680
Delshams A., Gonchenko M. S., Gonchenko S. V.
We study bifurcations of nonorientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on nonorientable twodimensional surfaces. We consider one- and two-parameter general unfoldings and establish results related to the emergence of elliptic periodic orbits.
Keywords: area-preserving map, non-orientable surface, elliptic point, homoclinic tangency, bifurcation
Citation: Delshams A., Gonchenko M. S., Gonchenko S. V.,  On Bifurcations of Area-preserving and Nonorientable Maps with Quadratic Homoclinic Tangencies, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 702-717
Delshams A., Gutierrez P., Koltsova O. Y., Pacha J. R.
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 $(n+2)$-degree-of-freedom 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).
Keywords: hyperbolic KAM tori, transverse homoclinic orbits, Melnikov method
Citation: 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, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 222-236

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