dc.contributor.author Delshams Valdés, Amadeu dc.contributor.author Fedorov, Yuri dc.contributor.author Ramírez Ros, Rafael dc.contributor.other Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I dc.date.accessioned 2007-05-07T14:41:16Z dc.date.available 2007-05-07T14:41:16Z dc.date.created 2000 dc.date.issued 2000 dc.identifier.uri http://hdl.handle.net/2117/895 dc.description.abstract The billiard motion inside an ellipsoid of ${\bf R}^{3}$ is completely integrable. If the ellipsoid is not of revolution, there are many orbits bi-asymptotic to its major axis. The set of bi-asymptotic orbits is described from a geometrical, dynamical, and topological point of view. It contains eight surfaces, called separatrices. The splitting of the separatrices under symmetric perturbations of the ellipsoid is studied using a symplectic discrete version of the Poincar\'e-Melnikov method, with a special emphasis in the following situations: close to the flat limit (when the minor axis of the ellipsoid is small enough), close to the oblate limit (when the ellipsoid is close to an ellipsoid of revolution around its minor axis) and close to the prolate limit (when the ellipsoid is close to an ellipsoid of revolution around its major axis). It is proved that any non-quadratic entire symmetric perturbation breaks the integrability and splits the separatrices, although (at least) sixteen symmetric homoclinic orbits persist. Close to the flat limit, these orbits become transverse under very general polynomial perturbations of the ellipsoid. Finally, a particular quartic symmetric perturbation is analyzed in great detail. Close to the flat and to the oblate limits, the sixteen symmetric homoclinic orbits are the unique primary homoclinic orbits. Close to the prolate limit, the number of primary homoclinic orbits undergoes infinitely many bifurcations. The first bifurcation curves are computed numerically. dc.language.iso eng dc.rights Attribution-NonCommercial-NoDerivs 2.5 Spain dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/2.5/es/ dc.subject.lcsh Hamiltonian systems dc.subject.lcsh Differentiable dynamical systems dc.subject.other Twist maps dc.subject.other billiards dc.subject.other separatrix splitting dc.subject.other Melnikov potential dc.title Homoclinic billiard orbits inside symmetrically perturbed ellipsoids dc.type Article dc.subject.lemac Hamilton, Sistemes de dc.subject.lemac Sistemes dinàmics diferenciables dc.subject.lemac Teoria ergòdica dc.contributor.group Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions dc.subject.ams Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems dc.subject.ams Classificació AMS::37 Dynamical systems and ergodic theory::37N Applications dc.rights.access Open Access
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