Homoclinic billiard orbits inside symmetrically perturbed ellipsoids
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Inclou dades d'ús des de 2022
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hdl:2117/895
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Data publicació2000
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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.
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