Organization and bifurcation of planar closed orbits of an atomic electron in crossed fields
Abstract
We describe the patterns of creation and splitting of planar closed orbits of electrons in hydrogen atoms in crossed electric and magnetic fields. These orbits lie in the plane perpendicular to the magnetic field, and they start and end at the nucleus. Using a Poincaré map to study the regular motions, we observe that the bifurcations of planar closed orbits fall into an ordered sequence as energy changes: a ``tangent bifurcation'' creates one closed orbit that splits into two; subsequently, one of them becomes periodic, and splits by a ``pitchfork bifurcation'' into two periodic orbits and one closed orbit. Based on these calculations, we classify the closed orbits that are involved in a sequence of bifurcations in a family, and we name the family by the winding ratio of the periodic orbits in the family. To understand this ordered sequence of bifurcations, we create a simple integrable Hamiltonian as a model of the Poincaré map. This model gives a simple interpretation of the sequence of the bifurcations. The model contains only general assumptions, so we expect that such sequences of bifurcations of closed orbits will be commonly found in physical systems.
 Publication:

Physical Review A
 Pub Date:
 April 2001
 DOI:
 10.1103/PhysRevA.63.043409
 Bibcode:
 2001PhRvA..63d3409W
 Keywords:

 32.80.t;
 05.45.a;
 03.65.Sq;
 32.60.+i;
 Photon interactions with atoms;
 Nonlinear dynamics and chaos;
 Semiclassical theories and applications;
 Zeeman and Stark effects