Chaos in a mean field model of coupled quantum wells; bifurcations of periodic orbits in a symmetric hamiltonian system

We analyze a discrete model of coupled quantum wells with short-range mean-field interaction in one site. The system evolves according to the time dependent Schrِdinger equation with a nonlinear electrostatic term. The simplest vector field that accounts for the chaotic dynamical behaviour present in the continuum case has four degrees of freedom and can be written as a classical hamiltonian system. It is invariant under diagonal rotations in C4, reversible, autonomous and nonintegrable. The conserved quantities are the energy and the total charge. The organizing centers of the dynamical behaviour are bifurcations of rotating periodic solutions. The global structure of the periodic behaviour is organized via subharmonic bifurcations in which the characteristic multipliers (CM) pass each other on the unit circle and a branch of torus filled with nonsymmetric periodic solutions is born. We have also found another kind of bifurcation in which two pairs of CM depart the unit circle and the symmetric periodic orbit becomes unstable.