Wave propagation in a nonlinear periodic medium

The propagation of electronic and electromagnetic waves in a periodic, nonlinear medium is described in terms of a discrete dynamical system represented by a universal, area-preserving map of a plane onto itself. The map is characterized by two control parameters: the wave vector k and the current density j. The dynamics of this Hamiltonian map is very complex admitting periodic, quasiperiodic, and chaotic orbits bifurcating and resonating at various points of the two-dimensional parameter space (k,j). The analysis of this dynamical system is based on the pattern of strong resonances and then applied to the problems of electromagnetic-wave propagation through a superlattice characterized by a strong excitonic nonlinearity and the ballistic transport of electrons in spatially periodic media.