Parametrically forced pattern formation

Pattern formation in a nonlinear damped Mathieu-type partial differential equation defined on one space variable is analyzed. A bifurcation analysis of an averaged equation is performed and compared to full numerical simulations. Parametric resonance leads to periodically varying patterns whose spatial structure is determined by amplitude and detuning of the periodic forcing. At onset, patterns appear subcritically and attractor crowding is observed for large detuning. The evolution of patterns under the increase of the forcing amplitude is studied. It is found that spatially homogeneous and temporally periodic solutions occur for all detuning at a certain amplitude of the forcing. Although the system is dissipative, spatial solitons are found representing domain walls creating a phase jump of the solutions. Qualitative comparisons with experiments in vertically vibrating granular media are made. (c) 2001 American Institute of Physics.