Parametrically forced pattern formation

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