Geometrical resonance in a driven symmetric bistable system subjected to strong or to weak damping

The response of a symmetric bistable system driven by a time-periodic rectangular force modeled by the Jacobian elliptic function sn is studied in two limiting situations: overdamping and weak damping. For the overdamping case, the appearance of responses with the same shape and period as the driving force is explained in terms of a geometrical resonance phenomenon. The distortion of the response under changes in the forcing period and shape is also considered. For weak damping, the reduction of homoclinic chaos as the driving shape approximates the geometrical resonance forcing shape is explained by means of Melnikov's analysis in the asymptotic case of infinite period driving.