Bifurcations of subharmonics

AbstractWe consider a piecewise linear second order ordinary differential equation which is topologically equivalent to the sine-Gordon equation. The system is subjected to a time harmonic disturbance and the behavior of the periodic solutions is examined. It is shown that subharmonic motions with period n times that of the disturbance appear via saddle-node bifurcations for all n = 1, 2, 3,…. These motions then undergo period-doubling or pitchfork bifurcations as parameters are varied beyond the saddle-node bifurcation values. The limit n → ∞ is considered and is compared with a Melnikov calculation for the system. Due to the piecewise linear nature of the system no approximations are necessary and thus the results are valid for “nonsmall” parameter values. Furthermore, this provides an answer to the uniformity question in the study of stability of subharmonics