Damped superlinear Duffing equation with strong singularity of repulsive type

Abstract

A damped Duffing equation with a singularity is considered in this paper, where the elastic restoring force g has a singularity at origin and satisfies superlinear condition at infinity. By applying the twist theorem of nonarea-preserving map, we obtain the equation has at least one period-mT solution, where the minimal period is mT. It is also shown by bifurcation analysis that for a explicit form, the equation undergoes fold bifurcation, period doubling bifurcation and Hopf bifurcation, which leads to different solutions, including harmonic solutions, subharmonic solutions and quasiperiodic solutions. At last, we give the phase portraits and correspond Poincare section of the solutions.