Symmetry and bifurcations of a two-degree-of-freedom vibro-impact system

A two-degree-of-freedom system with impact is considered. The symmetry of the system and its Poincaré map is described. The symmetric period n−2 motion corresponding to the symmetric fixed point of the Poincaré map is obtained. If the Jacobian matrix of the Poincaré map at the fixed point has a real eigenvalue crossing the unit circle at +1, the symmetric fixed point will bifurcate into two antisymmetric fixed points, which have the same stability via pitchfork bifurcation. The numerical simulation shows that the symmetric fixed points may have pitchfork bifurcations and Hopf bifurcations. While the control parameter changes continuously, the two antisymmetric fixed points will give birth to two synchronous bifurcation sequences.