Neimark-Sacker-pitchfork bifurcation of the symmetric period fixed point of the Poincare map in a three-degree-of-freedom vibro-impact system

A three-degree-of-freedom vibro-impact system with symmetric two-sided rigid constraints is considered. Since the symmetric period n - 2 motion of the vibro-impact system corresponds to the symmetric fixed point of the Poincare map of the vibro-impact system, we investigate bifurcations of the symmetric period n - 2 motion by researching into bifurcations of the associated symmetric fixed point. The Poincare map of the system has symmetry property, and can be expressed as the second iteration of another unsymmetric implicit map. Based on both the Poincare map and the unsymmetric implicit map, the center manifold technique and the theory of normal forms are applied to deduce the normal form of the Neimark-Sacker-pitchfork bifurcation of the symmetric fixed point. By numerical analysis, we obtain the Neimark-Sacker-pitchfork bifurcation of the symmetric fixed point of the Poincare map in the vibro-impact system. (c) 2012 Elsevier Ltd. All rights reserved.