Neimark–Sacker-pitchfork bifurcation of the symmetric period fixed point of the Poincaré 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 Poincaré 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 Poincaré map of the system has symmetry property, and can be expressed as the second iteration of another unsymmetric implicit map. Based on both the Poincaré 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 Poincaré map in the vibro-impact system.

► This paper considers the Neimark–Sacker-pitchfork bifurcation of a vibro-impact system.
► The symmetry property of the Poincaré map is deduced.
► The normal form of the Neimark–Sacker-pitchfork bifurcation is determined.
► The numerical results of the Neimark–Sacker-pitchfork bifurcation are obtained.