Symmetry of the Poincaré map and its influence on bifurcations in a vibro-impact system

Symmetric period n−2 motion of a three-degree-of-freedom (3-dof) vibro-impact system with symmetric rigid constraints is considered. The Poincaré map of the system is established, and the symmetric fixed point of the Poincaré map corresponds to the associated symmetric period n−2 motion. It is shown that the Poincaré map exhibits some symmetry property, and can be expressed as the second iteration of another unsymmetric implicit map. The symmetry of the Poincaré map influence bifurcation behaviors in vibro-impact system significantly, and suppresses not only period-doubling bifurcation, but also Hopf–flip bifurcation and pitchfork–flip bifurcation. Based on the second iteration of another unsymmetric implicit map, the normal forms in the case of Hopf–Hopf bifurcation and Hopf bifurcation satisfying 1:2 resonant conditions are obtained. By numerical simulation, general Hopf bifurcation, Hopf–Hopf bifurcation and Hopf bifurcation satisfying 1:2 resonant conditions of the symmetric period n−2 motion are represented. However, period-doubling bifurcation, Hopf–flip and pitchfork–flip bifurcation have not been obtained, which reflects upon the effect of the symmetry property on possible bifurcations. It is interesting that the system can exhibit both the characteristic of 1:2 resonance and that of torus T2 under some parameter combination, and 2×T1 torus is also obtained.