LOCAL DYNAMICAL BEHAVIOR OF TWO-PARAMETER FAMILY NEAR THE NEIMARK-SACKER-PITCHFORK BIFURCATION POINT IN A VIBRO-IMPACT SYSTEM

A three-degree-of-freedom vibro-impact system with symmetry is considered. Due to the symmetry, the Poncar é map Pis the second iteration of another virtual implicit map Q. It is shown that the symmetric period n-2 motion of the vibro-impact system corresponds to the symmetric fixed point of the Poncaré map. Then we can investigate bifurcations of the symmetric period n-2 motion by researching into bifurcations of the associated symmetric fixed point. Based on the symmetry of the system, it is shown that the Neimark-Saker-pitchfork bifurcation of the symmetric fixed point of the Poncaré map Pcorresponds to the Neimark-Saker-flip bifurcation of the map Q. By using the map Q, according to the two-parameter unfolding of the normal form, we reveal the possible local dynamical behaviors of the symmetric fixed point of the Poncaré map P near the Neimark-Saker-pitchfork bifurcation point in detail. Near this codimension two bifurcation point, the dynamic behaviors of the vibro-impact system can be expressed by a single symmetric fixed point, a pair of conjugate fixed points, a pair of conjugate quasi-periodic attractors or a single symmetric quasi-periodic attractor in the projected Poncaré section. The numerical simulation represents various possible cases near the Neimark- -Saker-pitchfork bifurcation point. It is shown that the interaction of the Neimark-Saker bifurcation and the pitchfork bifurcation may lead into the creation of some new results. The symmetric fixed point bifurcates into a pair of conjugate unstable fixed points firstly, and the two conjugate fixed points will bifurcate into the same symmetric quasi-periodic attractor finally.