一类碰撞振动系统在内伊马克沙克-音叉分岔点附近的局部两参数动力学

考虑一类具有对称性的三自由度碰撞振动系统.系统的庞加莱映射在一定条件下存在对称不动点,对应于系统的对称周期运动.根据对称性导出庞加莱映射P是另外一个隐式虚拟映射Q的二次迭代.推导了庞加莱映射对称不动点的解析表达式.根据映射不动点的稳定性及分岔理论,映射P的对称不动点发生内伊马克沙克-音叉(Neimark--Saker-pitchfork)分岔对应于映射Q发生内伊马克沙克-倍化(Neimark--Sakerflip)分岔.利用隐式虚拟映射Q,通过对范式作两参数开折分析,研究了映射P的对称不动点在内伊马克沙克-音叉分岔点附近的局部动力学行为.碰撞振动系统在这个余维二分岔点附近的局部动力学行为可能表现为投影后的庞加莱截面上的单一对称不动点、一对共轭不动点、单一对称拟周期吸引子以及一对共轭拟周期吸引子.数值模拟得到了内伊马克沙克-音叉分岔点附近的各种可能情况.内伊马克沙克-分岔和音叉分岔互相作用可能产生新的结果:对称不动点虽然首先分岔为两个共轭不动点,但是这两个共轭不动点是不稳定的,最终收敛到同一个对称拟周期吸引子.

LOCAL DYNAMICAL BEHAVIOR OF TWO-PARAMETER FAMILY NEAR THE NEIMARK-SACKER-PITCHEORK 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 P is 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 P corresponds 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.