摩擦系统倍周期分岔细结构的研究

Study on the Fine Structure of the Period-doubling Bifurcation in Friction System

本文研究了服从双状态变量本构律的单自由度摩擦系统的准静态运动。通过对相平面上运动轨迹的直接观察，以及运用Ｐｏｉｎｃａｒｅ截面和功率谱分析方法，对该系统的倍周期分岔和混沌行为进行了分析。研究结果表明，这样一个简单的摩擦系统可具有一个非常复杂的耗散结构。仅改变弹簧刚度，这个摩擦系统便可以朝平衡态、准周期态和混沌态等不同的状态演化。同时，与罗辑斯蒂映象类似，这些状态可以通过系统参数──弹簧刚度的连续改变而相继转换。即，本文对于服从双状态变量本构律的单自由度摩擦系统找到了一条由平衡态经过准周期态到达混沌态的倍周期分岔道路。对于这个信周期分岔序列，仿照Ｆｅｉｇｅｎｂａｕｍ，确定了累进比δｎ：δ１＝５．７５，δ２＝６．０３，δ３＝３．７７，δ４＝３．９４另外，在相平面上给出了混沌吸引子的图形。

The quasi-static behavior in a single degree of freedom elastic system with a two-state variable friction law has been studied in this paper. The period-doubling bifurcation and chaotic phenomena of the system are analyzed by investigating phase trajectories, Poincare maps and analyzing power spectra.We found that such a simple friction system can have a complex dissipative structure. As changing the spring stiffness, the system may be evolved to different states such as equilibrium, quasi-period and chaos,etc. Simultaneously, the states can be transformed one after another similar as logistic map when system parameter-spring stiffness is continuously changed. The road is found that a period-doubling bifurcation route to chaos from equilibrium state cross quasi-period state.For the period-doubling bifurcation sequence, following feigenbaum, the ratio δn is determined as δ= 5. 75, δ2= 6. 03,δ3= 3. 77, δ4=3. 94and chaotic attractor is illustrated in phase plane.