分段线性齿轮系统的Lyapunov指数计算方法

A Method for Calculating Lyapunov Exponents of Piecewise-linear Spur Gear Systems

齿轮动力系统是一个分段线性的非光滑系统,由于系统的Jacobian矩阵在非光滑点处不存在,不能直接应用光滑动力系统的Lyapunov指数的计算方法来定量化地判别系统的动力学行为。本文利用M LLER算法,根据非光滑点处的检测函数和碰撞函数,找到齿轮系统不同状态之间偏移矢量的转换关系,得到了非光滑齿轮系统的最大Lyapunov指数的统一算法。通过算例,将系统的最大Lyapunov指数,与系统的相图和Poincaré截面进行比较,验证了齿轮动力系统最大Lya-punov指数计算方法的有效性。

A spur gear dynamic system is a piecewise -linear non-smooth one, so the methods for calculating the Lyapunov exponents of smooth dynamic systems cannot be applied to spur gear dynamic systems directly. In this paper, MULLER algorithm is employed to solve this problem. First, the transition condition at the discontinuities （singular points） is obtained. Then, the maximum Lyapunov exponent of spur gear systems can be computed to discrimimate whether an attractor is periodic or chaotic. An example is given to verify this method. By comparing the results with system phase plot and Poincare map, the validity of this method is proved.