一个分段连续系统中的胖奇异集激变

A fat strange set crisis in a piece wise-continuous system

报道一种有特色的激变 .这种激变是在一类分段连续力场作用下的受击转子模型中观察到的 .描述系统的二维映象定义域中的函数不连续边界随离散时间发展振荡 ,从而使这个边界的向前象集构成一个承载混沌运动的胖分形 .在控制参数的一个阈值下 ,一个椭圆周期轨道突然出现在此胖混沌奇异集中 ,使得迭代向它逃逸 ,胖混沌奇异集因此突然变为一个胖瞬态集 .在这种情况下 ,有可能根据椭圆周期轨道逃逸孔洞 ,以及胖分形奇异集的测度随参数变化的规律 ,估算迭代在奇异集中的平均生存时间所遵循的标度规律 .直接数值计算和由此估算所得标度因子值可以很好地互相印证 .

This article reports a characteristic crisis observed in a kicked rotor subjected to a piecewise continuous force field. The discontinuity border in the definition range of the two-dimensional mapping, which describes the system, oscillates as the discrete time develops so that the forward images of the border form a fat fractal. With a chosen group of parameters the iterations on the fat fractal display chaotic motion, and the transient iterations from the initial values in a certain region of the phase space are attracted to the fat fractal. At a threshold of a control parameter, an elliptic periodic orbit suddenly appears inside the fat strange set, inducing an escaping of the iterations to the elliptic islands around it. The fat chaotic attractor thus suddenly transfers to a fat transient set. The influence of the feature of the crisis on the dependence of the lifetime in the transient set on the control parameter has been analyzed. It is shown that the dependence follows a universal scaling law suggested by Grebogy, Ott and Yorke, and the scaling exponent can be approximated according to the variation rules of the elliptic islands and the measure of the fat fractal as control parameter changes. It is possible to compare the results of the direct numerical computation on the scaling exponent and that obtained with the approximation. They are in good agreement.