论文标题
关于齐格勒摆的动力和集成性
On the dynamics and integrability of the Ziegler pendulum
论文作者
论文摘要
我们证明，只要位于摆在摆在枢轴点的弹性弹簧的刚度为零，并且系统中没有摩擦，则可以集成Ziegler的摆（具有自助力的双摆）。我们表明，系统的整合性来自于周期性解决方案的两参数家族。我们解释了一种从整合动力学的过渡的机制，在该动力学中存在两个第一个积分和解决方案，属于四维相位空间中的二维托里，再到更复杂的动力学。简要研究了两个弹簧的刚度非零的情况。我们表明，常规动力学与混乱的动力学并存。
We prove that the Ziegler pendulum -- a double pendulum with a follower force -- can be integrable, provided that the stiffness of the elastic spring located at the pivot point of the pendulum is zero and there is no friction in the system. We show that the integrability of the system follows from the existence of two-parameter families of periodic solutions. We explain a mechanism for the transition from integrable dynamics, for which there exist two first integrals and solutions belong to two-dimensional tori in a four-dimensional phase space, to more complicated dynamics. The case in which the stiffnesses of both springs are non-zero is briefly studied numerically. We show that regular dynamics coexists with chaotic dynamics.
