Beletskii V. V., Pivovarov M. L., Starostin E. L.
M. V. Keldysh Institute of Applied Mathematics, 4 Miusskaya Sq., Moscow, 125047, RussiaSpace Research Institute, 84/32 Profsoyuznaya, Moscow, 117810, RussiaM. V. Keldysh Institute of Applied Mathematics, 4 Miusskaya Sq., Moscow, 125047, Russia.
Chaos. 1996 Jun;6(2):155-166. doi: 10.1063/1.166160.
Periodic and regular motions, having a predictable functioning mode, play an important role in many problems of dynamics. The achievements of mathematics and mechanics (beginning with Poincare) have made it possible to establish that such motion modes, generally speaking, are local and form "islands" of regularity in a "chaotic sea" of essentially unpredictable trajectories. The development of computer techniques together with theoretical investigations makes it possible to study the global structure of the phase space of many problems having applied significance. A review of a number of such problems, considered by the authors in the past four or five years, is given in this paper. These include orientation and rotation problems of artificial and natural celestial bodies and the problem of controlling the motion of a locomotion robot. The structure of phase space is investigated for these problems. The phase trajectories of the motion are constructed by a numerical implementation of the Poincare point map method. Distinctions are made between regular (or resonance), quasiregular (or conditionally periodic), and chaotic trajectories. The evolution of the phase picture as the parameters are varied is investigated. A large number of "phase portraits" gives a notion of the arrangement and size of the stability islands in the "sea" of chaotic motions, about the appearance and disappearance of these islands as the parameters are varied, etc. (c) 1996 American Institute of Physics.
具有可预测功能模式的周期性和规则运动，在许多动力学问题中起着重要作用。数学和力学（从庞加莱开始）的成就使得确定这样的运动模式成为可能，一般来说，这些运动模式是局部的，并且在本质上不可预测的轨迹的“混沌海洋”中形成规则性的“岛屿”。计算机技术的发展与理论研究一起，使得研究许多具有应用意义问题的相空间全局结构成为可能。本文给出了作者在过去四五年中考虑的一些此类问题的综述。这些问题包括人造天体和自然天体的定向与旋转问题以及移动机器人运动控制问题。针对这些问题研究了相空间的结构。通过庞加莱点映射方法的数值实现来构建运动的相轨迹。区分了规则（或共振）、准规则（或条件周期）和混沌轨迹。研究了随着参数变化相图的演化。大量的“相图”给出了关于混沌运动“海洋”中稳定岛的排列和大小、随着参数变化这些岛屿的出现和消失等情况的概念。（c）1996美国物理研究所。
