Regular and chaotic motions in applied dynamics of a rigid body

摘要

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 Poincaré) 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 Poincaré 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.