Abstract
We introduce the concepts of the degree and the order of synchronism on the basis of a mathematical model of the emergence of synchronization in the form of an asymptotically stable integral torus in the phase plane. We investigate the existence conditions for synchronisms in a dynamic system described by differential equations with rapidly rotating phases. As an application we examine synchronisms in a system of quasi-Hamiltonian objects. In recent years the phenomena of synchronization and resonance in dynamic systems have been subjected to intensive study, in particular, in connection with the question of the synchronization of satellites [1, 2] and of mechanical vibrators [3]. On the mathematical side the appearance of synchronization is closely connected with the theory of differential equations with rapidly rotating phases. Here in the first place we must mention the works specified in [4–9].
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