A.F. Gribov – Ph.D.(Phys.-Math.), Associate Professor, Department «Mathematic Modeling», Bauman Moscow State Technical University

E-mail: alexandr-gribov@list.ru

A.D. Tsisarsky – Ph.D.(Eng.), Associate Professor, Department «Autonomous information and operating systems», Bauman Moscow State Technical University

E-mail: alex22081952@gmail.ru

B.I. Shakhtarin – Dr.Sc.(Eng.), Professor, Department «Autonomous information and operating systems», Bauman Moscow State Technical University

E-mail: shakhtarin@mail.ru

The article deals with the system of phase-locked frequency of arbitrary dimension with piecewise linear approximation of the nonlinear function. This approximation allows to obtain analytical conditions for the existence of different types of behavior of this system. Analytical conditions for the existence of homoclinic trajectories for piecewise linear systems in the case of the saddle-focus equilibrium state are obtained, thus the existence of the spiral chaos of Shilnikov is established. The analytical conditions of bifurcation of the birth and existence of multi– pass rotational cycles in a piecewise linear phase system on the basis of which the criterion of the bifurcation transition to chaos can be obtained are determined.

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