The paper investigates the possibility of cyclic behavior of dynamic synergetic systems taking into account nonlinear processes of different increasing orders. The systems are represented in the form of dynamic, nonlinear, differential equations in phase spaces. Phase spaces are formed from the essential variables characterizing the system. Essential variables form a system of "order parameters" for models. Bifurcations in the behavior of a number of models of dynamic synergetic systems have been studied and the processes of the emergence of cyclic behavior in dynamic synergetic systems have been studied. Nonlinear processes in dynamical systems and their changes at special points of phase diagrams are studied. The behavior and stability of synergetic models in the areas of simple elementary catastrophes such as "assembly" and "fold" have been studied. Cyclic processes in the event of an “assembly” type disaster are investigated. A model of cyclical logistic revolutions in regional economies is considered. Cycles in the "soft" Arnold disaster "dovetail" have been studied. The occurrence of cyclic processes, as well as the stability of cycles, has been studied. Methods for model verification and model management capabilities are determined. The areas of phase diagrams for complex nonlinear dynamical systems with cyclic behavior are investigated. Dynamics of cycles in different regions of the phase space for synergetic systems is discussed. The problems of verification and control of models with the possible appearance of cycles are identified, and the emergence of higher order cycles is discussed.
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Lebedev Viktor Ivanovich
Doctor of Physical and Mathematical Sciences, Professor
North Caucasus Federal University
Stavropol, Russian Federation
