NUMERICAL AND ANALYTICAL STUDY OF NONLINEAR DYNAMICS OF A MULTIDIMENSIONAL CONTROLLED OBJECT

The paper studies a multidimensional dynamic system that models oscillations of various technical controlled objects (transport, aviation, fire, underwater and other autopilots), the mathematical description of which can be presented in the form of piecewise continuous or discontinuous autonomous ordinary differential equations with a characteristic of object control in the form of a piecewise linear continuous or discontinuous function. The structure of the phase space of the considered dynamic system is made up of three subspaces, in each of which the dynamics is determined by “its own” the nonlinear dynamics of the system and phase portraits of the full phase space is studied using the mathematical apparatus of bifurcation theory and the method of point mappings of two-dimensional Poincare? surfaces. Analytical equations of point mappings, equations for determining the coordinates of fixed points corresponding to stable periodic symmetrical movements of the object are given. The existence of a separatrix loop going from saddle to saddle is proved, the parameter range where the separatrix loop certainly exists is determined, and also it is asserted that in any neighborhood of the saddle-focus separatrix loop in the selected parameter range there is a countable set of periodic motions of the piecewise linear dynamic system under study. A lemma on the existence in the parameter space of a non-empty range of values for which a separatrix loop exists in the system is presented and proved. A detailed algorithm for numerical calculation of the separatrix loop is developed. It is proved that in any neighborhood of the saddle-focus loop in the found parameter region, there is a countable set of periodic motions of the system. Calculations of complex types of motion of the controlled object are carried out according to the developed software product in Python using the Plotly mathematical package. It is shown that in this parameter range in the system, there are both symmetric stable periodic motions of the object and chaotic motions of the object.