Construction of Liapunov functions for highly nonlinear dynamic systems with feedback
Annotatsiya
Abstract The paper provides stability and qualitative research of oscillations of a highly nonlinear dynamic system with feedback. For a system satisfying Liapunov theorem, definite-positive recurrent functions of order 1 with negative derivatives are constructed. Sufficient stability conditions are established for the considered cases of power nonlinearities. Surface diagrams of Liapunov functions and their derivatives are constructed for various values of the feedback parameter. Using the Liapunov criterion, the behavior of the trajectories of dynamic system on the state planes and near the singular points is investigated. Possible limit cycles are determined based on the Poincaré method of contact curves. Integral curves and phase trajectory diagrams are constructed numerically using the Mathcad 13 software package. A transition process from an unstable focus to self-oscillating and relaxation vibration modes is established, as well as the corresponding limit cycles consistent with analytical definition of rings containing the indicated limit cycles.