On Two-Frequency Oscillations of a DC Electric Drive with Pulse Control

Abstract

Purpose of research is of the paper is to analyze bifurcations of two-frequency oscillations of a DC electric drive with pulse-width control.Methods. The research is based on the construction of a stroboscopic Poincare map, the calculation of saddle periodic orbits and their stable and unstable invariant manifolds.Results. The study of the mechanisms of the occurrence of two-frequency oscillations from a periodic motion that loses stability in a DC electric drive with pulse-width control was carried out. A non-local saddle-node bifurcation leading to resonance (synchronization) on a torus characterized by a pair of independent frequencies when their ratio becomes a rational number, was studied.Conclusion. A bifurcation analysis of the control system of a DC electric drive, the dynamics of which is described by non-smooth nonautonomous differential equations, was carried out. The research was conducted on an iterable map obtained from the specified vector field in an analytical form. It is shown that the system under consideration demonstrates two-frequency oscillations that occur through the Neimark-Sacker bifurcation. In the phase space of the discrete model, a closed invariant curve corresponds to oscillations with two independent frequencies. It is shown that if these frequencies are correlated multiply, then a resonance occurs when the dynamics becomes periodic. But at the same time, the closed curve remains invariant, and the limit points of the orbit form a pair of periodic cycles – stable and saddle, corresponding to a rational frequency ratio. A closed invariant curve is formed by unstable manifolds of a saddle cycle. If the frequency ratio is irrational, then the dynamics is quasi-periodic. The orbits of such motion fill the closed curve everywhere densely.