Investigation of the Bifurcation Process in the RL Diode Circuit by Employing Mathieu and Hill Equations

Abstract

The work deals with the dynamic processes of the RL diode circuit at the bifurcation points. It is shown, that just at the first bifurcation point (slightly beyond it) the circuit dynamics can be modeled by the Mathieu equation. Furthermore, the role of the voltage source there is not in imposing forcing function to the system equation. It has been discovered that the role of the source is that of a pumping source in the parametric sense. Hence, the resonator behaves at the bifurcation point as if it were a resonator, whose reactor is periodically time-varying. Further investigations of the dynamics at higher bifurcation points reveal that a similar mechanism is repeated and the processes there can be modeled by the Hill equation (which is a generalized version of the Mathieu equation). The theoretical investigation for the first bifurcation point is supported by experiments. Practical half-tone generators are shown to be closely related to the RL diode circuit.