Mechanisms of chaos generation in an atypical single-transistor oscillator

Abstract

A recently-introduced autonomous electronic chaos generator which has an elementary circuit topology not intentionally derived from canonical oscillators is considered. It comprises a bipolar junction transistor, a resistor, two inductors and one capacitor. Though its notable generative potential has been remarked, its functioning principles have thus far remained unclear. Here, we investigate them systematically. First, the diversity of available behaviours in relation to the component values is illustrated. Taking the resistor value as the primary control parameter, a period-doubling route to chaos is evident. Second, the circuit is viewed as a relaxation oscillator exciting a damped resonator which in turn perturbs it via influencing its reset, and an approximation which considerably simplifies the non-linear term is introduced. It is shown that chaos arises because of the reciprocal interaction between these two aspects of the dynamics, namely, at parameter settings in between the extremes for which one or the other dominates. Third, the Lur’e form is derived, representing the circuit as a non-linear feedback block alongside a linear transfer function. Its Bode plots further clarify the role of each component in shaping the frequency spectrum of the broadband oscillatory dynamics. Finally, an attempt is made to apply the harmonic balance method towards predicting the system behavior; while several factors hinder this, partial approximations for the oscillation amplitude, frequency and distortion are given. Besides illustrating the dynamical richness characterizing this particular circuit, several considerations of possibly general validity are offered.