Competition and Cooperation: Mechanism of Chaos Generation in the Smallest Double-Scroll Transistor-Based Oscillator

An autonomous double-scroll chaos generator was recently introduced, intentionally designed using a circuit topology that does not stem from canonical oscillators. The circuit comprises two bipolar junction transistors, one resistor, one capacitor, and two inductors. To date, the underlying operational principles of this circuit remained unclear. Here, the original circuit is initially simulated using realistic transistor models, and the key features of its dynamics are revealed. A simplified mathematical model reproducing the observed phenomena is then introduced based on an analysis of the functioning of each transistor in the circuit, which hinges on the coupling role of one of the inductors. When one transistor saturates, the other enters cut-off mode. This creates competitive dynamics, resulting in the transistors continuously switching between saturation and cut-off, eventually leading to chaotic behavior. The circuit is further analyzed in terms of two distinct components: a damped resonator and a relaxation oscillator, which consists of two transistors. A simplified switching model reproducing the double-scroll dynamics is subsequently put forward and proposed as a means of generalizing the results. It is found that, at the heart of double-scroll generation, there lies a peculiar motif of partially symmetric interconnection between the two bipolar transistors.