On rational phase-locking oscillations of a simple sawtooth oscillator, with UJT

There are some reports about sawtooth oscillations on various systems. The authors constructed the sawtooth oscillator circuit exploiting a Uni-Junction Transistor (UJT), and investigated various nonautonomous-like oscillations induced by an external sinusoidal input. Different from the cases of ordinary differential equation (ODE) systems driven with some periodic oscillation, this input periodically alternates only the threshold of the mode transition. Many systems in various fields of electrical circuits, biology and so on utilize threshold variation or modulation between operational modes, and our system is a very simple representative. Though our circuit behaves as a piecewise-continuous ODE system, we will suggest one-dimensional map depending on the phase of the external sinusoidal input. And we will study the background about the existence of various rational locking mode in the analytic method, numerical investigations and circuit experiments, identifying our system as a discrete dynamical system. This paper reveals not only the existing regions of the respective modes, but also reports about respective itineraries of periodic solutions geometrically. Readers will see that the mode distribution constitutes a Cantor set structure.