Stability and bifurcation analysis in transformer coupled oscillators

We derive a simplified mathematical model to describe the nonlinear dynamics of a system consisting of a chaotic transformer-coupled oscillator. This well-known chaotic oscillator is currently modelled by very complex equations which are intractable analytically and numerically as well due to stiffness. The model derived is simple, flexible, accurate and efficient to provide full insight of the dynamics of the oscillator while compared with the well-known classical models. The electronic structure of the oscillator is proposed and sets of nonlinear ordinary differential equations are derived to describe the dynamics of the oscillator. Using the Routh-Hurwitz theorem, the stability analysis of critical points is carried out and conditions for the occurrence of Hopf bifurcations are derived. Various bifurcation scenarios are obtained numerically showing several striking routes to chaos. The biasing current is considered as bifurcation control parameter to highlight the effect of the bias (i.e. power supply) on the dynamics of the oscillator. This is a relevant contribution which enriches the literature as the effect of the bias on the dynamics of such oscillator has not been considered so far by the relevant state-of-the-art. The real physical implementation (i.e. use of electronic components) of the oscillator is considered to validate the simplified model proposed through several comparisons between experimental and numerical results. The bifurcation analysis reveals the possibility for the oscillator to move from near sinusoidal regime to chaos via the usual path of period doubling and sudden transition when monitoring the bias in very tiny windows. Regions of multistability are depicted in which coexist various types of attractors.