Complex dynamics and chaos in electronic self-oscillator with saturation mechanism provided by parametric decay

We consider an electronic oscillator based on two LC-circuits, one of which includes negative conductivity (the active LC-circuit), where complex dynamics and chaos occur corresponding to the model of wave turbulence of Vyshkind–Rabinovich. The saturation effect for the self-oscillations and their chaotisation take place due to parametric mechanisms due to the presence of a quadratic nonlinear reactive element based on an operational amplifier and an analog multiplier. The study is based on combination of circuit simulation with the use of the software product Multisim and of numerical computations with equations that directly describe the oscillations of voltages and currents in the oscillatory circuit, amplitude equations, and the equations represented in the form suggested by S.Y. Vyshkind and M.I. Rabinovich. For all these models, time dependences of dynamic variables are presented as well as portraits of attractors, and Lyapunov exponents depending on parameters. For the Vyshkind-Rabinovich model we additionally present a chart of dynamic regimes in the parameter plane. It is shown that all models demonstrate transitions to chaos through period-doubling bifurcation scenario observed under decrease in the supercriticality parameter in the active LC-circuit. The resulting chaotic attractor is similar in structure to the R.ossler attractor. The proposed scheme allows observing in the electronic device chaotic dynamics of the resonant triplet under instability of the high-frequency mode, considered in due time by Vyshkind and Rabinovich and interpreted as a model of a certain type of wave turbulence in dissipative media. The presented results testify a possibility of using the considered electronic circuit for analog simulation of oscillatory and wave phenomena in systems to which the Vyshkind–Rabinovich model is applicable.