Lorenz type attractor in electronic parametricgenerator and its transformation outside the accurate parametric resonance

The paper deals with a parametric oscillator composed of three LC -circuits and a quadratic nonlinear reactive element built on the basis of an operational amplifier and an analog multiplier; the equations for amplitudes of the interacting modes are derived. Motivation is a desire to implement the mechanism of parametric interaction of oscillatory modes giving rise to emergence of a strange attractor of Lorenz type without distortions introduced by nonlinearities of order three and higher. The study is based on a combination of circuit simulation using Multisim software and numerical integration of the dynamic equations of the system both in its original form and in the form of reduced equations for the slowly varying complex amplitudes. The proposed scheme for the first time allows demonstrating the decay mechanism of chaos generation described earlier by Pikovsky, Rabinovich and Trahtengertsin concern to the waves in magnetized plasma, in an electronic device in purified form. In addition to observation of the Lorenz-type attractor and characteristic features of the respective dynamics by means of the circuit simulation and on the basis of numerical integration of equations in the case of precise parametric resonance conditions, a study of transformation of the attractors is carried out with detuning frequencies, and the corresponding chart of dynamical regimes on the parameter plane is presented. It is shown that instead of the quasi-hyperbolic Lorenz-type attractor, with frequencies deviating from the exact parametric resonance, distinct types of attractors arise, although similar in shape to the original one, but lacking robustness: under variations of the parameters chaos may disappear with emergence of regular oscillatory regimes.