Parametric generator of hyperbolic chaos based on two coupled oscillators with nonlinear dissipation

Abstract

The design of a parametric chaos generator based on two coupled Q-modulated oscillators with working frequencies differing by a factor of two and pumped by pulses at triple frequency with a repetition period equal to the modulation period of the Q factor is considered. We analyze a model in which the time evolution comprises four periodically repeated stages of the same duration. At the first stage, the oscillators are excited parametrically in the presence of linear dissipation; at the second stage, the second oscillator is damped; at the third stage, the oscillators interact via a quadratic nonlinearity; while at the fourth stage, the first oscillator is damped. The transformation of the oscillation phase over four stages is defined by an expanding circle map. In the phase space of the four-dimensional mapping describing the variation of the state over the modulation period, the Smale-Williams attractor occurs. We consider the results of numerical analysis of chaotic dynamics determined by the presence of the attractor as well as the result of calculations confirming its hyperbolic nature on the basis of the cone criterion known from the mathematical literature.