A system of two coupled nonautonomous oscillators is considered. Dynamics of complex amplitudes is governed by differential equations with periodic piecewise continuous dependence of the coefficients on time. The Poincaré map is derived explicitly. With the exclusion of the overall phase, on which the evolution of other variables does not depend, the Poincaré map is reduced to three–dimensional (3D) mapping. It possesses an attractor of Plykin-type located on an invariant sphere. Computer verification of the cone criterion confirms the hyperbolic nature of the attractor in the 3D map. Some results of numerical studies of the dynamics for the coupled oscillators are presented, including the attractor portraits, Lyapunov exponents, and the power spectral density.
A special comment is needed to the work of Halbert and Yorke (Ref. 19) announcing a physical realization of the Plykin attractor. As a physical object, the taffy-pulling machine they discuss is not a low-dimensional system, but contains a piece of continuous medium undergoing deformations in such way that the motion of local elements of the medium obeys a map with the Plykin-type attractor. In other words, it is an ensemble of elements, each of which carries out motion on the Plykin-type attractor. Thus, referring to the physical realization of the attractor, the authors stand for another meaning than that we have in mind here (as well as other authors, Refs. 9 and 20–22).