Effect of noise in a nonautonomous system of alternately excited oscillators with a hyperbolic strange attractor

We study the effect of noise for a physically realizable flow system with a hyperbolic chaotic attractor of the Smale-Williams type in the Poincaré cross section [Kuznetsov, Phys. Rev. Lett. 95, 144101 (2005)]. It is shown numerically that, by slightly varying the initial conditions on the attractor, one can obtain a uniform approximation of a noisy orbit by the trajectory of the system without noise, which is called the "shadowing" trajectory. We propose an algorithm for locating the shadowing trajectories in the system under consideration. Using this algorithm, we show that the mean distance between a noisy orbit and the approximating one does not depend essentially on the length of the time interval of observation, but only on the noise intensity. This dependence is nearly linear in a wide interval of the intensity of the noise. It is found that for weak noise the Lyapunov exponents do not depend noticeably on the noise intensity. However, in the case of strong noise the largest Lyapunov exponent decreases and even becomes negative, indicating the suppression of chaos by external noise.