The effect of noise on a hyperbolic strange attractor in the system of two coupled van der Pol oscillators

TL;DR

This study investigates how noise influences a hyperbolic chaotic attractor in coupled van der Pol oscillators, demonstrating shadowing of noisy orbits, the linear relation between noise and orbit deviation, and noise-induced chaos suppression.

Contribution

It introduces an algorithm to locate shadowing trajectories in a hyperbolic chaotic system and analyzes the effects of noise on Lyapunov exponents and system dynamics.

Findings

Shadowing trajectories can be approximated with minimal dependence on observation time.

Mean distance between noisy and shadowing trajectories depends linearly on noise intensity.

Strong noise reduces the largest Lyapunov exponent, indicating chaos suppression.

Abstract

We study the effect of noise for a physically realizable flow system with a hyperbolic chaotic attractor of the Smale - Williams type in the Poincare cross-section [S.P. Kuznetsov, Phys. Rev. Lett. 95, 2005, 144101]. It is shown numerically that 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, that is called as 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 dependance is nearly linear in a wide interval of the intensities of noise. It is found out that for weak noise the Lyapunov exponents do not depend noticeably on the noise intensity. However, in the case of a strong noise the largest Lyapunov exponent decreases and even becomes negative indicating the suppression of chaos by the external noise.