The effect of interactive noise on the driven Brusselator model

Abstract

We examine the effects of interactive noise, i.e., noise which is processed by the system, on the Brusselator, a nonlinear oscillator. The Brusselator is investigated for three types of motion: periodic, quasiperiodic, and chaotic. Fluctuations are imposed on the system variables (Type V noise). The average fluctuation amplitudes are chosen between 10 and 10 000 ppm (1%) and they are Gaussian distributed. The simulated time series are analyzed by autocorrelation functions, Fourier spectra, Poincaré sections, one-dimensional maps, maximum Lyapunov exponents, and correlation dimensions. As a result, noisy periodic and quasiperiodic motion can be distinguished from deterministic chaos if the fluctuation amplitude is sufficiently small. The generic structure of the attractor can be recognized when Lyapunov exponents or correlation dimensions are extrapolated to zero fluctuation amplitude. Quasiperiodic attractors in the Brusselator are obscured even by small amounts of noise. Chaos in the Brusselator, on the other hand, is found to be robust against noise. For periodic motion we show that points close to a bifurcation exhibit a stronger sensitivity towards noise than points far away. In the log–log plots for the correlation dimension we observed break points for noisy periodic and quasiperiodic motion. They separate the noise from the purely deterministic part of the motion. For increasing noise levels the break points move to higher length scales of the attractor. Break points were not found for chaos in the Brusselator nor in the Lorenz and Rössler models. In the Brusselator very large noise levels beyond 1% obscure the deterministic structure even of a chaotic attractor so that any clear distinction between chaos and noise induced (statistical) aperiodicity is no longer possible. Implications on experimental systems are discussed.