Effects of noise on the chaotic dynamics of nonlinear systems

Abstract

The influence of weak external noise on the dynamics of the logistic map and the driven double-well oscillator which undergo period-doubling bifurcations has been studied for both case of white and colored noise in parameter range where the transition from a periodic solution to a chaotic one occurs. The effects of noise on the dynamics of the systems are analyzed from the standpoint of the effects on the attractors and the maximal characteristic Lyapunov exponent. The noise-induced gap in the cascade bifurcation sequence and the scaling behavior of the characteristic Lyapunov exponent in the vicinity of the transition have been investigated through power spectrum analyses of the solutions. It is seen from the bifurcation diagram that the noise destroys all the miscellaneous structures observed in the absence of noise, such as periodic windows between chaotic band. Also the injection of noise leads to a change in the sign of Lyapunov exponent from negative which is characteristics of periodic solutions in the absence of noise to the positive which is characteristics of the chaos. This is so called noise-induced early chaos. The logistic map (discrete system) and the driven double-well oscillator (continuous one) show the same universal scaling behavior in the response to the external noise. The reason is because both systems follow the same route to chaos. We confirm that the external noise plays an important role as a disordering field which produces noise-induced chaos and observe that the effects of it can be described by two parameters, one is the noise intensity and the other the noise correlation time in the case of colored noise.