The transitional time scale from stochastic to chaotic behavior for solar activity

Following the progression of nonlinear dynamical system theory, many authors have used varied methods to calculate the fractal dimension and the largest Lyapunov exponent lambda(1) of the sunspot numbers and to evaluate the character of the chaotic attractor governing solar activity. These include the Grassberger-Procaccia algorithm, the technique provided by Wolf et al., and the nonlinear forecasting approach based on the method of distinguishing between chaos and measurement errors in time series described by Sugihara and May. In this paper, we use the Grassberger-Procaccia algorithm to estimate the other character of the chaotic attractor. This character is time scale of a transition from high-dimensional or stochastic at shorter times to a low-dimensional chaotic behavior at longer times. We find that the transitional time scale in the monthly mean sunspot numbers is about 8 yr; the low-dimensional chaotic behavior operates at time scales longer than about 8 yr and a high-dimensional or stochastic process operates at time scales shorter than about 8 yr.