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In this study, the nonlinear analysis of the sunspot index is embedded in thenon-extensive statistical theory of Tsallis. The triplet of Tsallis, as well asthe correlation dimension and the Lyapunov exponent spectrum were estimated forthe SVD components of the sunspot index timeseries. Also the multifractalscaling exponent spectrum, the generalized Renyi dimension spectrum and thespectrum of the structure function exponents were estimated experimentally andtheoretically by using the entropy principle included in Tsallis non extensivestatistical theory, following Arimitsu and Arimitsu. Our analysis showedclearly the following: a) a phase transition process in the solar dynamics fromhigh dimensional non Gaussian SOC state to a low dimensional non Gaussianchaotic state, b) strong intermittent solar turbulence and anomalous(multifractal) diffusion solar process, which is strengthened as the solardynamics makes phase transition to low dimensional chaos in accordance toRuzmaikin, Zeleny and Milovanov studies c) faithful agreement of Tsallis nonequilibrium statistical theory with the experimental estimations of i)non-Gaussian probability distribution function, ii) multifractal scalingexponent spectrum and generalized Renyi dimension spectrum, iii) exponentspectrum of the structure functions estimated for the sunspot index and itsunderlying non equilibrium solar dynamics.

Tsallis non-extensive statistics, intermittent turbulence, SOC and chaos in the solar plasma. Part one: Sunspot dynamics