Abstract. The problem of prediction of a given time series is examined on the basis of recent nonlinear dynamics theories. Particular attention is devoted to forecast the amplitude and phase of one of the most common solar indicator activities, the international monthly smoothed sunspot number. It is well known that the solar cycle is very difficult to predict due to the intrinsic complexity of the related time behaviour and to the lack of a successful quantitative theoretical model of the Sun's magnetic cycle. Starting from a recent previous work, we checked the reliability and accuracy of a forecasting model based on concepts of nonlinear dynamical systems applied to experimental time series, such as embedding phase space, Lyapunov spectrum, chaotic behaviour. The model is based on a local hypothesis of the behaviour on embedding space, utilising an optimal number of neighbour vectors to predict the future evolution. The performances of this method for the current 23rd solar cycle suggest its valuable insertion in the set of the so-called non-precursor statistical-numerical prediction techniques. The main task is to set up and to compare a promising numerical nonlinear prediction technique, essentially based on an inverse problem, with the most accurate prediction methods, like the so-called "precursor methods" which appear now reasonably accurate in predicting "long-term" Sun activity, with particular reference to "solar" and "geomagnetic" precursor methods based on a solar dynamo theory.

Abstract. Using wavelet analysis approach, we can derive a measure of the disorder content of solar activity, following the temporal evolution of the so-called wavelet entropy. The interesting feature of this parameter is its ability to extract a dynamical complexity information, in terms of frequency distribution of the energy content, avoiding restrictions, common in the nonlinear dynamics theory, such as stationarity. The analysis is performed on the monthly time series of sunspot numbers. From the time behaviour of the wavelet entropy we found a clear increase in the disorder content of solar activity for the current 23 th solar cycle. This result suggests general low accuracies for current solar cycle prediction methods. Moreover, we pointed out a possible connection between wavelet entropy behaviour and solar excursion phases of solar dipole.

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We have examined the long-term trends in the solar variability that can be deduced from some indirect data and from optical records. We analyzed the radiocarbon measurements for the last 4500 years, based on dendrochronology, the Schove series for the last 1700 years, based on auroral records, and the Hoyt-Schatten series of group sunspot numbers. Focusing on periodicities near one and two centuries, which most likely have a solar origin, we conclude that the present epoch is at the onset of an upcoming local minimum in the long-term solar variability. There are some clues that the next minimum will be less deep than the Maunder minimum, but ultimately the relative depth between these two minima will be indicative of the amplitude change of the quasitwo-century solar cycle.