We have written a new equation to study the statistics of earthquake distributions. We call this equation "the generalized logistic equation". The Gutenberg-Richter frequency-magnitude formula was derived from the solution of the generalized logistic equation as an asymptotic case for the approximation of large magnitude earthquakes. To illustrate how the solution of the generalized logistic equation works, it was used to approximate the observed cumulative distribution of earthquakes in four different geological provinces: the Central Atlantic (40N -25N, 5W -35W), Canary Islands, Magellan Mountains (20N -9S, 148E -170E), and the Sea of Japan. This approximation showed an excellent correlation between the theoretical curves and observed data for earthquakes of magnitudes 1 < m < 9.

The generalized logistic equation is derived to model kinetics and statistics of natural processes such as earthquakes, forest fires, floods, landslides, and many others. This equation has the form

We have studied long period, 206 and 412 day, variations in tidal sea level corresponding to various moon phases collected from five observatories in the Northern and Southern hemispheres.Variations in sea level in the Bay of Fundy, on the eastern Canadian seaboard, with periods of variation 206 days, and 412 days, have been discovered and carefully studied by C. Desplanque and D. J. Mossman (2001, 2004). The current manuscript focuses on analyzing a larger volume of observational sea level tide data as well as on rigorous mathematical analysis of tidal force variations in the Sun-Earth-Moon system. We have developed a twofold model, both conceptual and mathematical, of astronomical cycles in the Sun-Earth-Moon system to explain the observed periodicity. Based on an analytical solution of the tidal force variation in the Sun-Earth-Moon system, it is shown that the tidal force can be decomposed into two components: the Keplerian component and the Perturbed component. The Perturbed component of the tidal force variation was 8  . It follows that the

The Late Pleistocene Antarctic temperature variation curve is decomposed into two components: ''cyclic'' and ''high frequency, stochastic.'' For each of these components, a mathematical model is developed which shows that the cyclic and stochastic temperature variations are distinct, but interconnected, processes with their own self-organization. To model the cyclic component, a system of ordinary differential equations is written which represent an auto-oscillating, self-organized process with constant period. It is also shown that these equations can be used to model more realistic variations in temperature with changing cycle length. For the stochastic component, the multifractal spectrum is calculated and compared to the multifractal spectrum of a critical sine-circle map. A physical interpretation of relevant mathematical models and discussion of future climate development within the context of this work is given.

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