The present yearbook (which is the fourth in the series) is subtitled Trends & Cycles. It is devoted to cyclical and trend dynamics in society and nature; special attention is paid to economic and demographic aspects, in particular to the mathematical modeling of the Malthusian and post-Malthusian traps' dynamics. An increasingly important role is played by new directions in historical research that study long-term dynamic processes and quantitative changes. This kind of history can hardly develop without the application of mathematical (...) methods. There is a tendency to study history as a system of various processes, within which one can detect waves and cycles of different lengths – from a few years to several centuries, or even millennia. The contributions to this yearbook present a qualitative and quantitative analysis of global historical, political, economic and demographic processes, as well as their mathematical models. This issue of the yearbook consists of three main sections: (I) Long-Term Trends in Nature and Society; (II) Cyclical Processes in Pre-industrial Societies; (III) Contemporary History and Processes. We hope that this issue of the yearbook will be interesting and useful both for historians and mathematicians, as well as for all those dealing with various social and natural sciences. (shrink)

Section 1 of this article presents a mathematical analysis of the longterm global urbanization dynamics and demonstrates that it could be described as a series of phase transitions between attraction basins. This makes it possible to suggest new approaches to the analysis of global social macroevolution. Section 2 presents a threestage model of the macroevolution of the World System statehood (early – developed – mature state) that, we believe, describes the main features of political macroevolution better than the twostage model (...) proposed by Claessen and Skalník (early – mature state). This model has been used as a basis for the analysis of changes in the development of political structures from the prestate polities and the most primitive early states (and their analogues) till the modern supercomplex states. In this section we also analyze the relationship between the growth of the territory controlled by the World System states and the growth of the statehood complexity. Finally, Section 3 attempts at the detection of the correlation between the global evolution of political structures and global urbanization processes. This study confirms the presence of a system of attraction basins and phase transitions within the World System macroevolutionary development field, as well as the presence of mathematically interpretable correlations and interrelations between the analyzed global processes – political, urbanizing, demographic, technological, and sociostructural. (shrink)