History & Mathematics: Processes and Models of Global Dynamics
A more and more 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. The history is studied more and more 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.
This issue is the third collective monograph in the series of History & Mathematics almanacs and it is subtitled Processes and Models of Global Dynamics. The contributions to the almanac present a qualitative and quantitative analysis of global historical, political, economic and demographic processes, as well as their mathematical models.
This issue of the almanac consists of two main sections: (I) Analyses of the World Systems and Global Processes, and (II) Models of Economic and Demographic Processes.
We hope that this issue of the almanac will be interesting and useful both for historians and mathematicians, as well as for all those dealing with various social and natural sciences.
Multidisciplinarity is one of the salient features of contemporary science. This seems to be congruent with the globalization process as the globalized world will need a "global" science that is able to integrate and to unite various fields in order to solve fundamental problems. It may be said that, in some sense, the History & Mathematics almanac is "genetically" interdisciplinary as it was initially designed as a means to contribute to the construction of a bridge be-tween the humanities, social, natural, and mathematical sciences (see the Intro-duction to its first Russian issue [Гринин, Коротаев, Малков 2006: 4–11]). That time this very combination of words – History and Mathematics – might have looked a bit artificial. However, it gradually becomes habitual; what is more, it becomes to be recognized as quite an organic and important scientific phenomenon. This appears to be supported by the point that the recent two years have evidenced the publication of eight issues of the History & Mathe-matics almanac in Russian and two issues in English.1 Various conferences in this direction are held now quite regularly, and, what is especially promising, they bring together representatives of very diverse fields of human knowledge. One of the most recent conferences of this kind was held in December 2009 in the Institute of History and Archaeology (Ekaterinburg, Russia). The confer-ence has confirmed the existence of a critical mass of researchers within the world science that apply mathematical and quantitative methods to the study of history. Against this background the current discussions on the establishment of the Mathematical History academic journal do not appear coincidental.
Though the issue of economic cycles has been subject to numerous studies, this problem has retained its high importance. What is more, the current crisis has confirmed in an extremely convincing way the point that, notwithstanding all the successes achieved by many states in their countercyclical policies, no economy is guaranteed against uncontrollable upswings and unexpected crises and recessions that tend to follow such upswings. In addition to this, the financial globalization has increased substantially the risks of such cyclical fluctuations.
The notion of economic cycles is regarded ambiguously in economic science. In modern theories, business cycles are frequently defined as fluctuations of actual output around its potential value which is achieved in full employment conditions (see, e.g., Fischer, Dornbusch, and Schmalensee 1988). However, quite frequently economics does not achieve on the rise the potential GDP volume when a recession phase starts (such situations are described in more detail in Гринин, Коротаев 2009а: ch. 1). Thus, economic cycle, in our opinion, can be defined as periodical fluctuation around medium line of production volume, where repeating phases of rise and decrease can be specified.
In the model that we propose below we have tried to briefly describe the main features of medium-term cycles of business activity, or business cycles (7–11 years) that are also known as Juglar cycles after the prominent 19th-century French economist Clement Juglar (1819–1905), who investigated these cycles in detail (Juglar 1862, 1889).
 Many economists maintain that business cycles are quite regular with the characteristic period of 7–11 years. However, some suggest that economic cycles are irregular (see, for example, Fischer, Dornbusch, and Schmalensee 1988). As we suppose, comparative regularity of business cycles is observed rather at the World System scale than in every country taken separately. This corroborates the important role of exogenous factors for the rise and progress of business cycles (for more detail see below).
 Medium-term cycles (7–11 years) were first named after Juglar in works by Joseph Schumpeter, who developed the typology of different-length business-cycles (Schumpeter 1939, 1954; see also Kwasnisсki 2008).