Cliodynamics (from Clio, the muse of history, and dynamics, the study of temporally varying processes) is a new multidisciplinary area of research focused at mathematical modeling of historical dynamics.^{[1]} It investigates dynamic processes in history, and ascends to such figures as Ibn Khaldun, Jack Goldstone, Randall Collins, Peter Turchin, John Komlos, Sergey Nefedov and Andrey Korotayev. The term was originally coined by Peter Turchin in 2003.
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History is the study of the past, with special attention to the written record of the activities of human beings over time. Scholars who write about history are called historians. History examines and analyzes a sequence of events and attempts to investigate objectively the patterns of cause and effect that determine events.^{[2]}^{[3]} In probability theory, an event is an outcome set to which a probability is assigned.
A mathematical model uses mathematical language to describe a system. The process of developing a mathematical model is termed 'mathematical modelling' (also modeling). Eykhoff (1974) defined a mathematical model as 'a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in usable form'.^{[4]} Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures.
A system is a set of interacting or interdependent entities, real or abstract, forming an integrated whole. The concept of an integrated whole can also be stated in terms of a system embodying a set of relationships which are differentiated from relationships of the set to other elements, and from relationships between an element of the set and elements not a part of the relational regime.
Dynamical system modeled as a mathematical formalization has fixed "rule" which describes the time dependence of a point's position in its ambient space. Small changes in the state of the system correspond to small changes in the numbers. The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule is deterministic: for a given time interval only one future state follows from the current state.
At the moment in this realm the main achievements have been made with respect to the mathematical modeling of longterm ("secular") cycles of sociodemographic dynamics (Turchin 2003, 2006, 2007 etc.), as well as the mathematical modeling of the extreme longterm ("millennial") trends of the World System dynamics (Korotayev et al. 2006a, 2006b etc.).
