One of the earliest instances where truly complex phenomena where observed as such was the discovery of a phenomenon that resides between predictability and thus order and total unpredictability and randomness. This phenomenon, called "deterministic chaos", involves a brand of mathematics called nonlinear dynamical systems (NDS) (Hirsch, Smale & Devaney, 2004).   

The field of NDS studies time-dependent phenomena (like epidemics, conflict, or ecological systems with both predators and prey) by using systems of differential equations to track the unfolding of a system's state as reflected in a set of its "state variables", populations of predator species versus prey as functions of time, for example. There are well-studied models in the field such as the Lotka-Volterra, predator-prey Model, and the Lanchester model of political conflict and war (Epstein, 1997).

In NDS models, these borderline conditions between predictability and pure randomness are what has been called "chaos," "strangeness" or "criticality." These conditions come about when time is modeled as unfolding in discrete steps rather than continuously, when the model of the system is stopped to take stock, and then is restarted once more to take action and see what unfolds.  In human systems, conditions where our models seem to reside in this place between predictability and uncertainty are called "criticalization." This term highlights the fact that the system may transition from stability to criticality depending upon the particulars that are unfolding (Goldstein, Hazy & Lichtenstein, 2010).

Criticalization is a key idea in the study of Complexity and Society because it signals the presence of potentiality for change. Absent this potential, social systems can be self reinforcing.and quite stable in their functioning at their own level of scale. Alternatively, systems can also go into states of instability where there is no real potential to change, where downward or upward spirals of reinforcing feedback loops and divergence dominate dynamics until all predictability and structure is lost. The "fog of war" pehnomenon when a battle is raging, might be an example of this.

To understand the boundaries of these conditions, one must understand some of the key features of dynamical systems, and in particular the complementary notions of dynamic stability and of attractors.

 

References

Epstein, J. M. (1997). Nonlinear dynamics, mathematical biology, and social science (Vol. IV). ISBN 9780201419887.

Goldstein, J., Hazy, J. K., & Lichtenstein, B. (2010). Complexity and the Nexus of Leadership: Leveraging nonlinear science to create ecologies of innovation. ISBN 978023062272.

Hirsch, M.W., Smale, S., & Devaney, R.L. (2004). Differential equations, dynamical systems, & and introduction to chaos, 2nd edition. ISBN 0123497035.

Holland, J. H. (1975). Adaptation in natural and artificial systems. ISBN 9780262581110.

Holland, J. H. (2001). Exploring the evolution of complexity in signaling networks. Retrieved August 22, 2002, 2002, from http://www.santafe.edu/sfi/publications/wplist/2001