_{i}: i=1,...,n} whose values uniquely characterise the state of a civilisation at a moment in time. All other properties of a civilisation are functions of these state variables.

The evolution in time of a civilisation is then governed by a set of n coupled non-linear differential equations, which the state variables must satisfy:

dX

_{i}/dt = F

_{i}(X

_{1},...,X

_{n};λ

_{1},...,λ

_{m}), (i=1,...,n).

The m parameters define external constraints upon a civilisation, such as those determined by the geography, geology and meteorology of the planet on which the civilisation exists.

Unfortunately, unlike Hari Seldon in Isaac Asimov's

*Foundation*books, we don't yet know what these equations are. We don't even know what the state variables are. One might, however, hypothesise the following candidates:

(i) Population size, and population rate of change.

(ii) Total amount of available free energy (i.e., energy resources), and rate of energy consumption (i.e., power output).

(iii) Amount of information stored, and amount of information processed.

Additional variables might then be required to characterise the hierarchies and organisations which define the political state of a civilisation. All economic variables, however, can be treated as functions of the variables in categories (i), (ii) and (iii).

Identifying the correct state variables, and the equations which govern the evolution of civilisations, is a task set for the reader. However, some progress towards understanding the dynamical processes in human history is being made by people such as Peter Turchin, who has launched his cliodynamics research programme, which aims to identify the mathematical patterns in human history. Turchin has identified, he claims:

*...long-term cycles that, it turns out, characterize the dynamics of agrarian states and empires. When we consider the long course of history of Western Europe from the days of the Roman empire until the Industrial Revolution, we observe waves of internal instability (widespread rebellions, state collapse, and persistent civil war) that recur every two or three centuries. The internal warfare cycles appear to be dynamically linked with cycles of population increase and decline or stagnation: population peaks are followed, after a time lag, with peaks of instability. This empirical pattern suggests some kind of a Malthusian explanation. However, the American sociologist Jack Goldstone showed that population growth beyond the means of subsistence does not directly bring the onset of civil wars. Instead, its effect is indirect, mediated through the social and political structures (elite overproduction and state fiscal insolvency), which is why there is a time lag between population and instability peaks. This pattern of linked population and internal warfare oscillations is not limited to Europe. Recently two Russian historians, Sergey Nefedov and Andrey Korotayev, showed that the same relationship holds for China during its two thousand year imperial history, for Egypt (from the Hellenistic period to the nineteenth century), and for Russia. It is remarkable that such complex, and very different, societies would all show similar dynamical patterns.*

Civilisation is a deterministic non-linear dynamical system, and it is also, presumably, one which exhibits sensitive dependence upon initial conditions: two different histories which begin in a very similar state can, one presumes, diverge from each other at an exponential rate. In terms of dynamical systems theory, such systems are said to be chaotic, hence civilisation is a chaotic dynamical system. Human history also appears to be an aperiodic trajectory in civilisation space, at least over the space of 10,000 years or so. A crucial question, however, is whether the laws of civilisation permit the exist of attractors.

An attractor is a subset of the set of all possible states of a dynamical system, which is such that once a trajectory of the system enters, it never leaves. This doesn't entail that the trajectories are doomed to become periodic once they enter the attractor; they can continue indefinitely, without repeating the same exact state. Whilst chaotic systems are defined by the sensitive dependence on initial conditions that they exhibit, they also tend to possess various chaotic attractors. Which begs the questions:

*Is civilisation a chaotic system which possesses attractors, and if so, is human history converging to such an attractor?*

The obvious candidate for such an attractor is secular, liberal, democratic, capitalist society. This is essentially the possibility Francis Fukuyama was raising in

*The End of History*. Whilst the current state of the world might not currently reside inside this attractor, and perturbations like 9/11 might be capable of temporarily taking it further away from the attractor, perhaps the current state of the world lies inside the basin of attraction for global secular, liberal, democratic, capitalist society.

Cliodynamics Deterministic nonlinear dynamical system Attractors