Chaos

It looks a little more mathematical and regular than it is; its exactitude is obvious, but its inexactitude is hidden; its wildness lies in wait. G. K. Chesterton

It is possible for a deterministic system to exhibit behaviour that is unpredictable. Such a system is in a state of chaos. Here is an example taken from “Chaos and Order in the Capital Markets” by Edgar E. Peters.

Suppose there is a stock with a price P_t at time t. It sells for less than $1. The behaviour of P is deterministic and given by:

P_t+1 = a*P_t – a*P_t*P_t

The first term might represent buyers influence on price and the second term sellers influence. The constant a is a rate parameter. We can now compute how prices evolve over time for different starting prices, P₀, and rate parameter values, a.

For small values of a, the price converges to zero. At a growth rate of a = 2.5 and P0 = 0.1, a fair price of 0.60 is eventually reached, for example (Chart 1).

Chart 1

At a growth rate above 2.5, there are suddenly two possible fair prices and the system oscillates between them (chart 2). At that critical level, buyers and sellers are not entering the market equally. There is a lag as the quadratic term becomes a bigger drag than the growth rate (a) but at the lower price, the growth factor dominates. There are two fair values: at one the sellers sell, at the other the buyers buy.

Chart 2

At a growth rate of 3.5, the price cycles through four prices (Chart 3).

Chart 3

Then, above a = 3.75, there are an infinite number of prices and prediction is impossible (Chart 4).

Chart 4

The features of this chaotic situation are:

A nonlinear deterministic relationship;
Critical levels where more than a single equilibrium exists;
A self-similarity in that at different intervals of time (hourly, daily, weekly) the system has similar characteristics (fractal structure);
Sensitive dependence on initial conditions – a slight change in P_t could cause a huge change in P_t+1.

The solar system is a chaotic system! Planets obey the (deterministic) law of gravity, but the future movements of a system can only be predicted accurately if there are two bodies. As soon as there are three or more bodies, the positions cannot be predicted precisely and the precision decreases the further ahead we predict.

Link to chaos books and sites

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