What is Chaos Theory?

Answer 1

Chaos theory is a field of applied mathematics with applications in many related fields, including meteorology, economics, and even philosophy. The basic premise is that small differences in the initial conditions of a system yield widely-varying outcomes, precluding attempts at longterm prediction. It is often instantiated using the "butterfly effect" to show how capricious chaotic systems can be, the critical things being (i) their complexity, and (ii) just how delicately balanced they can be.

The term "butterfly effect" is also reflective of the Lorenz Attractor plot, which is the fractal structure that corresponds to the long-term behavior of a Lorenz Oscillator, a 3-D dynamical system, whose motion is chaotic. The plot looks like the wings of a butterfly.

Some of the common misunderstandings of this theory:

1. Chaos is not simply how complicated a system is, or how many different variables there are to take into account.
2. Chaos does not mean a system is not deterministic. By this I mean there may be a well defined set of rules the system follows, and in principle you can predict, as accurately as you want, what is going to happen - assuming you have the instruments to measure the current state of the system accurately enough, and the computing power to do sufficiently accurate calculations.

A chaotic system is one that is so sensitive to initial conditions that it's impractical to predict its behavior far in advance, because of limited accuracy of measurement and computing power.

A good example is the logistic map. You start with a constant r. (This isn't chaotic for all values of r, but for most values between about 3.5 and 4 it is chaotic.) You pick a number x_0, between 0 and 1. Then you define the sequence x_0, x_1, x_2, ... as follows:

x_1 = r * x_0 * (1-x_0)

x_2 = r * x_1 * (1-x_1)

x_3 = r * x_2 * (1-x_2)

etc. This isn't a complicated system - the rule for calculating the next term in the sequence is easy to understand, and there is only one variable (x). It's also deterministic: If you know exactly what the value of x_0 is, you can calculate x_1000 exactly. (More accurately: if you know x_0 to enough decimal places and have a big enough computer, you can calculate x_1000 to, say, 10 decimal places.) But in practice any attempt to calculate x_1000 is doomed to failure.

Ed Lorenz' seminal paper in 1963, Deterministic Nonperiodic Flow, can be found at the related link below.