The Theory behind Chaos

Chaos theory is the study of dynamical systems, which are extremely sensitive to initial conditions. If we were to change these initial conditions slightly, the outcome would be extremely different, rendering long-term predictions almost impossible. In this post I will give a brief introduction on chaos theory and what it is used for. 

History

Chaos theory originates from the studies of Henri Poincaré concerning the three-body problem (the problem of the motion of three objects in mutual gravitational attraction). In these studies, Poincaré concluded that there can be orbits which are non-periodic and yet they’re not forever increasing nor approaching a fixed point.

Later studies on non-linear differential equations were carried out by G.D. Birkhoff, A.N Kolmogorow, M.L Cartwright, J.E Littlewood, and S. Smale. Apart from Smale, who was probably the first pure mathematician to study nonlinear dynamics, all of these studies were inspired by physics; experiments had shown turbulence in fluid motion and non-periodic oscillation in radio circuits, but there was no theory to describe it.

Although there were some insights in the first half of the 20th century, chaos theory only became formalised after mid-century, after it became clear that linear theory could not describe the behaviour observed in some experiments. The main catalyst for this development was the electronic computer, as much of the mathematics in chaos theory requires repeated iteration of mathematical formulas, something that computers made practical and easy to do.

A pioneer of chaos theory was Edward Lorenz, whose interest was peaked through his work on weather prediction in 1961. He discovered that small changes in the initial conditions produced large changes in the long-term outcome. His discovery, named Lorenz attractors, showed that even detailed atmospheric modelling cannot make precise long-term weather predictions.

In 1982, Benoit Mandelbrot published ‘The Fractal Geometry of Nature‘ (more about fractals here), which became a classic of chaos theory. Following this, in 1987 James Gleick published the book ‘Chaos: Making a New Science‘, which became a best-seller and introduced the principles of chaos theory, as well as its history, to the general public (although it has been said that his description of its history under-emphasised important Soviet contributions).

What is a non-linear dynamical system?

A non-linear dynamical system will exhibit one or more of these behaviours:

• it is forever at rest
• it is forever expanding
• it is in periodic motion
• it is in quasi-periodic motion
• it is in chaotic motion

The latter is the non-periodic complex motion which chaos theory is concerned with.

Chaotic Movement

For a system to be considered to be chaotic it must satisfy the following properties:

• it must be sensitive to initial condition: this is also called ‘the butterfly effect’

Source: J.L. Westover

• it must be topologically mixing: This means it will evolve in the same region for a certain period of time, then eventually overlap with any other given region. This highlights that two adjacent points in a complex system will eventually end up in very different positions after some time has passed.
• its periodic orbits must be dense: explained here

Attractors

Some systems seem to be too chaotic to recognise a pattern, however mathematicians began to discover that complex systems often seem to run through a sort of cycle, although situations are rarely repeated. Plotting systems in simple graphs showed that there often seems to be a situation that the system tried to achieve – an ‘equilibrium’. This point is called an attractor.

There is also such a thing as a strange attractor, which is a more dynamic ‘equilibrium’. The difference between these two is that an attractor represents a state to which the system finally settles, whilst a strange attractor represents a trajectory upon which the system runs from situation to situation without ever settling down. The Lorenz attractor described before is an example of a strange attractor.

Source: theory.org

The discovery of attractors was extremely important as it revealed the concept of self-similarity.

Random or Chaotic?

Chaotic sequences are generated deterministically from a dynamical system:

Random processes are fundamentally different; two successive realisations of a random process will give different sequences, even if the initial state is the same. A random process is non-deterministic.

“All methods for distinguishing deterministic and stochastic processes rely on the fact that a deterministic system always evolves in the same way from a given starting point.”

Applications

Chaotic theory has applications to many scientific areas including geology, biology, computer science, economics, engineering, meteorology, physics and politics.

Chaotic behaviour has been observed in many different man-made systems (electrical circuits, lasers, computer models of chaotic processes etc.), as well as in nature (weather, population growth in ecology and movement etc.).

Currently, chaotic theory is being applied to medical studies of epilepsy in order to predict seemingly random seizures by observing initial conditions.

Sources: 1 | 2 | 3

Hope you enjoy this post and have a nice Easter! M x