Chaos Theory – A Brief Overview

A lot of us were probably first introduced to the idea of chaos theory in the film “Jurassic Park” wherein Jeff Goldbloom is constantly telling us that the essence of chaos theory is that “Life finds a way”. This is a very acute  way of describing the broader world of complex systems and chaotic behaviour although the essence of the theory is there. In this blog we will break down chaotic behaviour into its most fundamental form using recursive relations, and looking at simple ‘complex systems’ such as a coupled pendulum.

Chaotic systems are applicable to almost every aspect of life, from population dynamics, to hyperbolic geometry to abstract functions of the brain. This makes it one of the most breakthrough developments in mathematics and physics in history.

The idea of chaos in physics is actually quite abstract and although we describe some systems as being “chaotic” they can still be modelled by the most simple formulas. For example if we take a recursive relation which is also periodic, where we choose specific initial conditions and go for an infinite amount of time we may perceive simple periodic behaviour although if we choose to start at some other point which may be very close to the other point we can observe that after very few recursions that the period of the function has changed and it is almost as if we are plotting two shifted functions which are out of phase.

 This phenomenon is called period doubling and the period of events can be doubled until the period is essentially infinite. This directly relates to the idea of “if it can happen then it has happened or will happen”, from which Murphy’s Law is derived.

Chaos is observed in almost every thing we observe and if we look close enough and for long enough it can be modelled. Some such chaotic behaviour turns out to be self similar, meaning that if we observe it on one scale that it looks near identical to if we observe it another scale. These are called fractals, and they can be used in many aspects of physics such as probability theory or Monte Carlo simulations.

Monte Carlo Simulations themselves are some of the most useful programmes as they can calculate to a high degree of accuracy the number pi, they can also determine the golden ratio using a fibonacci sequence (two numbers added together determine the next number), which is prevalent in electrodynamics and quantum physics, Monte Carlo simulations are used in almost all aspects of computational physics.

References