Chaos Theory

Chaos theory, also known as the butterfly effect, is used to describe the reactions of certain dynamical systems, especially where a slight change in initial conditions can cause a large difference in a dynamical system. Confused? Try this layman’s chaos theory definition — a butterfly flapping its wings in Brazil can cause a tornado over Texas. Chaotic systems often appear to exhibit random behavior; the behavior is called deterministic chaos, or simply chaos.

Although the word chaos implies complete disorder, that is not technically correct. Each system is deterministic, which means that whatever happens in the future is caused by the initial conditions – random elements are not involved. Chaotic systems usually display a clear boundary or structure, at odds with the usual use of the word chaos. In recent years, new fields in physics include quantum Chaos theory and relativistic chaos, ensuring that the study of chaos will continue unabated.

A variety of systems have demonstrated chaotic behavior under laboratory conditions. These include lasers, electrical circuits, fluids dynamics, and much more. Nature also shows evidence of chaotic behavior – satellites in the solar system, population growth in ecology, molecular vibrations. Weather provides a perfect opportunity to watch the daily activity of a chaotic system.

The first hints of chaos theory emerged around 1900. Scientists had noticed a few interesting anomalies. Astronomers studying the movement of heavenly bodies noticed that when three objects shared a mutual gravitational field, there is often a “wobble” in their trajectories. Turbulence had been observed in fluid dynamics, and radio circuits oscillated strangely. No fear he could account for their observations.

Early computers provided a push for Chaos theory. ENIAC was one of the first computers. It was used to perform the mathematical calculations required in performing simple weather forecasting. A scientist named Edward Lorenz wanted a closer look at a sequence of data generated by the computer. To save time, he decided to start the program in the middle, using his printout to enter the information for the timeslot he had chosen. Surprisingly, the computer forecast completely different weather. The problem lay in the printout itself – the printout rounded off digits to a three digit number, while the computer used six digit numbers. Through this, Lorenz discovered that small changes early on could produce large changes in the final outcome.

If a dynamical system is going to be called chaotic, it must meet three conditions. First, it must be sensitive to the initial conditions. This is now known popularly as the “butterfly effect”, a term coined by Lorenz in his 1972 paper entitled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? The wing’s motion creates a very small change to the initial condition of the dynamical system, one that causes a very large and noticeable change.

Second, the system must be “topologically mixing”. This means the dynamical system will change over time, swirling and moving through space.

Finally, the periodic orbits must be dense. For example, you may trace the path created by a pendulum swinging in a circle. As you do, you will notice that there are small variations in the trajectory. These variations, or periodic orbits, are packed closely together. There is no wild swaying from one side to the other; all for the orbits follow roughly the same path.

Today, Chaos theory remains a hot topic in the fields of physics and mathematics. No longer relegated to splinter cells, chaos theory has cracked through the education barrier. Once reviled as nonsense, it is fast becoming a major topic of research in universities and laboratories around the world.

Phase diagram for a damped driven pendulum, with double period motion, courtesy Wikipedia.