Chaos Theory is a distinct branch within the tree of Mathematics which deals with and looks at the behaviour of dynamical systems which are highly sensitive to initial conditions.
One of the most simple examples of this comes from the double pivot pendulum scenario. The double pivot pendulum simulates chaotic behaviour as shown that if starting the pendulum from a slightly different angle or initial velocity then the initial conditions are slightly changed resulting in a totally different trajectory.
A key concept of Chaos Theory is the butterfly effect which was first named by Edward Lorenz. This concept was brought about due to the metaphoric use of a flutter of a butterflies wings displacing air molecules which sometime later could cause a tornado. Lorenz as a meteorologist looked at weather systems and found that the slightest change in the start of a dynamic weather system could result in a totally different outcome.
An illustrative look on the butterfly effect can be seen when we look at the Lorenz Attractor. The Lorenz System is a series of Ordinary Differential Equations which has chaotic solutions for particular parameter values. Where the Lorenz Attractor is the set of chaotic solutions to the Lorenz Systems we end up with the iconic Butterfly wing resemblance.
This occurs for the Lorenz Equations:
Where: ρ = 28, σ = 10, and β = 8/3
Likewise as Chaos Theory goes if we change the initial conditions we end up with dramatically different results such as when: ρ = 14, σ = 10, and β = 8/3:
Applications of Chaos Theory include in stock markets, economics, dynamical weather systems and also within modeling in all different areas of science to better understand and predict how models will react.
Chaos Theory is a deep and wonderful area of Mathematics and one which I am looking forward to study further at Higher Education. If you do have anything to share regarding Chaos Theory then please let me know but until next time…