The Lorentz System
Chaos! A nice part of mathematics.
Find below a solution to the Lorentz system of equations that has been animated using MATLAB.
The Lorentz system consists of three equations:
\[\begin{aligned} \frac{\partial x}{\partial t} &= \sigma(y-x)\\ \frac{\partial y}{\partial t} &= x(\rho-z)-y\\ \frac{\partial z}{\partial t} &= xy-\beta z \end{aligned}\]The system can be classed as chaotic due to the sensitive dependence on initial conditions. By changing the starting position of the simulation by a minute amount, the system diverges significantly. The below animation gives you an idea of what this looks like.
The red and blue indicate different paths where the \(y\)-position of each starting point was altered by \(+1\times 10^{-4}\).
For the MATLAB code used to create the above, click here.