I’m a big fan of the Lorenz Attractor, which, when plotted, resembles the half open wings of a butterfly. This attractor was derived from a simplified model of convection in the earth’s atmosphere. One simple version of the Lorenz attractor is pictured below:
The Lorentz system is a set of ordinary differential equations notable for its chaotic solutions (see below). Here $x$, $y$ and $z$ make up the system state, $t$ is time, and $sigma, row, beta$ are the system parameters.
The Lorentz attractor is a chaotic solution to this system found when $row = 28, sigma = 10. beta = 8/3$.
The series does not form limit cycles nor does it ever reach a steady state.
We can calculate and render the aforementioned chaotic solution to this ODE as follows:
function loren3 clear;clf global A B R A = 10; B = 8/3; R = 28; u0 = 100*(rand(3,1) - 0.5); [t,u] = ode45(@lor2,[0,100],u0); N = find(t>10); v = u(N,:); x = v(:,1); y = v(:,2); z = v(:,3); plot3(x,y,z); view(158, 14) function uprime = lor2(t,u) global A B R uprime = zeros(3,1); uprime(1) = -A*u(1) + A*u(2); uprime(2) = R*u(1) - u(2) - u(1)*u(3); uprime(3) = -B*u(3) + u(1)*u(2);
This results in the figure: