Liapunov plot in MATLAB

I seem to do a lot of MATLAB coding… In a paper I’m studying I came across the following:

“… in which case it is a dissipative system possessing a Liapunov function … thus, released from an initial (non-equilibrium) state, typical solutions approach asymptotically stable fixed points (sinks).”

(Brown & Holmes (2001), Modelling a simple choice task: Stochastic dynamics of mutually inhibitory neural groups, Stochastics  and Dynamics, 1:2 (2001), 159-191))

So I’ve been doing a bit of background reading on system stability (in linear systems) and Liapunov functions, which can be used to prove the stability of an equilibrium of an ordinary differential equation. I plotted a simple one in MATLAB:

Liapunov plot

Plot of a simple Liapunov function

There isn’t really a lot of code behind it, so I’ll just post it here (I also might update this later with slightly more complicated functions):

x = [-1:.05:1];
y = x;
[x,y] = meshgrid(x,y);
z = x.^2 + y.^2;
title('Liapunov equation: f(x) = x^2 + y^2')