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:

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;

figure(1)

colormap(bone)

surf(x,y,z)

title('Liapunov equation: f(x) = x^2 + y^2')

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