Evans function example

The parameters I've been using:

eps = 0; h = 0.3; Z = 6; sigma = 0.25; w_star = 0.003;  

Set up:

e = evans(eps, h, Z, sigma, w_star);  

Now it's a function:

e(1+2j)  

We can plot it evaluated over a circle:

plot(e(0.3 * exp(2*pi*j*[0:0.05:1])))  

Or a grid:

k = e(cgrid(0:2:10, 0:2:10),10);  
plot(k); hold on; plot(k.')  

The second parameter increases accuracy. Values needed to reach stability, for given inputs:

10: 10
40: 16
100: 22
150: 27
200: 30
400: 41

50j: 13
100j: 17
150j: 20

10+10j: 10
40+40j: 16 or 17

You can find this with e.g. arrayfun(@(x) e(10+10j,x),7:11) and looking for the value where the result stops changing.

Old stuff

This is a small Matlab script to calculate fronts for the following system:

U' + cU - cz = 0
eps*V' + cV - cz = 0
z' = -(1/c)(1-V)*F(hU + (1-h)V)

where F is a mess. (2.3 in the Gordon paper, with h for (1-γ⁻¹))

This system was produced by integration of

U" + cU' + (1-V)F(W) = 0
eps*V" + cV' + (1-V)F(W) = 0

where W = hU + (1-h)V.

It shoots from near (1,1) and adjusts c to land at (0,0).

Usage:

[x,u,v] = integrated_find_c(eps,h,Z,sigma,w_star);

Or with some fairly arbitrary numbers:

[x,u,v] = integrated_find_c(0.01,0.3,6,0.25,1e-3);

Then you can do things like:

plot(u,v);
plot(x,[u v]);

There are a few numbers one could change in integrated_solve.m and integrated_find_c.m. Let me know what else you'd like to do, or if there are problems. This is far from a masterwork yet.

Acknowledgements

This is part of joint work with Dr. Anna Ghazaryan, Miami University, and Dr. Stephane Lafortune, College of Charleston.