How to Use Nested For Loops to Create a Coupled System of Equations

I am trying to write a system of coupled ODEs to eventually pass through ode45(). The equations are:

When I try the following code in an attempt to produce those equations and solve then for n = 3, I get the following error:

options = odeset('RelTol',3e-14,'AbsTol',1e-15,'Stats','on');

[t,y]=ode45(@hmmPD,[0 5000],[0.1 0.2 0.3],options);

figure

plot(t,y(:,1),'b')

hold on

plot(t,y(:,2),'r')

hold on

plot(t,y(:,3),'g')

title('Compare')

hold off

function dydt = hmmPD(t,y)

k=1/5;

a=1.25;

r=1/4;

n=3;

m=4;

for i = 1:n

for j = 1:n

dydt = [(k/m)*(sin(-y(i)+a)-r*sin(-2*y(i)) + sin(y(j)-y(i)+a)-r*sin(2*(y(j)-y(i))));];

end

end

end

Error using odearguments (line 95)

HMMPD returns a vector of length 1, but the length of initial conditions vector is 3. The vector

returned by HMMPD and the initial conditions vector must have the same number of elements.

Error in ode45 (line 106)

odearguments(FcnHandlesUsed, solver_name, ode, tspan, y0, options, varargin);

Error in LargeSystem (line 4)

[t,y]=ode45(@hmmPD,[0 5000],[0.1 0.2 0.3],options);

This appears, to me at least, to be a dimensionality issue. The for loops are likely not doing what I expect and either not constructing the equation I was attempting to or the size of the dydt vector is not right. Either way, I have tried quite a few things but haven't been able to fix it. Any suggestions?

Answer 1

[Alan Stevens]

More like this perhaps (you've very tight tolerances!):

options = odeset('RelTol',3e-14,'AbsTol',1e-15,'Stats','on');
[t,y]=ode45(@hmmPD,[0 5000],[0.1 0.2 0.3],options);

21891 successful steps
9 failed attempts
131401 function evaluations

figure
plot(t,y(:,1),'b')
hold on
plot(t,y(:,2),'r')
hold on
plot(t,y(:,3),'g')
title('Compare')
hold off

function dydt = hmmPD(~,y)
k=1/5;
a=1.25;
r=1/4;
n=3; 
m=4;
for i = 1:n
    term = sin(-y(i)+a)-r*sin(-2*y(i));
    summation = 0;
    for j = 1:n
      summation = summation + sin(y(j)-y(i)+a)-r*sin(2*(y(j)-y(i)));
    end
    dydt(i,1) = (k/m)*(term + summation);
end
end

[Torsten]

@Cameron3332 comment moved here:

Hey Alan, thanks so much for the answer - it did exactly what I was looking for! As far as the tolerances go, unfortunately the extended system has some pretty nasty heteroclinic orbits that have required me to knock the tolerances up a bit. That has also done the trick through.

Thanks again!