Weird bugs when solving three coupled second order differential equations using ode45

Hi guys, I am new to ode45 so I am very confused about its results. I have good results when solving two coupled differential equations (say, x1 and y ), but when I add one more equation (say x2), then the results are very different and even if I set x2 equation the same as x1, which doesn't make much sense for me. For example, if eq3 is changed to eq2, and also deval(~,2), the answer for y is different.

Here are the codes I used:

clear
close all
syms x1(t) x2(t) y(t)  
dx1=diff(x1,t);
dx2=diff(x2,t);
dy=diff(y,t);

omega_ir1 = 0.52*2*pi;% 
omega_ir2 = 0.28*2*pi;
omega_r = 0.425*2*pi;%

gamma_ir = 0;%
gamma_r = 0;%
t0 = [-40 50];%
tNum = 2000;
L = length(t0);
eq1 = diff(x1,2) == -2*gamma_ir*omega_ir1*dx1 - omega_ir1^2*x1  + y + 2*x2*y;
eq2 = diff(x2,2) == -2*gamma_ir*omega_ir2*dx2 - omega_ir2^2*x2  + y + 2*x1*y;
eq3 = diff(y,2) == -2*gamma_r*omega_r*dy - omega_r^2*y + x1*x2;
V = odeToVectorField(eq1,eq2,eq3);
M = matlabFunction(V,'vars', {'t','Y'});
interval = t0;    
ySol = ode45(M,interval,y0);
tValues = linspace(interval(1),interval(2),tNum);
sol = deval(ySol,tValues,3);

Thanks in advance！

Answer 1

[Torsten]

syms x1(t) x2(t) y(t)
t0 = [0 1];
% y0 must be given in the order as prescribed by S (see below), thus
% y0 = (x2(0),dx2/dt(0),x1(0),dx1/dt(0),y(0),dy/dt(0))
% The solution is obtained in the same order.
y0 = [1 2 4 7 8 4];
tNum = 100;
gamma_ir = 1;
omega_ir1 = 1;
omega_ir2 = 0.5;
gamma_r = 2;
omega_r = 3;
eq1 = diff(x1,2) == -2*gamma_ir*omega_ir1*diff(x1) - omega_ir1^2*x1  + y + 2*x2*y;
eq2 = diff(x2,2) == -2*gamma_ir*omega_ir2*diff(x2) - omega_ir2^2*x2  + y + 2*x1*y;
eq3 = diff(y,2) == -2*gamma_r*omega_r*diff(y) - omega_r^2*y + x1*x2;
[V,S] = odeToVectorField(eq1,eq2,eq3)

V = 

S = 

M = matlabFunction(V,'vars', {'t','Y'})

M = function_handle with value:
    @(t,Y)[Y(2);Y(1).*(-1.0./4.0)-Y(2)+Y(5)+Y(3).*Y(5).*2.0;Y(4);-Y(3)-Y(4).*2.0+Y(5)+Y(1).*Y(5).*2.0;Y(6);Y(5).*-9.0-Y(6).*1.2e+1+Y(1).*Y(3)]

interval = t0;    
ySol = ode45(M,interval,y0);
tValues = linspace(interval(1),interval(2),tNum);
sol = deval(ySol,tValues,3);
plot(tValues,sol)
grid on

[chris shen]

So the fifth row of the Sol is y? I thought it was the third one.

[Torsten]

Yes, the fifth row is y. The third row is x1. As said, you must be careful when supplying the initial values and evaluating the solution.

I'm not sure whether you can somehow prescribe the ordering, but I doubt there is a direct way to do so. Of course, you could permute the rows of V according to your personal S after the odeToVectorField command.

[chris shen]

Thanks Torsten.