Variable change (diffeomorphism) in Matlab

Hi,

I want to know if there's a startand procedure to do the following steps in Matlab:

I have this set of differential equations:

and I want to make the following variable change (a diffeomorphism, in fact):

In order to obtain the first derivatives of the new variables (which I already know):

I know that I can define the first set of like this:

f = @(t,x) [-2*x1+a*x2+sin(x1);
            -x2*cos(x1)+u*cos(2*x1)];

And my question is: how to make the variable substitution and the next steps in order to obtain the second set of (,)? Is this even possible in Matlab?

I have tried functions like subs, solver, diff, ode, etc., but I wasn't able to get any results.

Thanks in advance!

Brito

Answer 1

[José Brito]

I found the solution:

syms x1 x2 z1 z2 a u

% Defining x_dot = f(x)
f(x1,x2) = [-2*x1+a*x2+sin(x1);
            -x2*cos(x1)+u*cos(2*x1)];  
        
% Defining z = g(x)
g(x1,x2) = [x1;
            a*x2+sin(x1)];
solver = formula(g);

% Defining J as the jacobian of g: J = dg(x)/dt = dz/dt
J(x1,x2) = jacobian(g);

% Defining x as a function of z: x = g^{-1}(z)   
[x1, x2] = solve(z1 == solver(1), ...
                 z2 == solver(2));
          
% Calculate z_dot = dg(x)/dx * dx/dt | x = g^{-1}(z)
% Note: dg(x)/dt = dz/dt
z_dot = J(x1,x2)*f(x1,x2);
simplify(z_dot)

which outputs:

ans =
 
                                              z2 - 2*z1
 cos(z1)*sin(z1) - 2*z1*cos(z1) + a*u*(2*cos(z1)^2 - 1)

And since:

 >> simplify(2*cos(z1)^2 - 1)
 
ans =
 
cos(2*z1)

Then the final solution is:

ans =
 
                                       z2 - 2*z1
 cos(z1)*sin(z1) - 2*z1*cos(z1) + a*u*cos(2*z1))

Which is the correct solution.