MATLAB: Variable change (diffeomorphism) in Matlab

diffeomorphismdifferential equationsmathematicsMATLABodesolve

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

Best Answer

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.
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