Solving Logistic ODE with Non-commuting Matrices

[Manchot]

I'm trying to find a general solution for the logistic ODE \frac{dU}{dx}=A(I-U)U, where A and U are square matrices and x is a scalar parameter. Inspired by the scalar equivalent I guessed that U=(I+e^{-Ax})^{-1} is a valid solution; however, U=(I+e^{-Ax+B})^{-1} is not when U and A don't commute. Any ideas?

 

[maajdl]

The general solution to the scaler equation is:

E^(A x)/(E^(A x) + E^C)

where C is a constant.
Maybe this can lead to a similar solution for the matricial version?

 

[maajdl]

If U is a function of A,
then U commutes with A.

 

[Manchot]

I tried all sorts of versions of the scalar equation, maajdl. They all run into the same commutation problem. Unfortunately, U is a function of both A and the initial condition, which means that it doesn't commute with A unless the initial condition does.

 

[maajdl]

Could that help?

Assuming:

A = M^-1DM where D is a diagonal matrix
V = MUM^-1

The ODE becomes:

dV/dx = D(I-V)V

 

[Manchot]

Yeah, I tried diagonalizing both A and the initial condition. No dice.

 

[maajdl]

What is your practical goal?
Why do you need a formal solution?