I've thought of something and I have some questions about it. Consider the following transformation:
$$Q(x,y,z)=(x^2 + 2yz, y^2 + 2xz, z^2 + 2xy)$$.
If for every nonnegative integer t,
$$(x_{t+1},y_{t+1},z_{t+1}) = Q(x_t,y_t,z_t),$$
$$X_t = $$ \begin{pmatrix} x_t & z_t & y_t \\ y_t & x_t & z_t \\ z_t & y_t & x_t \\ \end{pmatrix}
and $$C = $$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{pmatrix}
then $$X_{t+1} = C \cdot X_t^2 \cdot C^{-1}$$
so $$X_t = C^t \cdot X_0^{2^t} \cdot C^{-t}.$$
It is always possible to find a closed-form solution for all of the components of the powers of a square matrix by reducing the matrix to Jordan canonical form. In this case as in almost all cases, it is possible to diagonalize the matrices C and $X_0.$ In this way, one can get a closed-form solution for $x_t,$ $y_t$ and $z_t.$ More generally, $X_0$ can be any nxn circulant matrix and C can be any nxn permutation matrix, allowing one to solve a discrete dynamical system in n variables. Even more generally, $X_0$ can be any nxn matrix and C can be any invertible nxn matrix, allowing one to solve a discrete dynamical system in n or $n^2$ variables.
Next consider the following system of differential equations,
dx/dt = $x^2$ + yz $\quad$ dy/dt = xy + yw
dz/dt = xz + zw $\quad$ dw/dt = $w^2$ +yz.
Let X(t) = $$ \begin{pmatrix} x(t) & y(t)\\ z(t) & w(t)\\ \end{pmatrix} $$
Then dX(t)/dt = $[X(t)]^2$ so $X(t) = X(0) [I - t X(0)]^{-1}.$
I have checked this and it works. Some other solvable matrix differential equations are
dX(t)/dt = $[X(t)]^2$ + X(t);
dX(t)/dt = $[X(t)]^2$ - X(t);
and
dX(t)/dt = $[X(t)]^2$ - I.
Again X(0) can be any nxn matrix, allowing one to solve a system of $n^2$ differential equations in $n^2$ variables.
So I have two questions about this. First, is all of this already known? And second, if it is, where can I read about it?