2 added 58 characters in body

Your equation can be re-written as a difference equation, if g(t) is the flow of f(x) then

$$g(t+2)=g(t)+g(t+1)^2$$

or

$$\Delta^2 g-(\Delta g+g)^2+2\Delta g-2g=0$$

This

It is second order non-linear ordinary difference equation. It looks weird but at least you can try to apply methods of discrete calculus to this equation (with not much hope though).

1

Your equation can be re-written as a difference equation, if g(t) is the flow of f(x) then

$$g(t+2)=g(t)+g(t+1)^2$$

or

$$\Delta^2 g-(\Delta g+g)^2+2\Delta g-2g=0$$

This looks weird but at least you can try to apply methods of discrete calculus to this equation (with not much hope though).