Suppose you have the autonomous ordinary differential equation $dx(t)/dt = f(x(t))$ with $x: \mathbb{R} \to \mathbb{R}$ and the initial condition $x(0)=x\_0$. Then, assuming some regularity conditions, you get as solution the
flow $\Phi(x\_0,t):=x(t)$. To give a trivial example: If $f(x)=x$, then $\Phi(x\_0,t)=x\_0 \exp(t)$.


Now, I'm not interested in the trajectories for a given initial condition, that is in $\Phi(x\_0,t)$ with $x\_0$ fixed and $t$ variable; but in the map $x\_0 -> \Phi(x\_0, t)$ for a fixed $t$ (say $t=1$). 


Given the function $f$, you can easily (at least in principle, by solving the ODE) get the function $\Phi(\cdot, 1)$. There are a lot of theorems about existence and uniqueness of this problem and analytical and numerical algorithms. 


But how can one get $f$ out of $\Phi(\cdot, 1)$? Is this a well posed problem? Are there any theorems? 


This problem is closely related to "interpolating" the $n$-fold functional iterates of $g$ (with $g^{[0]} = \mathrm{Id}, g^{[1]} = g, g^{[2]} = g \circ g, g^{[n+m]} = g^{[n]} \circ g^{[m]}$ for $n,m \in \mathbb{N}$) from $n \in \mathbb{N}$ to real values. If such an interpolation succeeds, on can get the ODE out of the flow $\Phi(\cdot, 1)$ by determining $\Phi(\cdot, 1)^{[\alpha]}$ for small $\alpha >0$. I have done some calculation, that give results, but lack
in rigor. 


For noninteger iterates of functions, a classical reference is <http://www.math-inst.hu/~p_erdos/1960-07.pdf>. 



Noninteger iterates of functions: How to get ODE from flow at a given time?  
If g is a real-analytic function defined near x0 with g(x0) = x0 and 0 < λ ≠ 1 where λ := g'(x0), then Koenigs proved that there exists a real-analytic homeomorphism h defined near x0 such that hgh-1(x) = L(x) where 


L(x) := x0 + λ(x-x0)


and h is unique up to a constant factor\*. 


This allows g to be embedded in a local flow: Φ(x,t) := h-1 Lt h(x), where 


Lt(x) := x0 + λt (x-x0),


such that Φ(x,1) = g(x) where defined.


Then Φ satisfies the differential equation dx(t)/dt = V(x(t)) where the velocity V (denoted by f in the Question) is given by


V(x) := ∂Φ(x,t)/∂t |t=0.


Note, however, that the flow Φ(x,t) into which the original function g embeds as the time-1 map need not be unique when there exist more than one fixed point of g. As an example, for x > 0 consider the function 


gc(x) := cx. 


Then for 1 < c < e1/e the function gc has two distinct fixed points each satisfying the hypotheses of the Koenigs theorem, and these give two distinct flows into which gc embeds as the time-1 map. 


Concretely, set c = √2, so that gc(x) = x for both x = 2 and x = 4, with derivatives 


gc'(2) = ln(2) and 


gc'(4) = ln(4). 


Calculating the respective real-analytic flows Φ2(x,t) and Φ4(x,t), both are defined for (x,t) = (3, 1/2). 


But Φ2(3, 1/2) and Φ4(3, 1/2) first differ in the 25th decimal place. Hence they are solutions of distinct differential equations. I.e., they have different velocity functions.




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\* This is actually true in greater generality; see J. Milnor's book *Dynamics in One Complex Variable*, 3rd ed., Princeton University Press, 2006.


