Noninteger iterates of functions: How to get ODE from flow at a given time?

Question

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.

Answer 1

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 L^t h(x), where

L^t(x) := x₀ + λ^t (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

g_c(x) := c^x.

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

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

g_c'(2) = ln(2) and

g_c'(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.

Answer 2

Being the time one map of a flow, there is an invariant one dimensional foliation where the flow "evolves" (assuming no singularities). For example, this implies that if there is a periodic point of $\phi(.,1)$ then there is an invariant circle. It is not hard to construct diffeomorphisms with periodic points and not having an invariant circle.

Even if there exists an invariant foliation of dimension one, and one can construct a flow for which the diffeomorphism is the time one map, uniqueness is not guarantied, as you can see by taking the time one map of a flow in $S^1$ with velocity $2\pi$ (this example can be made more interesting).

Another thing, which is related to what you say is that if you can take "square roots" of $\phi(-,1)$ indefinitely then one can recreate the ODE, but this is not true in general.

Answer 3

Aldrovandi and Freitas' article Continuous iteration of dynamical maps begins to lay the groundwork for viewing systems in physics as continuously iterated functions, providing an alternative to ODEs and PDEs. The function $f(x)$ that is continuously iterated describes the "physics" of $\Phi(x,t)$. If the scale of time $t$ is known, then setting $t=1$ trivially gives $f(x)=\Phi(x,1)$. If the scale of time $t$ is not known, then finding $f(x)$ probably becomes a difficult, if not intractable problem.

Consider a similar problem, but using cellular automata. Stephen Wolfram has searched for 25 years for a cellular automata rule that accurately models the laws of physics, but he hasn't released any results indicating he is able to model with fidelity any interesting physical process.

For more references see continuous iteration.