Noninteger iterates of functions: How to get ODE from flow at a given time? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T09:10:58Z http://mathoverflow.net/feeds/question/26462 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/26462/noninteger-iterates-of-functions-how-to-get-ode-from-flow-at-a-given-time Noninteger iterates of functions: How to get ODE from flow at a given time? Andreas Rüdinger 2010-05-30T15:47:55Z 2010-10-17T21:22:50Z <p>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 $\Phi(x_0,t)=x_0 \exp(t)$.</p> <p>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 $t=1$). </p> <p>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. </p> <p>But how can one get $f$ out of $\Phi(\cdot, 1)$? Is this a well posed problem? Are there any theorems? </p> <p>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. </p> <p>For noninteger iterates of functions, a classical reference is <a href="http://www.math-inst.hu/~p_erdos/1960-07.pdf" rel="nofollow">http://www.math-inst.hu/~p_erdos/1960-07.pdf</a>. </p> http://mathoverflow.net/questions/26462/noninteger-iterates-of-functions-how-to-get-ode-from-flow-at-a-given-time/26472#26472 Answer by rpotrie for Noninteger iterates of functions: How to get ODE from flow at a given time? rpotrie 2010-05-30T17:18:08Z 2010-05-30T17:18:08Z <p>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. </p> <p>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).</p> <p>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. </p> http://mathoverflow.net/questions/26462/noninteger-iterates-of-functions-how-to-get-ode-from-flow-at-a-given-time/26495#26495 Answer by Daniel Asimov for Noninteger iterates of functions: How to get ODE from flow at a given time? Daniel Asimov 2010-05-30T20:20:58Z 2010-05-30T21:47:27Z <p>If g is a real-analytic function defined near x<sub>0</sub> with g(x<sub>0</sub>) = x<sub>0</sub> and 0 &lt; λ &ne; 1 where λ := g'(x<sub>0</sub>), then Koenigs proved that there exists a real-analytic homeomorphism h defined near x<sub>0</sub> such that hgh<sup>-1</sup>(x) = L(x) where </p> <p>L(x) := x<sub>0</sub> + λ(x-x<sub>0</sub>)</p> <p>and h is unique up to a constant factor*. </p> <p>This allows g to be embedded in a local flow: Φ(x,t) := h<sup>-1</sup> L<sup>t</sup> h(x), where </p> <p>L<sup>t</sup>(x) := x<sub>0</sub> + λ<sup>t</sup> (x-x<sub>0</sub>),</p> <p>such that Φ(x,1) = g(x) where defined.</p> <p>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</p> <p>V(x) := ∂Φ(x,t)/∂t |<sub>t=0</sub>.</p> <p>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 </p> <p>g<sub>c</sub>(x) := c<sup>x</sup>. </p> <p>Then for 1 &lt; c &lt; e<sup>1/e</sup> the function g<sub>c</sub> has two distinct fixed points each satisfying the hypotheses of the Koenigs theorem, and these give two distinct flows into which g<sub>c</sub> embeds as the time-1 map. </p> <p>Concretely, set c = &radic;2, so that g<sub>c</sub>(x) = x for both x = 2 and x = 4, with derivatives </p> <p>g<sub>c</sub>'(2) = ln(2) and </p> <p>g<sub>c</sub>'(4) = ln(4). </p> <p>Calculating the respective real-analytic flows Φ<sub>2</sub>(x,t) and Φ<sub>4</sub>(x,t), both are defined for (x,t) = (3, 1/2). </p> <p>But Φ<sub>2</sub>(3, 1/2) and Φ<sub>4</sub>(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.</p> <hr> <p>&#42; This is actually true in greater generality; see J. Milnor's book <em>Dynamics in One Complex Variable</em>, 3rd ed., Princeton University Press, 2006.</p> http://mathoverflow.net/questions/26462/noninteger-iterates-of-functions-how-to-get-ode-from-flow-at-a-given-time/42543#42543 Answer by Daniel Geisler for Noninteger iterates of functions: How to get ODE from flow at a given time? Daniel Geisler 2010-10-17T21:22:50Z 2010-10-17T21:22:50Z <p>Aldrovandi and Freitas' article <a href="http://arxiv.org/abs/physics/9712026" rel="nofollow">Continuous iteration of dynamical maps</a> 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.</p> <p>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.</p> <p>For more references see <a href="http://tetration.org/Dynamics/index.html" rel="nofollow">continuous iteration</a>.</p>