## Olivier Bournez ; Amaury Pouly - A Universal Ordinary Differential Equation

lmcs:4437 - Logical Methods in Computer Science, February 28, 2020, Volume 16, Issue 1 - https://doi.org/10.23638/LMCS-16(1:28)2020
A Universal Ordinary Differential Equation

Authors: Olivier Bournez ; Amaury Pouly

An astonishing fact was established by Lee A. Rubel (1981): there exists a fixed non-trivial fourth-order polynomial differential algebraic equation (DAE) such that for any positive continuous function $\varphi$ on the reals, and for any positive continuous function $\epsilon(t)$, it has a $\mathcal{C}^\infty$ solution with $| y(t) - \varphi(t) | < \epsilon(t)$ for all $t$. Lee A. Rubel provided an explicit example of such a polynomial DAE. Other examples of universal DAE have later been proposed by other authors. However, Rubel's DAE \emph{never} has a unique solution, even with a finite number of conditions of the form $y^{(k_i)}(a_i)=b_i$. The question whether one can require the solution that approximates $\varphi$ to be the unique solution for a given initial data is a well known open problem [Rubel 1981, page 2], [Boshernitzan 1986, Conjecture 6.2]. In this article, we solve it and show that Rubel's statement holds for polynomial ordinary differential equations (ODEs), and since polynomial ODEs have a unique solution given an initial data, this positively answers Rubel's open problem. More precisely, we show that there exists a \textbf{fixed} polynomial ODE such that for any $\varphi$ and $\epsilon(t)$ there exists some initial condition that yields a solution that is $\epsilon$-close to $\varphi$ at all times. In particular, the solution to the ODE is necessarily analytic, and we show that the initial condition is computable from the target function and error function.

Volume: Volume 16, Issue 1
Published on: February 28, 2020
Accepted on: February 28, 2020
Submitted on: April 11, 2018
Keywords: Mathematics - Classical Analysis and ODEs,Computer Science - Computational Complexity,Computer Science - Logic in Computer Science,Electrical Engineering and Systems Science - Systems and Control,Mathematics - Dynamical Systems