A universal ordinary differential equation

An astonishing fact was established by Lee A. Rubel (1981): there exists a fixed non-trivial fourthorder polynomial differential algebraic equation (DAE) such that for any positive continuous function ℓ on the reals, and for any positive continuous function ∈(t), it has a C∞p solution with |y(t) - ℓ(t)| < ∈(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, while these results may seem very surprising, their proofs are quite simple and are frustrating for a computability theorist, or for people interested in modeling systems in experimental sciences. First, the involved notions of universality is far from usual notions of universality in computability theory because the proofs heavily rely on the fact that constructed DAE does not have unique solutions for a given initial data. Indeed, in general a DAE may not have a unique solution, given some initials conditions. But Rubel's DAE never has a unique solution, even with a countable number of conditions of the form y(ki)(ai) = bi. This is very different from usual notions of universality where one would expect that there is clear unambiguous notion of evolution for a given initial data, for example as in computability theory. Second, the proofs usually rely on solutions that are piecewise defined. Hence they cannot be analytic, while analycity is often a key expected property in experimental sciences. Third, the proofs of these results can be interpreted more as the fact that (fourth-order) polynomial algebraic differential equations is a too loose a model compared to classical ordinary differential equations. In particular, one may challenge whether the result is really a universality result. The question whether one can require the solution that approximates ℓ 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 fixed polynomial ODE such that for any ℓ and ∈(t) there exists some initial condition that yields a solution that is ∈-close to ℓ at all times. The proof uses ordinary differential equation programming. We believe it sheds some light on computability theory for continuous-time models of computations. It also demonstrates that ordinary differential equations are indeed universal in the sense of Rubel and hence suffer from the same problem as DAEs for modelization: a single equation is capable of modelling any phenomenon with arbitrary precision, meaning that trying to fit a model based on polynomial DAEs or ODEs is too general (if it has a sufficient dimension).