The outcomes of this article are twofold. Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class P of languages computable in polynomial time in terms of differential equations with polynomial right-hand side. This result gives a purely continuous elegant and simple characterization of P. We believe it is the first time complexity classes are characterized using only ordinary differential equations. Our characterization extends to functions computable in polynomial time over the reals in the sense of Computable Analysis. Our results may provide a new perspective on classical complexity, by giving a way to define complexity classes, like P, in a very simple way, without any reference to a notion of (discrete) machine. This may also provide ways to state classical questions about computational complexity via ordinary differential equations. Continuous-Time Models of Computation. Our results can also be interpreted in terms of analog computers or analog models of computation: As a side effect, we get that the 1941 General Purpose Analog Computer (GPAC) of Claude Shannon is provably equivalent to Turing machines both in terms of computability and complexity, a fact that has never been established before. This result provides arguments in favour of a generalised form of the Church-Turing Hypothesis, which states that any physically realistic (macroscopic) computer is equivalent to Turing machines both in terms of computability and complexity.

Let $(\alpha ,\beta )\subseteq \mathbb {R}$ denote the maximal interval of existence of solutions for the initial-value problem \[ $\left \{ \begin {array} [c]{l}\frac {dx}{dt}=f(t,x), x(t_{0})=x_{0}, \end {array} \right .$ \] where $E$ is an open subset of $\mathbb {R}^{m+1}$ , $f$ is continuous in $E$ and $(t_{0},x_{0})\in E$ . We show that, under the natural definition of computability from the point of view of applications, there exist initial-value problems with computable $f$ and $(t_{0},x_{0})$ whose maximal interval of existence $(\alpha ,\beta )$ is noncomputable. The fact that $f$ may be taken to be analytic shows that this is not a lack of the regularity phenomenon. Moreover, we get upper bounds for the “degree of noncomputability” by showing that $(\alpha ,\beta )$ is r.e. (recursively enumerable) open under very mild hypotheses. We also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable.