  We provide an implicit characterization of polynomial time computation in
terms of ordinary differential equations: we characterize the class
$\operatorname{PTIME}$ of languages computable in polynomial time in terms of
differential equations with polynomial right-hand side.
  This result gives a purely continuous (time and space) elegant and simple
characterization of $\operatorname{PTIME}$. This is the first time such 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. This extends to deterministic
complexity classes above polynomial time.
  This may provide a new perspective on classical complexity, by giving a way
to define complexity classes, like $\operatorname{PTIME}$, 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, i.e.~by using the framework of analysis.
