Upper and lower bounds on the computational complexity of ODEs

We investigate the computational complexity of the solution of y′ = f(t, y) when f : R → R is primitively recursive computable. In particular, we show that there is some f and initial conditions with complexity on the level E, n ≥ 3, of Grzegorczyk’s hierarchy, such that the complexity of the unique solution y does not collapse to the previous level E. As a corollary, we present some connections between Shannon’s General Purpose Analog Computer and Computable Analysis.