The connection between computability of a nonlinear problem and its linearization: The Hartman-Grobman theorem revisited

As one of the seven open problems in the addendum to their 1989 book Computability in Analysis and Physics [21], Pour-El and Richards asked, “What is the connection between the computability of the original nonlinear operator and the linear operator which results from it?” Yet at present, systematic studies of the issues raised by this question seem to be missing from the literature. In this paper, we study one problem in this direction: the Hartman-Grobman linearization theorem for ordinary differential equations (ODEs). We prove, roughly speaking, that near a hyperbolic equilibrium point x0 of a nonlinear ODE ẋ = f(x), there is a computable homeomorphism H such that H ◦ φ = L ◦ H, where φ is the solution to the ODE and L is the solution to its linearization ẋ = Df(x0)x.