A simple geometric construction of an ODE with undecidable blow-ups

TL;DR

This paper constructs a smooth ODE in real space where the occurrence of finite-time blow-up encodes the halting problem, linking differential equations with computability theory.

Contribution

It introduces a novel geometric method to embed the halting problem into the dynamics of a smooth ODE, demonstrating undecidable blow-up behavior.

Findings

Finite-time blow-up can be made equivalent to Turing machine halting.

The construction is simple and geometric, making the undecidability explicit.

Links between dynamical systems and computability are established.

Abstract

We present a simple construction of an ODE on $\mathbb{R}^{n}$ where the vector field is smooth, and finite-time blow-up is equivalent to the halting problem for a universal Turing machine.