Solvable Initial Value Problems Ruled by Discontinuous Ordinary Differential Equations

TL;DR

This paper introduces a class of discontinuous differential equations with unique solutions, demonstrating that these solutions can be obtained analytically and relating their behavior to ordinal Turing computations.

Contribution

It defines solvable initial value problems for discontinuous ODEs and proves they have analytically obtainable solutions via transfinite recursion, linking to computability theory.

Findings

Unique solutions exist for the class of solvable initial value problems.

Solutions can be obtained analytically through transfinite recursion.

Example shows connection to the halting problem and ordinal Turing computations.

Abstract

We study initial value problems having dynamics ruled by discontinuous ordinary differential equations with the property of possessing a unique solution. We identify a precise class of such systems that we call solvable intitial value problems and we prove that for this class of problems the unique solution can always be obtained analytically via transfinite recursion. We present several examples including a nontrivial one whose solution yields, at an integer time, a real encoding of the halting set for Turing machines; therefore showcasing that the behavior of solvable systems is related to ordinal Turing computations.