The global symmetry group of first order differential equations and the global rectification theorem

TL;DR

This paper extends the local rectification theorem to a global version for Lipschitz differential equations and characterizes their symmetry group as a smooth wreath product of diffeomorphism groups.

Contribution

It proves a global rectification theorem for Lipschitz differential equations and describes their symmetry group as a universal structure independent of the specific equation.

Findings

Global rectification theorem holds under Lipschitz condition.

Symmetry group is a smooth wreath product of diffeomorphism groups.

The symmetry group is independent of the specific differential equation.

Abstract

Symmetry analysis can provide a suitable change of variables, i.e., in geometric terms, a suitable diffeomorphism that simplifies the given direction field, which can help significantly in solving or studying differential equations. Roughly speaking this is the so-called rectification theorem. The local version of this result is a well-known theorem in the field of ordinary differential equations. In this note we prove a global counterpart when the equation fulfils the Lipschitz condition. Then we use this result to determine the global symmetry group of such an ordinary differential equation. It turns out that, assuming the Lipschitz condition, the full symmetry group is a smooth wreath product of two diffeomorphism groups, and does not depend on the form of the equation, at all.