Normal form for second order differential equations

TL;DR

This paper develops a complete convergent normal form for second order differential equations, solving the local equivalence problem and enabling classification and analysis of these equations, including Painlevé equations.

Contribution

It introduces a unique, optimal normal form for second order ODEs, solving a long-standing problem and providing tools for their classification and analysis.

Findings

Provides a complete convergent normal form for second order ODEs

Solves the local equivalence problem for these equations

Identifies distinguished curves called chains associated with the equations

Abstract

We solve the local equivalence problem for second order (smooth or analytic) ordinary differential equations. We do so by presenting a {\em complete convergent normal form} for this class of ODEs. The normal form is optimal in the sense that it is defined up to the automorphism group of the model (flat) ODE $y"=0$. For a generic ODE, we also provide a unique normal form. By doing so, we give a solution to a problem which remained unsolved since the work of Arnold. The method can be immediately applied to important classes of second order ODEs, in particular, the Painlev\'e equations. As another application of the convergent normal form, we discover distinguished curves associated with a differential equation that we call {\em chains}.