Web geometry of a system of n first order autonomous ordinary differential equations

TL;DR

This paper applies E. Cartan's equivalence method to analyze the geometric invariants of systems of first order autonomous ODEs under specific diffeomorphisms, revealing their underlying geometric structure.

Contribution

It introduces a geometric approach using Cartan's method to identify invariants of autonomous ODE systems under certain transformations.

Findings

Identified invariants characterizing the system's geometry

Provided a framework for classifying systems based on invariants

Enhanced understanding of the geometric structure of autonomous ODEs

Abstract

Let $dx_i/dt=f_i(x_1,\cdots,x_n)$, ($i=1,\cdots,n$) be a system of $n$ first order autonomous ordinary differential equations. We use E. Cartan's equivalence method to study the invariants of this system under diffeomorphisms of the form $\Phi(t,x_1,\cdots,x_n)=(\varphi_0(t),\varphi_1(x_1),\cdots,\varphi_1(x_1))$.