A proof of the $C^r$ closing lemma and stability conjecture

TL;DR

This paper introduces a new perturbation method for the $C^r$ closing lemma, proving that generic dynamical systems have finite periodic orbits, applicable to both time-varying and time-invariant vector fields.

Contribution

A novel perturbation technique for the $C^r$ closing lemma that addresses the stability conjecture in dynamical systems.

Findings

Proves the $C^r$ closing lemma for generic systems.

Applicable to both time-varying and time-invariant vector fields.

Supports the stability conjecture in dynamical systems.

Abstract

It has long been conjectured that generic dynamical systems has finite periodic orbits, ever since the time of Poincar\'e. In this article, a perturbation method is proposed for the $C^r$ closing of periodic orbits. This method is applicable to both time-varying and time-invariant vector fields.