Nonautonomous gradient-like ODEs on the circle: classification, structural stability and autonomization

TL;DR

This paper classifies a special class of scalar differential equations on the circle, called gradient-like, focusing on their structural stability, foliation behavior, and almost periodic cases, providing a comprehensive framework for understanding their dynamics.

Contribution

It introduces a complete invariant for the classification of gradient-like equations on the circle and describes their global behavior and standard models.

Findings

Describes the global foliation behavior of gradient-like equations.

Provides a classification for almost periodic gradient-like equations.

Introduces a complete invariant for uniform equivalency of these equations.

Abstract

We study a class of scalar differential equations on the circle $S^1$. This class is characterized mainly by the property that any solution of such an equation possesses exponential dichotomy both on the semi-axes $\R_+$ and $\R_+$. Also we impose some other assumptions on the structure of the foliation into integral curves for such the equation. Differential equations of this class are called gradient-like ones. As a result, we describe the global behavior of the foliation, introduce a complete invariant of uniform equivalency, give standard models for the equations of the distinguished class. The case of almost periodic gradient-like equations is also studied, their classification is presented.