Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization

We study a class of scalar differential equations on the circle \begin{document}$ S^1 $\end{document} . This class is characterized mainly by the property that any solution of such an equation possesses an exponential dichotomy both on the semi-axes \begin{document}$ \mathbb R_+ $\end{document} and \begin{document}$ \mathbb R_+ $\end{document} . 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 a foliation, introduce a complete invariant of the uniform equivalency, give standard models for the equations of this distinguished class. The case of almost periodic gradient-like equations is also studied, their classification is presented.