  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.
