On a certain class of dynamical systems on the torus with rational rotation number
The authors consider the ordinary differential equation \[ \begin{aligned} \frac{dy}{dt} =& P(x) \cos y + Q(x) \sin y+ R(x) \\ \frac{dx}{dt} =& \hat{P}(x) \cos y + \hat{Q}(x) \sin y+ \hat{R}(x) \end{aligned} \] on the two-dimensional torus where \(P(x), Q(x), R(x), \hat{P}(x), \hat{Q}\) and \(\hat{R}(x)\) are trigonometric polynomials. They classify the set of \(\mu\in\mathbb R\) for which there exists a solution of the form \(y\equiv \mu x +y_0(\mod 2\pi)\) with \(\mu\geq 0\) and \(y_0\in\mathbb R\). In particular, the question of uniqueness of these partial linear solutions is discussed. In the special case \(\hat{P}(x)=\hat{Q}(x)\equiv 0\) and \(\hat{R}\equiv 1\) the system of differential equations simplifies to the single equation \[ \frac{dy}{dx} = P(x)\cos y + Q(x)\sin y+ R(x) \] and a description of the topological structure of the flow on the torus is possible.
scientific article; zbMATH DE number 5659044