About the integration of a class of differential systems on the two-torus (Q2763765)

The authors study differential systems of the form NEWLINE\[NEWLINE\dot x= X(x,y),\qquad \dot y= Y(x,y),\tag{1}NEWLINE\]NEWLINE where \(X\neq 0\) and \(X\), \(Y\) are continuous periodic functions in \(x\) and \(y\). For solutions to system (1), a Bohl-type representation theorem is obtained which holds also in several nonergodic cases (e.g. for systems with invariant integral). NEWLINENEWLINENEWLINEThe following theorem is proved: The Poincaré map of the differential equation NEWLINE\[NEWLINE \frac{dy}{dx}=\frac{Y(x,y)}{X(x,y)}\in C^2(T^2) \tag{2}NEWLINE\]NEWLINE is conjugated to a rotation \(\rho\) (\(\rho\in\mathbb{R}\)) if and only if there exists a continuous function \(\Omega (x,y)\) periodic in \(x\) and \(y\), such that any solution to (2) can be represented in the form \(y=\rho x+c+\Omega (x,\rho x+c)\), \(c=\text{const}\). NEWLINENEWLINENEWLINEThis result enables one to carry out a method for the numerical integration of such systems, that preserves qualitative features of integral curves in the whole domain of definition. The numerical study concerns two main problems: numerical approximation to the rotation number and the function \(\Omega (x,y)\). An estimation on the approximation error is given.