Two-Dimensional Autonomous Systems of Differential Equations with a First Integral

摘要

A geometry of orbits is studied for an autonomous system x = f(x, y), y = g(x, y) of the undamped type. A theorem is proved which shows that a autonomous system with a finite number of critical points is of the undamped type if and only if, all but a finite number of orbits of the system are closed curves. A variant is developed of the poincare-Bendixson theory which is adapted to systems with this property. This theory is then used to study the pattern of open-arc orbits and critical points of a system of undamped type. These results are used to obtain a complete geometric theory for such a system. (DDC Abstract)