Limit Cycles of a Planar Vector Field

This paper presents a solution to the problem to find isolated closed trajectories of two-dimensional dynamic systems. In contrast to the method of Bendixson’s ring regions, the new method is constructive. It allows the determination of the location of closed trajectories and therefore gives an upper bound for their number. The method is based on the idea to use inherent geometrical and physical extremal properties of these trajectories to transform the problem into an optimization task (isoperimetric problem of variational calculus) that can be solved by numerical algorithms, e.g., by hillclimbing.