A nonexistence criterion for closed orbits in planar flows and its application to fast limit cycle detection

Few analytical criteria exist in the literature for ruling out closed orbits and, thus, limit cycles of a given two-dimensional dynamical system. Well-known results are the gradient and Dulac criteria. In this work, an overview of recent advances on this topic is given. Furthermore, as the main result, a new, generally applicable criterion is presented. The derivation of this criterion is motivated by a reformulation of the time differential using the Moore-Penrose pseudoinverse of the velocity field, which holds exactly along trajectories of the system. This result can be used to enhance limit cycle detection, which is computationally expensive, by providing one-dimensional manifolds from which probing trajectories can be launched. These manifolds are guaranteed to intersect existing limit cycles in the phase plane. To illustrate a direct application of the proposed criterion, we consider a planar system with three limit cycles and demonstrate a drastic increase in efficiency in sampling random initial conditions required to identify all stable limit cycles, particularly when a limit cycle exists whose region of attraction is much smaller than the domain size.