Time delay in the swing equation: A variety of bifurcations

The present paper addresses the swing equation with additional delayed damping as an example for pendulumlike systems. In this context, it is proved that recurring sub- and supercritical Hopf bifurcations occur if time delay is increased. To this end, a general formula for the first Lyapunov coefficient in second order systems with additional delayed damping and delay-free nonlinearity is given. Insofar, the paper extends the results about the stability switching of equilibria in linear time delay systems from Cooke and Grossman. In addition to the analytical results, periodic solutions are numerically dealt with. The numerical results demonstrate how a variety of qualitative behaviors are generated in the simple swing equation by only introducing time delay in a damping term.