Time Delay in the Swing Equation: A Variety of Bifurcations

TL;DR

This paper investigates how increasing time delay in the swing equation's damping term causes recurring bifurcations, leading to complex oscillatory behaviors, supported by analytical formulas and numerical simulations.

Contribution

It provides a general formula for the first Lyapunov coefficient in second order systems with delayed damping, extending previous stability results.

Findings

Recurring sub- and supercritical Hopf bifurcations occur with increased delay.

Time delay induces a variety of qualitative behaviors in the swing equation.

Analytical and numerical methods together reveal complex dynamics due to delay.

Abstract

The present paper addresses the swing equation with additional delayed damping as an example for pendulum-like 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. In so far the paper extends results about 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 is generated in the simple swing equation by only introducing time delay in a damping term.