Time delay in the swing equation: A variety of bifurcations

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 Grossmann and complements an analysis of Campbell et al., who consider time delay in the restoring force. 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.The swing equation is, e.g., at the core of any power system model. By delayed frequency control it becomes a pendulum equation with delayed damping -a system which is shown to exhibit highly complex dynamics. Repeating Hopf bifurcations lead again and again to the emergence of limit cycles as delay is increased. These limit cycles undergo further bifurcations including period doubling cascades and the birth of invariant tori.

Two-Parameter Dynamics of An Autonomous Mechanical Governor System With Time Delay

“…( 7), the state transition matrix in a single integral step can be theoretically determined by Eq. (16).…”

“…Time-delay effect is widely encountered in many dynamical systems including biological systems [1][2][3], social and economic systems [4,5], neural networks [6,7], chemical reactions [8,9], and mechanical systems [10][11], due to the essential dependency of present states on past states. Compared to the dynamical systems without time-delay effect, systems involving inherent time delays can exhibit more rich and complicated dynamics, including the instability of equilibrium, multiple stability switches induced by time-delay [12,13], high co-dimension bifurcations [14][15][16], coexistence of attractors [17,18], and chaotic motions [19,20]. In addition to the understanding of delay-induced bifurcations and transitions, the beneficial applications of time-delay effect have also attracted great interest from the research community.…”

On norm-based estimations for domains of attraction in nonlinear time-delay systems

“…In physical or technical problems, it is common that some state components occur in a delayed form, but others contribute only by their instantaneous values (see e.g. [23,61,73] or Example 2 below). Hence, the history of some state variables…”

“…Already for simple lower order systems, the infinite dimensionality of time-delay systems frequently results in complex phenomena and a rich variety of bifurcations [52,62,63,73,79]. Of course, bifurcations of the above described limit cycles might occur with further delay parameter variation.…”

“…A bifurcation analysis of (29) reveals that each regain of stability is accompanied by a subcritical Hopf bifurcation (cmp. [73]). Vertical slices in Fig.…”