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.
For nonlinear time-delay systems, domains of attraction are rarely studied despite their importance for technological applications. The present paper provides methodological hints for the determination of an upper bound on the radius of attraction by numerical means. Thereby, the respective Banach space for initial functions has to be selected and primary initial functions have to be chosen. The latter are used in timeforward simulations to determine a first upper bound on the radius of attraction. Thereafter, this upper bound is refined by secondary initial functions, which result a posteriori from the preceding simulations. Additionally, a bifurcation analysis should be undertaken. This analysis results in a possible improvement of the previous estimation. An example of a time-delayed swing equation demonstrates the various aspects.
Lyapunov-Krasovskii functionals are found to be related to Lyapunov functions that prove partial stability in finite dimensional system approximations. These approximations are ordinary differential equations, which, in the present paper, originate from the Chebyshev (pseudospectral) collocation or the Legendre tau method. Lyapunov functions that prove partial stability are simply obtained by solving a Lyapunov equation. They approximate the Lyapunov-Krasovskii functional. A formula for the partial positive definiteness bound on the Lyapunov function is derived. The formula is also applied to a numerical integration of the known Lyapunov-Krasovskii functional. An example shows that both approaches converge to identical results, representing the largest quadratic lower bound on complete-type or related functionals.
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