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We consider a broad class of second-order dynamical systems and study theimpact of damping as a system parameter on the stability, hyperbolicity, andbifurcation in such systems. We prove a monotonic effect of damping on thehyperbolicity of the equilibrium points of the corresponding first-ordersystem. This provides a rigorous formulation and theoretical justification forthe intuitive notion that damping increases stability. To establish thisresult, we prove a matrix perturbation result for complex symmetric matriceswith positive semidefinite perturbations to their imaginary parts, which may beof independent interest. Furthermore, we establish necessary and sufficientconditions for the breakdown of hyperbolicity of the first-order system underdamping variations in terms of observability of a pair of matrices relatingdamping, inertia, and Jacobian matrices, and propose sufficient conditions forHopf bifurcation resulting from such hyperbolicity breakdown. The developedtheory has significant applications in the stability of electric power systems,which are one of the most complex and important engineering systems. Inparticular, we characterize the impact of damping on the hyperbolicity of theswing equation model which is the fundamental dynamical model of power systems,and demonstrate Hopf bifurcations resulting from damping variations.

The Impact of Damping in Second-Order Dynamical Systems with Applications to Power Grid Stability