  We consider a broad class of second-order dynamical systems and study the
impact of damping as a system parameter on the stability, hyperbolicity, and
bifurcation in such systems. We prove a monotonic effect of damping on the
hyperbolicity of the equilibrium points of the corresponding first-order
system. This provides a rigorous formulation and theoretical justification for
the intuitive notion that damping increases stability. To establish this
result, we prove a matrix perturbation result for complex symmetric matrices
with positive semidefinite perturbations to their imaginary parts, which may be
of independent interest. Furthermore, we establish necessary and sufficient
conditions for the breakdown of hyperbolicity of the first-order system under
damping variations in terms of observability of a pair of matrices relating
damping, inertia, and Jacobian matrices, and propose sufficient conditions for
Hopf bifurcation resulting from such hyperbolicity breakdown. The developed
theory has significant applications in the stability of electric power systems,
which are one of the most complex and important engineering systems. In
particular, we characterize the impact of damping on the hyperbolicity of the
swing equation model which is the fundamental dynamical model of power systems,
and demonstrate Hopf bifurcations resulting from damping variations.
