  Many dynamic processes involve time delays, thus their dynamics are governed
by delay differential equations (DDEs). Studying the stability of dynamic
systems is critical, but analyzing the stability of time-delay systems is
challenging because DDEs are infinite-dimensional. We propose a new approach to
quickly generate stability charts for DDEs using continuation of characteristic
roots (CCR). In our CCR method, the roots of the characteristic equation of a
DDE are written as implicit functions of the parameters of interest, and the
continuation equations are derived in the form of ordinary differential
equations (ODEs). Numerical continuation is then employed to determine the
characteristic roots at all points in a parametric space; the stability of the
original DDE can then be easily determined. A key advantage of the proposed
method is that a system of linearly independent ODEs is solved rather than the
typical strategy of solving a large eigenvalue problem at each grid point in
the domain. Thus, the CCR method significantly reduces the computational effort
required to determine the stability of DDEs. As we demonstrate with several
examples, the CCR method generates highly accurate stability charts, and does
so up to 10 times faster than the Galerkin approximation method.
