Stability and Bifurcation Analysis of Differential–Difference–Algebraic Equations

This paper treats a nonlinear dynamical system with both continuous-time and discrete-time variables as a differential–difference–algebraic equation (DDA) or a hybrid dynamical system, presents a fundamental analyzing method of such a DDA system for local sampling, asymptotical stability, singular perturbations and bifurcations, and further shows that there exist four types of generic codimension-one bifurcations at the equilibria in contrast to two types in continuous-time dynamical systems and three types in discrete-time dynamical systems. Finally the theoretical results are applied to digital control of power systems as an example. Numerical simulations demonstrate that our results are useful.