Abstract
In this chapter, the basic theory of nonlinear time-delay dynamical systems will be introduced. Local theory and bifurcation theory for nonlinear time-delay dynamical systems will be discussed. The stability switching and bifurcation on specific eigenvectors of the linearized, time-delayed system at equilibrium will be discussed. The higher-order singularity and stability for nonlinear time-delay systems on the specific eigenvectors will be developed. The lower dimensional, nonlinear, time-delay systems will be discussed for a better understanding of the stability and bifurcation theory.
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