ABSTRACT
The results which we will present in the remainder of this book (including the results of the present chapter) will usually be consequences of stability preserving mapping results. In particular, in this chapter we will utilize the results of Chapter 3 to establish the Lyapunov stability theory for the motions of arbitrary continuous time and discrete time dynamical systems. To accomplish this, the domain of definition of the stability preserving mappings will always be an arbitrary dynamical system (the object of inquiry) while the range of the stability preserving mappings will generally be a specific dynamical system (the comparison system) determined by the specific problem on hand. A majority of the results of this chapter will be direct consequences of the comparison theorems given in Section 3.4. The remaining results will not be consequences of comparison results; however, in all of the results which we will consider, stability preserving maps will play a central role.
