Determining asymptotic properties of dynamical systems, including the formulation of a qualitative picture of the system's trajectories over large intervals of time, is one of the central questions of modern theory for adaptive systems. This is not surprising, for the very reason for adaptation is the lack of available measurement information. If such information is not available a priori, and carrying out numerical or physical experiments is not a feasible option, assessment of the qualitative properties of the system's behavior is often the only way to characterize the system. What are these qualitative properties? Informally, from these properties we should be able to tell, for example, how a system might respond to external perturbations, or how the system's variables behave over long intervals of time. Formally, we may wish to know whether the system is stable in some sense, whether its trajectories are bounded, and to what sets these trajectories will be confined with time.

In this chapter we shall provide a brief summary and necessary background about these qualitative properties of dynamical systems. We do not wish, however, to present an exhaustive review of all concepts. There are many excellent texts devoted to detailed analysis of every single issue mentioned above. Here we will rather review these concepts with a level of detail and generality just sufficient for developing a qualitative understanding of the problem of adaptation and the basics of methods of adaptive regulation.