Qualitative Methods in Continuous and Discrete Dynamical Systems

This chapter gives a general and friendly overview to the qualitative theory of continuous and discrete dynamical systems, as well as some applications to simple dynamic economic models, and is concluded by a section on basic principles and results of optimal control in continuous time, with some simple applications. The chapter aims to introduce some general concepts, notations and a minimal vocabulary concerning the study of the mathematical theory of dynamical systems that are used in the other chapters of the book. In particular, concepts like stability, bifurcations (local and global), basins of attraction, chaotic dynamics, noninvertible maps and critical sets are defined, and their applications are presented in the following sections both in continuous time and discrete time, as well as a brief introduction to optimal control together with some connections to the qualitative theory of dynamical systems and applications in economics.