ABSTRACT
This chapter describes some of the basic tools of the mathematical and geometrical theories of dynamical systems. It focuses on that those tools characterized by chaos and singular perturbation phenomenon. The paradigm of deterministic chaos is certainly one of the most interesting phenomena observed and studied during the second half of the twentieth century. Singular perturbation theory provides the mean to decompose such systems into slow and fast dynamics which greatly simplifies their structural analysis and any subsequent control design. The most significant development in the analysis and control of nonlinear singularly perturbed systems has been the integral manifold approach. For nonlinear singularly perturbed systems, the composite state feedback law is the sum of slow and fast parts. The fast one steers the fast system states to the slow integral manifold. Another important concept of the theory of dynamical systems is the occurrence of bifurcations.
