Foundations of Dynamical Systems Theory
In this chapter the issues of dynamical systems theory, which is used for the further analysis, are presented. In the first section some basic facts are recalled. In the subsequent sections some advanced topics are considered. Then, the Euler method on a manifold is discussed. Then linear, weakly nonlinear and gradient dynamical systems are elaborated. In three last sections of this chapter topological conjugacy of cascades, pseudo-orbit tracing property and dynamical systems with control are presented. It should be mentioned that both topological conjugacy and shadowing property are the topics that are far from being worked out completely and a lot of problems in this field are open.