As this series cover from low-dimensional to high-dimensional, from linear to nonlinear, a continuous system is finally capable of generating and we finally have enough background knowledge to talk about chaos.
Because high-dimensional space is difficult to understand intuitively (von Neumann and his sister: we don’t), high-dimensional dynamics generally discusses linear systems first, and then regards the effects of nonlinearity as distortions of linear systems.
Phase portraits are often used to describe the qualitative and semi-quantitative behavior of dynamical systems, such as stationary points and their stability.
The ultimate goal of dynamics is to give a function of the physical quantity of interest changing with time, where time is at least one of the independent variables of this function.