This work is about the qualitative theory of autonomous ordinary differential equation (ODE) systems. The purpose of the work is threefold. First, it is intended to familiarize the reader with the essential theory of autonomous systems in dimension n. Second, it is hoped that the reader will learn the importance of planar autonomous systems, such the beautiful result of the Poincaré-Bendixson theorem. Third, since the theory is utilised in applied science, considerably space has been devoted to analytical methods that are used widely in applications.
The fundamental theory of existence and uniqueness of solutions to ODE systems are presented in Chapter 2. Then, Chapter 3 treats with the essential theory of autonomous systems in dimension n, such as the orbits and the limit sets of solutions. In Chapter 4 we consider planar autonomous systems. What makes planar systems different from higher dimensions is the existence of Jordan Curve theorem, which has made it possible for the theory to go much further. In particular, the Poincaré-Bendixson theorem, which is a statement about the long-term behavior of solutions to an autonomous system in the plane. Note that the Jordan Curve theorem is stated without proof, since the proof is terribly difficult but the result is obvious.
Lastly, in order not to lose sight of the applied side of the subject, Chapters 5 and 6 are devoted to analytical methods of autonomous systems. First, Chapter 5 treats with local stability analysis of an equilibrium. Then, in Chapter 6 we work with a relatively large study of an abnormal competing species model based on the science fiction movie The Terminator (1984), which should be taken with a pinch of salt. In its dystopian world there are two powerful forces of Men and the Terminator cyborgs trying to get completely rid of one another. Lack of space has, however, forced us to simplify some of the individual behaviour. These simplifications are partly justified by the fact that the purpose is to present how the theory can be applied even in a (hopefully) fictional situation and, of course, to answer the puzzling question whether the human race would stand a chance against the Terminators.
