Limit Cycles in Planar Systems of Ordinary Differential Equations

The idea of a dynamical system is predicting the future of a given system with respect to some initial conditions. If the dynamical system is formulated as a differential equation, then there is usually a direct relation between the dynamical system and the processes involved. Today, we can easily say that dynamical systems can predict a huge number of phenomena, including chaos. The real question is, therefore, not whether complicated phenomena may occur, but whether restrictions on the possible dynamics exist. In this chapter, we commence with major theorems that are frequently used for justifying phase space analysis. We continue with simple examples that either possess limit cycles and classes of differential equations that never possess limit cycles. We end up with the ideas behind two major theorems that put bounds for the number of limit cycles from above: Sansone’s (Annali di Matematica Pura ed Applicata, Serie IV 28:153–181, 1949) theorem and Zhang’s (Appl Anal 29:63–76, 1986) theorem. Both theorems apply to systems that have a clear mechanistic interpretation. We outline the major arguments behind the quite precise estimates used in these theorems and describe their differences. Our objective is not to formulate these theorems in their most general form, but we give references to recent extensions.