Jibin Li

Planar dynamical systems : selected classical problems

This book presents in an elementary way the recent significant developments in the qualitative theory of planar dynamical systems. The subjects are covered as follows: the studies of center and isochronous center problems, multiple Hopf bifurcations and local and global bifurcations of the equivariant planar vector fields which concern with Hilberts 16th problem. This book is intended for graduate students, post-doctors and researchers in the area of theories and applications of dynamical systems. For all engineers who are interested the theory of dynamical systems, it is also a reasonable reference. It requires a minimum background of an one-year course on nonlinear differential equations.

Chapter 5. Theory of Center-Focus and Bifurcation of Limit Cycles at Infinity of a Class of Systems

We have mentioned in Preface of this book that a real planar polynomial vector field V can be compactified on the sphere. The vector field p(V ) restricted to the upper hemisphere completed with the equator Γ∞ is called Poincaré compactification of a polynomial vector field. If a real polynomial vector field has no real singular point in the equator Γ∞ of the Poincaré disc and Γ∞ can be seen a trajectory, all trajectories in a inner neighborhood of Γ∞ are spirals or closed orbits, then Γ∞ is called the equator cycle of the vector field. Γ∞ can be become a point by using the Bendixson reciprocal radius transformation. This point is called infinity of the system. In this chapter, we discuss the center-focus problem of infinity (i.e., to distinguish when the trajectories in a inner neighborhood of Γ∞ are either closed orbits or spirals) and the bifurcation of limit cycles at infinity for a class of systems.