ABSTRACT
This chapter discusses some basic, and very powerful, results from the qualitative theory of nonlinear autonomous differential dynamical systems, primarily in the plane. In a field as vast as nonlinear dynamics, any essay of the present length must be selective. The chapter begins with linearized stability analysis for hyperbolic equilibria and proceeds to develop some diagnostic tools for nonhyperbolic cases. The distinctly nonlinear phenomenon of the limit cycle is then discussed and Hilbert's—still unsolved—16th Problem is stated. The Poincaré-Bendixson and Hopf Bifurcation Theorems are presented, as well as an introduction to Poincaré maps, which beautifully connect the world of continuous systems to that of discrete systems. Tools for precluding periodic orbits—the Bendixson and Bendixson-Dulac negative tests—are then presented and applied to a Kolmogorov system which, naturally, provides a forum for Kolmogorov's Theorem on cycles. The chapter presents some of the fundamental results, and gives an index theoretic proof of Brouwer's famous fixed point theorem on the disk.
