Nonlinear Dynamics

Abstract

This is perhaps the most challenging chapter of this book in that we attempt to show how homology can be used in nonlinear analysis and dynamical systems. As mentioned in the Preface, algebraic topology was developed to solve problems in these subjects. Thus, on one hand, some of the material (most notably Sections 10.4 and 10.5) is classical and has a rich history. On the other hand, the focus of Section 10.6, combining numerical analysis with homology via the computer to obtain mathematically rigorous results, is a cutting-edge topic. We assume that the background and interest of our readers are equally varied, ranging from an individual with no background in dynamics hoping to learn the mathematical theory to others whose primary goal is to understand how to apply these ideas to specific nonlinear systems.