Odes

In the next three chapters some of the more important mathematical properties of nonlinear dynamical systems as well as the diagnostic tools for analyzing such systems will be introduced. This is a vast subject, so we will only present enough so that you can appreciate and understand the various topics that will be presented in subsequent chapters as we explore the various domains of our nonlinear world. Where needed to further our understanding, we will later expand on these nonlinear mathematical concepts, and even introduce some new ones. In this chapter the properties of nonlinear ODE systems are examined, the subsequent two chapters dealing with nonlinear difference equations (commonly referred to as nonlinear maps) and, much more briefly, with nonlinear PDEs and cellular automata. We will begin by discussing the "breakdown" of the linear superposition principle for nonlinear ODEs. Because of this breakdown, many of the "bread and butter" mathematical techniques (such as Laplace transforms and Fourier analysis) for solving linear ODEs no longer work or are useful for attempting to solve nonlinear ODEs. This necessitates the introduction of new mathematical approaches, many of which apply only to certain classes of nonlinear equations. Some of these mathematical methods are beyond the scope and level of this introductory text and will not be covered. Our intention here is to provide a simple, yet sufficient, mathematical framework that the reader can understand and analyze the various nonlinear models that will be presented in ensuing chapters. Our goal is to give you a glimpse of the nonlinear world, not to teach you all the mathematical tricks that exist for solving nonlinear dynamical equations. It should also be mentioned that the frontiers of nonlinear dynamics are constantly being pushed out with new ideas and applications continually appearing on a regular basis in various research publications. At present there is a somewhat "piecemeal" approach to tackling nonlinear dynamical equations, but, undoubtedly, as the subject matures, new mathematical techniques and concepts will be discovered and further "unification" will occur.