Abstract
The word “chaos” appeared for the first time in the field of mathematics in an article of Li and Yorke entitled “Period Three Implies Chaos.” This short and elegant paper caused a great sensation in the world of mathematical physics. This chapter discusses the discretizations of ordinary differential equations (O.D.E.) from the viewpoint of chaos. Depending on the types of discretization, continuous dynamical systems may change into complicated discrete dynamical systems, specifically chaos. An interesting example of population dynamics as an introduction is presented in the chapter. The chapter describes the Euler discretization of O.D.E. and the fundamental theorem proposed by Yamaguti and Matano. The original differential equation has only one asymptotically stable equilibrium point in the case of large mesh size discretization. The cases of sufficiently small mesh size discretization are illustrated in the chapter. Even in these cases, the same chaotic dynamical systems are found by applying the Euler discretization. The chapter presents one interesting example for which every discretization causes chaotic phenomenon.
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