Discrete Dynamical Systems: Maps

Many physical systems displaying chaotic behavior are accurately described by mathematical models derived from well-understood physical principles. For example, the fundamental equations of fluid dynamics, namely the Navier–Stokes equations, are obtained from elementary mechanical and thermodynamical considerations. The simplest laser models are built from Maxwell’s laws of electromagnetism and from the quantummechanics of a two-level atom. Except for stationary regimes, it is in general not possible to find closed-form solutions to systems of nonlinear partial or ordinary differential equations (PDEs and ODEs). However, numerical integration of these equations often reproduces surprisingly well the irregular behaviors observed experimentally. Thus, thesemodels must have some mathematical properties that are linked to the occurrence of chaotic behavior. To understand what these properties are, it is clearly desirable to study chaotic dynamical systems whose mathematical structure is as simple as possible. Because of the difficulties associated with the analytical study of differential systems, a large amount of work has been devoted to dynamical systems whose state is known only at a discrete set of times. These are usually defined by a relation