ABSTRACT
Density evolution in one-dimensional maps is examined, associating the macroscopic state of the dynamical systems with a phase-space density. This chapter examines the discovered property of asymptotic periodicity in the density evolution of one-dimensional maps. The concepts of asymptotic periodicity are illustrated through different example systems. The irregular and apparently unpredictable nature of trajectory evolution in many nonlinear dynamical systems can be greatly simplified if one looks at their behavior in terms of density evolution. Following the numerical demonstration of the existence of highly irregular dynamical behavior in relatively simple nonlinear systems, the study of so-called "chaotic" systems has captured the attention of literally thousands of scientists in the physical, mathematical, biological, and social sciences. The chapter illustrates asymptotic periodicity in the hat map and the quadratic map at the parameters where one-dimensional maps generate banded chaos or quasi-periodicity.
