Evolution of Chaos and Power Spectra in One-Dimensional Maps

Abstract

Low-dimensional maps have turned out to be useful for discovering and understanding new properties of dynamical systems. Outstanding examples are the Bernoulli shifts [1] and the β transformations [2] in ergodic theory and the Lorenz map [3] and the quadratic model for the onset of fluid turbulence [4]. In fact these discrete processes have led us not only to a deeper understanding of chaotic orbits in terms of the topological entropy and the Lyapunov exponent, but also to the discovery of a band-splitting transition [5] and a dynamic scaling law near a chaotic transition point [4,6].