Permutation complexity in dynamical systems. Ordinal patterns, permutation entropy and all that
This book describes a study of permutation complexity, a subject that might be thought of as a kind of symbolic dynamics where the basic elements are ordinal patterns. Ordinal patterns are permutations defined by the order relations among points in the orbits of dynamical systems. The primary topics of the book are ordinal patterns, permutations, entropy and complexity. The book is aimed primarily at researchers in nonlinear dynamics and complex systems, but it is also accessible to graduate students. The common element among the topics studied here is a hypothetical order structure on the state space of a dynamical system substantiated in the form of ordinal patterns. The state space must be totally ordered, but is otherwise arbitrary. Thus it includes discrete sets and \(n\)-dimensional intervals, for example. The class of dynamical systems that are studied include stochastic sytems. Under some mild assumptions it can be shown that some ordinal patterns cannot be realized, contrary to the situation with symbol patterns. The existence of forbidden ordinal patterns, it turns out, can be used as a finger print of determinist orbit generation since, with probability one, random dynamics has no forbidden order patterns. The first three chapters of the book introduce the main concepts, show how they can be linked, and describe some applications of ordinal analysis. Mappings and their relationship with the structure of their admissible and forbidden patterns are taken up in the next two chapters. Here there is also a detailed study of the shift and signed-shift transformations. The remaining chapters discuss permutation entropy (metric and topological), discrete entropy, the detection of determinacy, and analysis of cellular automata and coupled map lattice systems.
scientific article; zbMATH DE number 5702310