This book is a comprehensive collection of known results about the Lozi map, a piecewise-affine version of the Henon map. Henon map is one of the most studied examples in dynamical systems and it attracts a lot of attention from researchers, however it is difficult to analyze analytically. Simpler structure of the Lozi map makes it more suitable for such analysis. The book is not only a good introduction to the Lozi map and its generalizations, it also summarizes of important concepts in dynamical systems theory such as hyperbolicity, SRB measures, attractor types, and more.
Preface
List of Figures
Introduction
Comprehensive Review of Hyperbolicity, Ergodicity, and Chaos
Entropies and statistical properties of chaotic attractors
Classification of strange attractors of dynamical systems
Reality of Chaos in the Hénon Mapping
Measuring chaos in the Hénon map
Bifurcations phenomena in the Hénon mappings
Hénon attractor is a quasi-attractor
Compound windows of the Hénon-map
The existence of infi nitely many period-doubling bifurcations
Dynamical Properties of the Lozi Mappings
Introduction to chaos via the Lozi maps
Ergodic properties of the Lozi mappings
Grammatical complexity of the Lozi mappings
Admissibility conditions for symbolic sequences of the Lozi map
Existence of strange attractor for b > 0
Existence of strange attractor for b < 0
Rigorous proof of chaos in the Lozi map using the theory of transversal heteroclinic cycles
Geometric structure of strange attractors in the Lozi map
Parameter-shifted shadowing property of Lozi maps
The basin of attraction of Lozi mapping
Relations between the forwards limit sets and the strange attractor
Topological entropy of the Lozi maps
Dynamical Properties of Modified and Generalized Lozi Mappings
Simple piecewise linear models for the zones of instability
Rigorous calculation of fractal dimension of a strange attractor
The piecewise linear noodle map
Generalized hyperbolic attractors
Global periodicity property of the generalization Lozi mappings
Generalized discrete Halanay inequality and the global stability of Lozi mapping
Global behaviors of some max difference equations
Generalized piecewise-linear area-preserving plane maps
Smooth versions of the Lozi mappings
Maps with border-collision period doubling scenario
Rigorous proof of chaos in a 2-D piecewise linear map
Occurrence of chaos via different routes
Generating multifold chaotic attractors
A new simple 2-D piecewise linear map
The discrete hyperchaotic double scroll
Piecewise smooth maps of the plane and robust chaos
Real and Mathematical Applications of Lozi Mappings
The Lozi mappings in control theory
The Lozi mappings in synchronization theory
The Lozi mappings and secure communications
The Lozi mappings in game theory
The Lozi mappings and evolutionary algorithms
The Lozi mappings and fuzzy modeling of an experimental thermal-vacuum system
Bibliography
Index
Biography
Elhadj, Zeraoulia