Over the past fifty years, the development of chaotic dynamical systems theory and its subsequent wide applicability in science and technology has been an extremely important achievement of modern mathematics. Chaotic attractors are not a fleeting curiosity, and their continued study is important for the progress of mathematics.
This book collects several of the new relevant results on the most important of them: the Lozi, Hénon and Belykh attractors. Existence proofs for strange attractors in piecewise-smooth nonlinear Lozi-Hénon and Belykh maps are given. Generalization of Lozi map in higher dimensions, Markov partition or embedding into the 2D border collision normal form of this map are considered. K-symbol fractional order discrete-time and relationship between this map and maxtype difference equations are explored. Statistical self-similarity, control of chaotic transients, and target-oriented control of Hénon and Lozi attractors are presented. Controlling chimera and solitary states by additive noise in networks of chaotic maps, detecting invariant expanding cones for generating word sets to identify chaos in piecewise-linear maps, and studying border collision bifurcations in a piecewise linear duopoly model complete this book.
This book is an essential companion for students and researchers in mathematics, physics, engineering, and related disciplines seeking to deepen their understanding of chaotic dynamical systems and their applications.
The chapters in this book were originally published in Journal of Difference Equations and Applications.
Introduction - Lozi, Hénon, and Belykh chaotic attractors: new results
fifty years on
1. Controlling chaotic transients in the Hénon and the Lozi
map with the safety function
2. On target-oriented control of Hénon and Lozi
maps
3. Controlling chimera and solitary states by additive noise in networks
of chaotic maps
4. Statistical self-similarity in Lozi and Hénon's strange
attractors
5. Markov partition in the attractor of Lozi maps
6. Lozi map
embedded into the 2D border collision normal form
7. A higher-dimensional
generalization of the Lozi map: bifurcations and dynamics
8. Existence proofs
for strange attractors in piecewise-smooth nonlinear Lozi-Hénon and Belykh
maps
9. On the relationship between Lozi maps and max-type difference
equations
10. K-symbol fractional order discrete-time models of Lozi system
11. Border collision bifurcations in a piecewise linear duopoly model
12.
Detecting invariant expanding cones for generating word sets to identify
chaos in piecewise-linear maps
René Lozi is Emeritus Professor at University Cote dAzur, France and Vice-President of the International Society of Difference Equations. His research areas include complexity and emergence theory, dynamical systems, bifurcations, control of chaos, cryptography based on chaos, and memristors
Lyudmila Efremova is Professor at Nizhny Novgorod State University and Moscow Institute of Physics and Technology, Russia. Her scientific interests include regular and chaotic properties of low-dimensional discrete dynamical systems.
Mohammed-Salah Abdelouahab is Professor at Abdelhafid Boussouf University Center of Mila, Algeria. He is the head of the research team in fractional calculus and its applications at the laboratory of mathematics and their interactions and the Editor-in-Chief of the Journal of Innovative Applied Mathematics and Computational Sciences.
Safwan El Assad is Professor at Polytech Nantes, France. Between 1988 to 2005, his research activities concerned radar imagery and digital communications. Nowadays his research largely focuses on chaos-based cryptography, encryption, crypto-compression, steganography, hash functions, authenticated encryption.
