Chaotic Systems Part 1 docx

CHAOTIC SYSTEMS Edited by Esteban Tlelo-Cuautle Chaotic Systems Edited by Esteban Tlelo-Cuautle Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2011 InTech All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Katarina Lovrecic Technical Editor Teodora Smiljanic Cover Designer Martina Sirotic Image Copyright mycola, 2010. Used under license from Shutterstock.com First published February, 2011 Printed in India A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org Chaotic Systems, Edited by Esteban Tlelo-Cuautle p. cm. ISBN 978-953-307-564-8 free online editions of InTech Books and Journals can be found at www.intechopen.com Part 1 Chapter 1 Chapter 2 Chapter 3 Chapter 4 Part 2 Chapter 5 Chapter 6 Chapter 7 Part 3 Chapter 8 Preface IX Chaotic Systems 1 Short-Term Chaotic Time Series Forecast 3 Jiin-Po Yeh Predicting Chaos with Lyapunov Exponents: Zero Plays no Role in Forecasting Chaotic Systems 25 Dominique Guégan and Justin Leroux Relationship between the Predictability Limit and Initial Error in Chaotic Systems 39 Jianping Li and Ruiqiang Ding Microscopic Theory of Transport Phenomenon in Hamiltonian Chaotic Systems 51 Shiwei Yan Chaos Control 117 Chaos Analysis and Control in AFM and MEMS Resonators 119 Amir Hossein Davaie-Markazi and Hossein Sohanian-Haghighi Control and Identification of Chaotic Systems by Altering the Oscillation Energy 135 Valery Tereshko Adaptive Control of Chaos 161 Hassan Salarieh and Mohammad Shahrokhi Synchronization 185 Chaotic Clustering: Fragmentary Synchronization of Fractal Waves 187 Elena N. Benderskaya and Sofya V. Zhukova Contents Contents VI Coexistence of Synchronization and Anti-Synchronization for Chaotic Systems via Feedback Control 203 Hammami Sonia and Benrejeb Mohamed Applications 225 Design and Applications of Continuous-Time Chaos Generators 227 Carlos Sánchez-López, JesusManuel Muñoz-Pacheco, Victor Hugo Carbajal-Gómez, Rodolfo Trejo-Guerra, Cristopher Ramírez-Soto, Oscar S. Echeverria-Solis and Esteban Tlelo-Cuautle Applying Estimation Techniques to Chaos-based Digital Communications 255 Marcio Eisencraft and Luiz Antonio Baccalá Emergence of Matured Chaos During, Network Growth, Place for Adaptive Evolution and More of Equally Probable Signal Variants as an Alternative to Bias p 281 Andrzej Gecow Chapter 9 Part 4 Chapter 10 Chapter 11 Chapter 12 Pref ac e This book represents a collection of major developments in the area of Chaotic Systems. Researchers from around the world summarize key guidelines on modeling, simula- tion, control synchronization, and applications of these systems. The organization of the book includes twelve chapters covering a wide variety of engineering areas regard- ing basic scientifi c research and technological development. The fi rst chapter presents a time series technique to forecast future events. The sec- ond chapter analyzes forecasting chaos, providing a deep insight on the meaning of Lyapunov exponents locally. The predictability of chaotic events considering initial error in chaotic systems is presented in chapter three. The fourth chapter introduces a microscopic theory of transport phenomena which is supported by applying Hamilto- nian forms. These four chapters motivate the reader to look for approaches to control and synchronization of chaotic systems. Furthermore, chapters fi ve, six and seven focus on controlling the chaos phenomenon. For instance, micro-electromechanical systems like resonators might present chaotic behavior, the prediction and control of chaos is then presented in chapter fi ve. Chaotic systems can be perturbed, and control and identifi cation approaches can be applied as shown in chapter six. Among all tech- niques for controlling chaos the adaptive one is very useful, as described in chapter seven. Chaos systems can be synchronized as shown in chapters eight and nine. In chapter nine a technique for chaos control called feedback control, is also presented. Some realizations of chaotic oscillators with electronic circuits are presented in chap- ter ten. In this chapter two chaos systems are synchronized by Hamiltonian forms and observer approach, to realize a secure communication system. Chapter eleven introduces estimation techniques for chaos systems to develop digital communica- tions. Some open research problems can be identifi ed in these eleven chapters, as well as in the fi nal chapter devoted to the emergence of matured chaos during network growth. This book includes recent advances in science and engineering applications. However, as highlighted in the majority of chapters, still many research and applications are very much in need of further development. In this way, the book is a good source for identifying basics on chaotic systems, techniques to control and synchronize them, and to acquire recent knowledge and ideas for future research in this area. Some ap- plications to engineering motivate the reader to develop experiments applied to real X Preface life problems. This book is intended for students, academia and industry since the col- lected chapters provide a rich cocktail while balancing theory and applications. Enjoy the book! Esteban Tlelo-Cuautle INAOE Department of Electronics Mexico . Layer Output Layer Output HN s x 1 x 2 x 3 x R b s a b HN 2 HN 1 b 2 b 1 a 1 a 2 a s W 11 W 12 W 1s w sR w 11 w 12 w 2R w 1R w 13 # # # # Fig. 2. The feedforward. single neuron uses the linear transfer function Chaotic Systems 6 ()afn n = = (8) where n=W 11 a 1 + W 12 a 2 +……+W is a s + b, W 11 ,W 12 , …,W is are the weights connecting the neurons. physical systems in the real world, such as rainfall systems (Hense, 19 87), chemical reactions (Argoul et al., 19 87), biological systems (Glass et al., 19 83) and traffic flow systems (Dendrinos, 19 94),