Nonlinear Systems: Design, Applications and Analysis

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Series: Computational Mathematics and Analysis
BISAC: MAT017000

A nonlinear system is a set of nonlinear equations, which may be algebraic, ordinary differential, partial differential, fractional, integral or a combination of these. Especially, nowadays, the term “dynamical system” is used as a synonym of nonlinear systems where the nonlinear equations represent the evolution of a solution over time. So, the notion of dynamical systems arose following the name of equations governing the motion of a system of particles, even though the nonlinear system may have no application to mechanics. Also, from an engineering point of view a nonlinear system may be represented with a feedback loop in which the output of an element is not proportional to its input.

Over the last few decades, nonlinear systems have been used to describe a great variety of phenomena, in social and life sciences as well as in physical sciences and engineering. The theory of nonlinear systems has applications to problems of population growth, economics, chemical reactions, celestial mechanics, physiology of nerves, onset of turbulence, regulation of heartbeats, electronic circuits, cryptography, secure communications and many others.

Nonlinear dynamical systems, which present chaotic behavior, are of great importance due to their applications in science and engineering. Chaotic systems are nonlinear dynamical systems and maps that are highly sensitive to initial conditions. The sensitivity of initial conditions is usually called the butterfly effect for dynamical systems and maps.

So, nowadays the design and analysis of nonlinear systems and especially chaotic systems has gained the interest of the research community due to the fact that many phenomena on financial, physical, biological, chemical, mechanical and engineering systems can be modeled and studied through the perspective of nonlinear dynamics. These nonlinear systems can be modeled by discrete-time or continuous-time mathematical models.

This book aims to bridge the gap between the design/analysis and applications, which are the two research stages on the progress of nonlinear systems and also which open up some new directions of real applications, where chaos can be put up to technological use, including secure communication systems, electronic circuits’ design, memristors and radar. This book can serve as an updated and handy reference for university professors, graduate students, laboratory researchers as well as physicists and applied mathematicians who are interested in studying the chaos and its applications through the field of nonlinear systems. (Imprint: Nova)

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Table of Contents

Table of Contents

Preface

Chapter 1. Volterra Series and Associated Frequency Domain Representation of a Class of Bilinear Partial Differential Equations
Yuzhu Guo, Ling-Zhong Guo and Stephen A. Billings (Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, England, and others)

Chapter 2. Particular Solutions of Nonlinear Ordinary Differential Equations and Applications
Haihong Liu and Linghai Zhang (Department of Mathematics, Yunnan Normal University, Kunming, People’s Republic of China, and others)

Chapter 3. Lyapunov Exponents and Parameter Planes of Hyperchaotic Regions of the Lü Model in 6D and Its Projections
L. A. Quezada-Téllez, S. Carrillo-Moreno, O. A. Rosas-Jaimes, G. Fernández-Anaya, E. Báez-Juárez and A. Zamora-Ramos (Department of Applied Mathematics and Systems, UAM-Cuajimalpa, Mexico City, México, and others)

Chapter 4. State-Feedback Nonlinear W1, ∞ Control and Related Minimax Designs
M. D. S. Aliyu and U. F. Sulaiman (Department of Electrical Engineering, King Faisal University, Al-ahsa, Saudi Arabia, and others)

Chapter 5. Comparative Performances of Anti-Synchronization between Different Chaotic Systems Using Three Control Schemes
Piyush Pratap Singh and Binoy Krishna Roy (Department of Electrical Engineering, National Institute of Technology Meghalaya, Shillong, Meghalaya, and others)

Chapter 6. The Control of a Hydraulic Servo System Using an Adaptive Sliding Mode Surface and Adaptive Gain
Moez Feki and Emna Kolsi Gdoura (Laboratory of Mathematics, Deterministic and Random Modeling, ESSTHS, University of Sousse, Tunisia, and others)

Chapter 7. Stabilizing Unstable Periodic Orbits of Chaotic Systems Using an Adaptive Fuzzy Time-Delayed Feedback Controller
Hanène Medhaffar, Moez Feki and Nabil Derbel (Research group CEMLab, National Engineering School of Sfax, University of Sfax, Tunisia, and others)

Chapter 8. Synchronization in Chaotic Systems by a Threshold Based Coupling Using the Poincaré Plane
L. J. Ontañón–García (Academic Coordination Western Altiplano Region, Autonomous University of San Luis Potosí, Salinas de Hidalgo, México)

Chapter 9. Dynamics, Synchronization and the Fractional Order Form of a Chaotic System without Equilibrium
Viet-Thanh Pham, Christos K. Volos, Sundarapandian Vaidyanathan and Ahmad Taher Azar (School of Electronics and Telecommunications, Hanoi University of Science and Technology, Hanoi, Vietnam, and others)

Chapter 10. Nonlinear Dynamics of Single and Multiple Bias-Tuned Colpitts Oscillators
Bishnu Charan Sarkar and Suvra Sarkar (Physics Department, BurdwanUniversity, Burdwan, India, and others)

Chapter 11. Chaos from an Active Band-Pass Filter: An Inductor Free Chua’s Circuit and More
Tanmoy Banerjee (Chaos and Complex Systems Research Laboratory, Department of Physics,University of Burdwan, Burdwan, India)

Chapter 12. The Dynamics of a Driven Double Scroll Chaotic Circuit with a Negative Impedance Convertor
S. Sabarathinam and K. Thamilmaran (Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli, India)

Chapter 13. An Experimental Study of Chaotic Phenomena in the RL-Diode Circuit Driven by a Square Wave
Nikolaos A. Gerodimos, Peter A. Daltzis, Christos K. Volos, Hector E. Nistazakis and George S. Tombras (Department of Electronics, Computers, Telecommunications and Control, Faculty of Physics, National and Kapodistrian University of Athens, Athens, Greece, and others)

Chapter 14. Dynamical Analysis of a Beta-Cell Biological System and Its Hardware Realization
D.K. Guevara-Flores, V. Fern´andez-Carre´on, J.M. Munoz-Pacheco, E. Zambrano-Serranoy, O.G. F´elix-Beltr´an, L.C. G´omez-Pav´on, A. Luis-Ramos and P. Zaca-Mor´anz (Faculty of Electronics Sciences, Benem’erita Autonomous University of Puebla, Puebla, Mexico, and others)

Chapter 15. Non-Coherent Chaos-Based Spread-Spectrum Systems Using Multi-Carrier Modulation for Wireless Communications
Nguyen Xuan Quyen and Christos K. Volos (School of Electronics and Telecommunications, Hanoi University of Science and Technology, Hanoi, Vietnam, and others)

Chapter 16. Chaos in Optimal Signals for Radar and Communications
Jonathan N. Blakely, Ned J. Corron, Aubrey N. Beal and Marko S. Milosavljevic, Charles M. Bowden Laboratory (U. S. Army Aviation and Missile Research, Development and Engineering Center, Redstone Arsenal, Alabama, USA)

Chapter 17. The Physical and Circuit-Theoretic Significance of the Memristor
Emanuel Gluskin (Engineering Department, Kinneret College in the Jordan Valley, Galilean Sea, Israel)

Index

This book can serve as an updated and handy reference for university professors, graduate students, laboratory researchers as well as physicists and applied mathematicians who are interested in studying nonlinear systems and their applications in various scientific.

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