Systems with Hidden Attractors (eBook)

Viet-Thanh Pham graduated in Electronics and Telecommunications in 2005 at Hanoi University of Technology, Vietnam. He received the PhD degree in Electronics, Automation, and Control of Complex Systems Engineering in 2013 from the University of Catania, Italy. Currently, he is Lecture at the School of Electronics and Telecommunications, Hanoi University of Science and Technology, Vietnam, where he has been involved in projects concerning the study of nonlinear circuits and systems. His scientific interest includes applications of nonlinear systems, analysis and design of analog circuits, and FPGA-based digital circuits. Dr. Volos received his Physics Diploma, his M.Sc. in Electronics and his Ph.D. in Chaotic Electronics, all from the Aristotle University of Thessaloniki. He currently serves as an Assistant Professor in the Physics Department of the Aristotle University of Thessaloniki. His research interests include, among others, the design of chaotic electronic circuits and their applications. Professor Kapitaniak is Head of Division of Dynamics, Technical University of Lodz. He is the corresponding member of the Polish Academy of Sciences.

Preface 6
Contents 8
1 Introduction 11
1.1 Self-Excited Attractors 11
1.2 Hidden Oscillations 14
1.3 Localization of Hidden Attractors 17
1.4 Control and Synchronization 17
1.5 Hidden Oscillations in Applied Models 19
1.5.1 Phase-Locked Loop Circuits 19
1.5.2 Automatic Control Systems 20
1.5.3 Chua's Circuit Oscillator 20
1.5.4 Electromechanical Systems 20
1.6 Families of Systems with Hidden Attractors 21
1.6.1 Systems Without Equilibrium 21
1.6.2 Systems with Stable Equilibrium 22
1.6.3 Systems with an Infinite Number of Equilibria 22
References 23
2 Systems with Stable Equilibria 30
2.1 Wang--Chen System with Only One Stable Equilibrium 30
2.2 Simple Flows with One Stable Equilibrium 32
2.3 Systems with Stable Equilibrium Points 35
2.4 Constructing a System with One Stable Equilibrium 38
2.5 Double-Scroll Attractors in Systems with Stable Equilibria 40
2.6 Fractional-Order Form of a System with Stable Equilibrium 41
References 43
3 Systems with an Infinite Number of Equilibrium Points 45
3.1 Simple Systems with Line Equilibrium 45
3.2 Systems with Closed Curve Equilibrium 48
3.3 Systems with Open Curve Equilibrium 52
3.4 Constructing a System with Infinite Equilibria 54
3.5 Multi-scroll Attractors in a System with Infinite Equilibria 55
3.6 Fractional-Order Form of Systems with Infinite Equilibria 56
References 57
4 Systems Without Equilibrium 59
4.1 Sprott A (Nose--Hoover) System 59
4.2 Wei System Without Equilibrium 60
4.3 Simple Systems with No Equilibrium 61
4.4 Constructing a System with No Equilibrium 64
4.5 Multi-scroll and Multi-wing Attractors in Systems Without Equilibrium 67
4.6 Fractional-Order Form of Systems Without Equilibrium 69
References 70
5 Synchronization of Systems with Hidden Attractors 72
5.1 Synchronization via Diffusion Coupling 72
5.1.1 Diffusion Coupling of Two Systems with One Stable Equilibrium 72
5.1.2 Diffusion Coupling of Two Systems with Infinite Equilibria 73
5.1.3 Diffusion Coupling of Two Systems Without Equilibrium 74
5.2 Synchronization via Nonlinear Control 75
5.2.1 Synchronization of Systems with One Stable Equilibrium 76
5.2.2 Synchronization of Systems with Infinite Equilibrium 78
5.2.3 Synchronization of Systems Without Equilibrium 81
References 83
6 Circuitry Realization 85
6.1 Basic Electronic Components and Electronic Circuits 85
6.2 Circuit Implementation of a System with One Stable Equilibrium 89
6.3 Circuit Implementations of Systems with Infinite Equilibria 92
6.3.1 Circuit Implementation of a System with Line Equilibrium 92
6.3.2 Circuit Implementation of a System with Closed Curve Equilibrium 95
6.3.3 Circuit Implementation of a System with Open Curve Equilibrium 98
6.4 Circuit Implementation of a System Without Equilibrium 101
6.5 Circuit Implementation of a System with Different Families of Hidden Attractors 104
References 107
7 Concluding Remarks 109
References 110
Index 111