ABSTRACT
This research present the synchronization and anti-synchronization of non-identical Lu and Lorenz chaotic system. Based on Lyapunov stability theory an adaptive control law is devised to make the states of two non-identical Lu and Lorenz chaotic systems with unknown system parameters asymptotically synchronized. Since the Lypunov exponents are not required for these calculations, the stability result for the synchronization and anti-synchronization schemes derived are established using Lyapunov Stability Theory. Finally, numerical simulations are presented using Runge –Kutta method to show the effectiveness of the proposed chaos synchronization and anti-synchronization scheme and the objective.
KEYWORDS: Adaptive Control, Chaotic system, Synchronization, Anti-synchronization, Master-Slave synchronization, Lyapunov stability theory
TABLE OF CONTENTS
TITLE PAGE i
ATTESTATION ii
CERTIFICATION iii
DEDICATION iv
ACKNOWELEDGEMENT v
ABSTRACT vi
TABLE OF CONTENT vii-viii
LIST OF FIGURES ix
CHAPTER ONE: INTRODUCTION 1
Background to study 1
Basic Definitions 2
Chaotic systems 2
Stability and Fixed Points 3
Chaotic orbit 3
Research Motivation 4
Statement of problem 5
Aim of Research 5
Research Objective 5
CHAPTER TWO: LITERATURE REVIEW. 6
2.1 Coullet chaotic system and synchronization 6
2.2 Principles and Applications 16
2.3 Analysis and prediction of chaotic systems 19
CHAPTER THREE: METHODOLOGY
3.1. System considered 21
3.2. Backstepping control method 21
3.3 Sliding mode control method 24
CHAPTER FOUR: RESULTS AND DISCUSSION
4.1 Design of backstepping control for synchronization 29
4.2 Design of sliding mode control for synchronization 33
4.3 Numerical results 35
CHAPTER FIVE: CONCUSION
5.0 Conclusion 39
REFERENCES 40
APPENDIX 44
LIST OF FIGURES
Figure 1: 3D Phase portrait of coullet chaotic system. 36
Figure 2: 2D Phase portrait of coullet chaotic system 36
Figure 3: Time series of coullet chaotic system. 36
Figure 4: Time response of state for synchronization of backstepping control 37
Figure 5: Time response of state for synchronization of sliding mode control 37
Figure 6: Time response of the error dynamics for backstepping synchronization 37
Figure 7: Time response of the error dynamics for sliding mode synchronization 38
CHAPTER ONE
INTRODUCTION
1.1 BACKGROUND TO STUDY
Chaotic systems have been widely studied by many scientists and engineers from different viewpoints. Chaos and non-linear dynamics are presently active fields of interdisciplinary research, attracting the attention of researchers on account of its wide applicability in the physical world. Chaotic systems are dynamical systems that are highly sensitive to initial conditions. This sensitivity is popularly known as the butterfly effect. Since chaos phenomenon in weather models which was first observed by Edward Lorenz in 1963, a large number of chaos phenomena and chaos behaviour have been discovered in social, economical, biological and electrical systems (El-Dessoky, 18 june 2007).
One of the key problems in the study of chaotic systems is the chaos control. The chaos control problem is concerned with stabilization of the chaotic attractor to a periodic orbit or a stable equilibrium point. The pioneering works of Ott et al (1998), are the most important works in this direction. It was promptly followed by application of various engineering control techniques like feedback control (Boris Kramer, august 17 2017), adaptive control (LOZANO, 2014), passive control (QI Dong-lian,2002), backstepping control (Vincent, january 2008), active control (Vincent, june 2008)etc. for achieving this objective.
Pecora and Carroll (1990) introduced a method to synchronize two identical chaotic systems and showed that it was possible for some chaotic systems to be completely synchronized. From then onwards, chaos synchronization has been widely explored in a variety of fields including physical systems, chemical systems, ecological systems, secure communications, etc. (Kenny Headington, 2004). Recently, the concept of synchronization has been extended to the scope, such as generalized synchronization, phase synchronization, lag synchronization, and anti-synchronization (AS), in which the state vectors of synchronized systems have the same absolute values but opposite signs. Therefore, the sum of two signals can converge to zero when AS appears (Singh Piyush Pratap, 2010).
Chaos has developed over time. For example, Ruelle and Takens (2000) proposed a theory for the onset of turbulence in fluids, based on abstract considerations about strange attractors (Sajad Jafari, 2009). In the early 1970‟s, May was working on a model that addressed how insect birth rate varied with food supply. He found that at certain critical values, his equation required twice time to return to its original state, the period having doubled in value. After several period-doubling cycles, his model became unpredictable, rather like actual insect populations tend to be unpredictable. Since May’s discovery with insects, mathematicians have found that this period-doubling is a natural route to chaos for many different systems (El-Dessoky, 18 june 2007).
In recent times, study on chaotic dynamics has entered a new phase. Coupled to the demonstration of chaos in a wide range of situations and its study, many researchers are interested in utilizing the basic knowledge of the theory of chaos either to analyze chaotic experimental time series data or to use the presence of chaos to achieve some practical goals like synchronization and control, which has applications in secure communications, description of financial times series prediction, techniques of neural networks and genetic algorithms(AZZAZ, may 2011). Chaotic dynamics essentially is one of the attributes of nonlinearity of an evolving process. There exists domains of operation in which apart from general nonlinear features one can expect the richness of chaotic behaviour. As a matter of fact every Natural system is a store house of such behaviours and in all probability chaos assumes the operating paradigm in such systems(Maiti, august 2013).
Chaos is one of the subtle behaviours of a nonlinear system. It has its intrinsic academic interest which involve a number of concepts definitions and can be studied from very elementary nonlinear differential or difference equations(Khaki-Sedigh, 6 april 2008). The essential mathematics is embedded in the dynamical systems theory which focuses on the qualitative and quantitative behaviours of first order differential /difference equations. Following the ideas of Poincare (2001) (First one to identify the richness of nonlinear system behaviour) on the qualitative description of system behaviour, some geometrical methods have evolved which provide information without explicitly solving the system (Maiti, august 2013). An interesting and useful fact, rather a dominating factor of a nonlinear system is its capability of producing multiple equilibrium states unlike linear system which sticks on to a unique steady state. How to realize or avoid a particular state invites the role of stability theory (Uğur Erkin Kocamaz, 2006).
In the context of synchronizing two non-identical oscillators, the most prevalent method is to couple the systems suitably, so that they asymptotically follow the same path on the attractor. The master-slave, or drive-response formalism is commonly used for synchronization and anti-synchronization problems involving non identical oscillators. Two chaotic systems (master- slave) are anti-synchronized, when sum of their states will converge to zero asymptotically and amplitude of states will be equal in magnitude but with opposite phase. State with equal magnitude but opposite in phase is also an important phenomenon in case of chaotic system synchronization (Singh Piyush Pratap, 2015).
In this research, Lu (Vaidyanathan1, 2011) and Lorenz (Spassova, 2013) Chaotic system is considered for anti-synchronization and synchronization using adaptive control as well as parameters estimation when system parameters are unknown. Stabilization and convergence of error dynamics is achieved using Lyapunov stability theory(Piyush Pratap Singh ⇑, 1 october 2014; Singh Piyush Pratap, 2015).
1.2 Research Motivation
Chaotic system has potential applications in physics, neurobiology, earth science, electrical engineering, and many other fields (Maiti, august 2013). Motivated by stated applications, chaotic systems are also being applied in field of communication with the use of synchronization concept. Secure communication is a major issue today. It is expected that synchronization of two chaotic systems may help towards secure communication. Motivated by such practical requirement, authors are motivated to consider synchronization and anti-synchronization between well-known Lu chaotic system, a member of unified chaotic system, and Lorenz system developed by Edward Lorenz. There are two methods for synchronization of chaotic systems known as drive-response scheme and coupling scheme. Drive-response is also called master–slave scheme which is widely used by researchers. In this scheme the output of master (drive) system is used to control the slave system (K. S. Ojo, 1 january 2014).
A very promising chaos control strategy that is lately being explored is the ‘adaptive’ approach. Adaptive control involves design of controllers, based on principle of engineering adaptive controllers(Karimi, 2016). As opposed to conventional controllers, they are capable of adjusting themselves with changes in the original system, that they are supposed to control. This flexibility enhances their effectiveness.
On this account, adaptive controllers are more appropriate for controlling chaos as opposed to many other commonly used control mechanisms like linear and non-linear feedback, tracking control, etc. Adaptive synchronization and anti-synchronization employs similar self-adjusting controllers to achieve complete synchronization and anti-synchronization between two similar or different chaotic systems(Singh Piyush Pratap, 2015).
In real world applications, the original system, that is, the system which has to be controlled, is often inaccessible and hence the parameters guiding these systems are either partially or completely unknown. Such errors might prove to be fatal due to the sensitive dependence of chaotic systems on initial conditions(Khan, 2021). Efficient controller designs should take this vital aspect into consideration. The controller should be robust enough to adjust itself accordingly in case of errors in measurement of system parameters. This aspect of controller design countered in this project by using estimators that approximate the system parameters effectively(LOZANO, 2014).
The synchronizing systems considered in this paper possess the same structural form but their parameters are fully unknown. The result of globally stable synchronization has been established with much weaker restrictions on the derivatives of the Lyapunov function in contrast to many traditional papers in the literature. All the results obtained in this paper are global in nature.
1.3 Statement of Problem
This research intends to achieve the synchronization and anti-synchronization between Lu and Lorenz non-linear chaotic system despite the imposition of various unknown parameters using adaptive control method.
1.4 Aims and Objectives
The aim of the research is to achieve synchronization and anti-synchronization between non-identical Lu and Lorenz chaotic systems using adaptive control method. The objectives of this research can be summarized as follows:
i. To investigate the synchronization of two non-identical chaotic systems, using adaptive control method in the presence of unknown system parameters.
ii. To investigate the anti-synchronization of two non-identical chaotic systems, using adaptive control method in the presence of unknown system parameters.
The rest of the research is organized as follows: In Chapter two, we give a literature review on synchronization, anti-synchronization and controls of chaos. Chapter three deals with the models and some useful tools for the analysis of chaotic system, while Chapter four contains the control formulation and numerical results. Chapter five deals with result and conclusion.
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