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GENERALIZED SYNCHRONIZATION OF MEMRISTOR BASED CHAOTIC OSCILLATOR VIA OPEN PLUS CLOSED LOOP
The memristor is also known as the fourth fundamental passive circuit element. When current flows in one direction through the device, the electric resistance increase and when current flows in the opposite direction the resistance decrease. When the current is stopped, the component retains the last resistance it had, and when the flow of charge starts again, the resistance of the circuit will be what it was when it was last active. It behaves as a non-linear resistor with memory.
Recently memristor have generated wide applications. In this research work, give an introduction to memristor and propose a mathematical model that is solved via open plus closed loop, the result of the model is simulated using C++ software and Gnuplot. The research work concludes with the uses of memristors.
TABLE OF CONTENTS
Front Page i
Table of Contents vii
CHAPTER ONE: INTRODUCTION
1.1 Deterministic Chaos 1
1.2 Differential Equations 3
1.3 Computer Program 4
1.4 Memristor 4
1.5 Objective of the Research 6
1.6 Research Motivation 6
1.7 Research Justification 6
1.8 Design and Implementation 7
1.9 Definition of Chaos 8
1.10 Lyapunov Exponent 8
1.11 Phase Space 9
1.12 Poincare Section 9
2.1 Literature Review 10
2.2 The historical concept of Chaos Synchronization 10
2.3 Research background on Chaos control and
2.4 Description of open plus closed loops controller
for general synchronization 16
2.5 Description of Memristor 17
2.6 Characteristic of a Memristor 20
3.1 Mathematical Model 26
3.2 OPCL method of synchronization 26
4.1 Numerical Result 32
4.2 Applications 32
4.3 Unidirectional coupled order 38
4.4 Bidirectional coupled order 44
5.1 Conclusion and Discussion 56
1.1 Deterministic Chaos
With the appearance of differential equations, the three laws of motion and universal gravitation discovered by Newton in the mid-1600s, dynamics has the become the most active research branch in Physics and Mathematics. The basic problem of dynamics is to predict the future state of a system given its initial state.
The system under consideration may be physical, chemical, or biochemical. Regardless of the context, many systems are modeled mathematically as differential equations with time as a continuous variable, or as difference equations where time takes on discrete integer values. Systems described by deterministic evolution equations are called deterministic dynamical systems. A basic problem in astronomy, the three body gravitational system, in 1887 challenged the understanding of scientists when they could not demonstrate the stability or any orbit of the solar system. A two hundred page paper written by the mathematician H. Poincare showed that the problem is “impossible” to solve because it may happen that small differences in the initial conditions produce very great ones in the final state. Prediction (of the future states) becomes impossible. The phenomenon Poincare discovered was an initial anticipation of modern deterministic chaos. However, this discovery wasn’t widely appreciated by the scientific society at the time because the mathematical works were difficult to read, and the theories weren’t explicit and general enough to convince scientists about the universality of chaos. This deterministic aperiodic behaviour therefore remained in the background as a curiosity of dynamical systems for the next 70 years, until high speed computer were invented in the 1950s and gave scientist like the metrologist E.N. Lorenz opportunities to work with differential equations in a way that was never before possible . While working on modeling the weather system, Lorenz discovered that a set of three first order, coupled and nonlinear differential equations could display solutions in which trajectories could be strongly divergent if the simulation is started from slightly different initial conditions.
This property is illustrated in the solution of the equations which never settle down to an equilibrium or periodic state, instead the solution continues to oscillate in an aperiodic fashion. Lorenz’s work provided the strong foundation for chaos theory in the 1970s when the speed of computer improved and refined experimental techniques were developed. With discovery after discovery, it has become clear that chaos is ubiquitous in nature and could appear in most branches of science. Besides the known example of the solar and weather systems, chaos could be seen in turbulent fluids and in population dynamics in biology, among others.
1.2 Differential Equations
The wide range of possible applications of chaos raised the interest in generating strong and well controlled chaotic dynamics. When one seeks ways to create chaotic behaviors, a natural question to ask is:
Where is chaos coming from? And what are the requirements for a dynamical system to exhibit such behaviour?
Analyzing the Lorenz equations, one can see that the deterministic chaotic behaviour is neither due to external sources of noise (there are none in the equations) nor to an infinite number of degree of freedom (there only three degree of freedom in the equations). The equations are purely classical. The erratic behaviour exhibited by the Lorenz system is instead due to properties often seen in nonlinear systems. These include exponentially fast separating initial close trajectories in a bounded region of phase space. However, nonlinearity is a necessary but not sufficient condition for the generation of chaotic motion since linear differential equations can be solved by Fourier transformation and they do not lead to chaos. For a dynamical system governed by a set of N first order autonomous, coupled, nonlinear, ordinary differential equation, it is known that N must be equal to or greater than three (3) for chaos to be possible.
1.3 Computer Program
Computers are idea tools for exploring nonlinear systems and demonstrating the intricate and often expected features of chaotic dynamics. The computer serves as a fast and efficient tool for generating numerical solutions of the equations of motion of chaotic systems and allows us to explore their behaviour and investigate details that are difficult or impossible to analyse by pure analytical method. The basic features of dynamics can only be imaged by a graphical plot of the iterated points in phase space. Therefore the use of a computer is unavoidable in studying or teaching chaotic dynamics. All programs used in this project were written with C++ under windows.
In 1971, Leon Chua postulated from symmetry argument that a fourth passive element which links the fundamental quantities of charge and magnetic flux must exit. This passive circuit element is named memristor, and behaves as a resistor with a memory effect that is functionally dependent upon charge.
L.Gmez-Guzman et al (2008) the HP lab showed the basic characteristics of the memristor in a nanoscale device. These two terminal physical models have been investigated in many applications, for example this new circuit element can be useful for low power computation and storage for information or data without the need of power. In addition, memristors can be used to implement programmable analog circuits.
Some of the research works on memristor are:
· Experimental implementation of the Memristor reported by Petrov et al. (1990);
· Adaptive synchronization of memristors-based Lorenz circuit reported by Wang et al. (1992);
· Fuzzy modeling and synchronization of different memistor-based chaotic circuits reported by Abed et al.(1992) and
· Modeling and Fuzzy synchronization of memristor-based Lorenz circuits with memristor-based Chua’s circuits reported by Kocarere et al. (1994).
We have discovered from part research works that only few papers are reported on memristor systems despite its rich complex dynamics and its importance in various fields of study like electronics, Physics, engineering, etc. This research is aimed at generalized and synchronization of memristor using the open plus closed loop method.
1.5 Objective of the Research
(i) To control chaotic dynamics of memristor in a Lorenz circuit system through an open plus closed loop design (bidirectional & unidirectional).
(ii) To synchronize chaotic dynamic in identical Lorenz oscillator systems through the open plus closed loop design (bidirectional & unidirectional coupling method).
1.6 Research Motivation
The motivations for this present research are as follows:
(i) The abundant and complex dynamical system of memristor based Lorenz and Chua’s systems and it’s application in various fields of study.
(ii) memristor have been widely used for modeling the behaviour of many engineering systems.
(iii) The increasing demand for memristors.
(iv) Open plus closed loop method has not been used to synchronize memristor-based Lorenz and Chua’s systems
1.7 Research Justification
From literature review most of the works reported on chaotic oscillator, especially those with identical and non-identical Lorenz and Chua’s systems on memristor using open plus closed loop are on the analysis of dynamics.
This is one of our motivation for this research since the the control and synchronization of Lorenz and Chua’s system using the open plus closed loop method with application to memristor has not been reported in the literature to the best of our knowledge.
The use of chaos to study memristor offers several advantages over conventional methods. For one, chaotic signals are much easier and faster to generate using simple circuits. Also, the non-periodic and bifurcation behaviour of the chaotic signal cannot easily be interpreted and predicted, thus an increase in the system security is obtained. In addition, large number of chaotic signals can be generated, which is useful in a multi user environment.
1.8 Design and Implementation
We have designed the controllers for both unidirectional and bidirectional synchronization via open plus closed loop techniques. The unidirectional method is to control chaos in slave while the bidirectional method is to control chaos in both drive and response. Having designed the controller for both the control and synchronization via open closed loop techniques we then solved the resulting equations using the 4th-order Runge-Kutta algorithm implemented through the C++ program. Results obtained from unidirectional and bidirectional were applied to the memristor. The effectiveness of the proposed systems was validated through numerical simulation.
1.9 Definition of Chaos
The following definition for chaos comes from S.H Strogatz (1994).
Chaos is aperiodic long-term behaviour in a deterministic system that exhibit sensitive dependence on initial conditions
Aperiodic long term behaviour means that the trajectories do not settle down to a periodic or quasi periodic motion.
A deterministic system is one whose time evolution of the system is uniquely determined by a set of initial conditions and not dependent on a random of noisy parameter.
Sensitive dependence on initial conation means that already slight differences in the initial conditions cause a strong divergence of the trajectories.
1.10 Lyapunov Exponent
LYAPUNOV EXPONENT of a trajectory gives the exponential rate of divergence of nearby trajectories. In this sense it provides a measure of how chaotic a system is, i.e., it can be used to diagnose chaos. For a chaotic motion, two trajectories with slightly different initial conditions tend to diverge exponentially.
1.11 Phase Space
The phase space of a dynamical system is a mathematical space with orthogonal co-ordinate directions representing each of the two variables needed to specify the instantaneous state of the system Phase space has two properties:
i) Non-crossing property: in phase space, two trajectories corresponding to similar energies may pass very close to each other, but the orbits will not cross each other. This is because past and future states of deterministic mechanical system are uniquely prescribed by the system state at a given time.
ii) Area preservation property: volume preservation means that all the points found in a given volume at one time move in such a way that at a later time the volume occupied by these points remain the same.
1.12 Poincare Section
Complex chaotic phase diagrams are simplified by the Poincare surface in which the phase portrait is sectioned or cut at some appropriate periodic rate. In this way, a three dimensional system can be reduced to a two dimensional system.