Chaos synchronization of the master-slave generalized Lorenz systems via linear state error feedback control

This paper provides a unified method for analyzing chaos synchronization of the generalized Lorenz systems. The considered synchronization scheme consists of identical master and slave generalized Lorenz systems coupled by linear state error variables. A sufficient synchronization criterion for a general linear state error feedback controller is rigorously proven by means of linearization and Lyapunov's direct methods. When a simple linear controller is used in the scheme, some easily implemented algebraic synchronization conditions are derived based on the upper and lower bounds of the master chaotic system. These criteria are further optimized to improve their sharpness. The optimized criteria are then applied to four typical generalized Lorenz systems, i.e. the classical Lorenz system, the Chen system, the Lv system and a unified chaotic system, obtaining precise corresponding synchronization conditions. The advantages of the new criteria are revealed by analytically and numerically comparing their sharpness with that of the known criteria existing in the literature.Comment: 61 pages, 15 figures, 1 tabl

Pulsive feedback control for stabilizing unstable periodic orbits in a nonlinear oscillator with a non-symmetric potential

We examine a strange chaotic attractor and its unstable periodic orbits in case of one degree of freedom nonlinear oscillator with non symmetric potential. We propose an efficient method of chaos control stabilizing these orbits by a pulsive feedback technique. Discrete set of pulses enable us to transfer the system from one periodic state to another.Comment: 11 pages, 4 figure

Delay time modulation induced oscillating synchronization and intermittent anticipatory/lag and complete synchronizations in time-delay nonlinear dynamical systems

Existence of a new type of oscillating synchronization that oscillates between three different types of synchronizations (anticipatory, complete and lag synchronizations) is identified in unidirectionally coupled nonlinear time-delay systems having two different time-delays, that is feedback delay with a periodic delay time modulation and a constant coupling delay. Intermittent anticipatory, intermittent lag and complete synchronizations are shown to exist in the same system with identical delay time modulations in both the delays. The transition from anticipatory to complete synchronization and from complete to lag synchronization as a function of coupling delay with suitable stability condition is discussed. The intermittent anticipatory and lag synchronizations are characterized by the minimum of similarity functions and the intermittent behavior is characterized by a universal asymptotic $-{3/2}$ power law distribution. It is also shown that the delay time carved out of the trajectories of the time-delay system with periodic delay time modulation cannot be estimated using conventional methods, thereby reducing the possibility of decoding the message by phase space reconstruction.Comment: accepted for publication in CHAOS, revised in response to referees comment

Synchronization of chaotic dynamical systems: a brief review

There are several reasons for the approach to chaos synchronization. This phenomenon is immediately interesting because of its high potential for applications. But, first of all, it is particularly interesting the study of a phenomenon that requires the adjustment of dynamic behaviors in order to obtain a coincident chaotic motion, being this possible even in chaotic dynamical systems in which sensitive dependence on initial conditions is one of the features. The possibility of applying techniques of chaos control in order to optimize the results of synchronization is alsoa motivating factor for the study of this phenomenon. It is presented a brief review of preliminary notions on nonlinear dynamics and then is considered in detail the synchronization of chaotic dynamical systems, both in continuous and discrete time

Complex Projective Synchronization in Drive-Response Stochastic Complex Networks by Impulsive Pinning Control

The complex projective synchronization in drive-response stochastic coupled networks with complex-variable systems is considered. The impulsive pinning control scheme is adopted to achieve complex projective synchronization and several simple and practical sufficient conditions are obtained in a general drive-response network. In addition, the adaptive feedback algorithms are proposed to adjust the control strength. Several numerical simulations are provided to show the effectiveness and feasibility of the proposed methods

Huygens' synchronization of dynamical systems : beyond pendulum clocks

Synchronization is one of the most deeply rooted and pervasive behaviours in nature. It extends from human beings to unconscious entities. Some familiar examples include the fascinating motion of schools of fish, the simultaneous flashing of fireflies, a couple dancing in synchrony with the rhythm of the music, the synchronous firing of neurons and pacemaker cells, and the synchronized motion of pendulum clocks. In a first glimpse to these examples, the existence of selfsynchronization in nature may seem almost miraculous. However, the main "secret" behind this phenomenon is that there exists a communication channel, called coupling, such that the entities/systems can influence each other. This coupling can be, for instance, in the form of a physical interconnection or a certain chemical process. Although synchronization is a ubiquitous phenomenon among coupled oscillatory systems, its onset is not always obvious. Consequently, the following questions arise: How exactly do coupled oscillators synchronize themselves, and under what conditions? In some cases, obtaining answers for these questions is extremely challenging. Consider for instance, the famous example of Christiaan Huygens of two pendulum clocks exhibiting anti-phase or in-phase synchronized motion. Huygens did observe that there is a "medium" responsible for the synchronized motion, namely the bar to which the pendula are attached. However, despite the remarkably correct observation of Huygens, even today a complete rigorous mathematical explanation of this phenomenon, using proper models for pendula and flexible coupling bar, is still missing. The purpose of this thesis is to further pursue the nature of the synchronized motion occurring in coupled oscillators. The first part, addresses the problem of natural synchronization of arbitrary self-sustained oscillators with Huygens coupling. This means that in the analysis, the original setup of Huygens’ clocks is slightly modified in the sense that each pendulum clock is replaced by an arbitrary second order nonlinear oscillator and instead of the flexible wooden bar (called here Huygens’ coupling), a rigid bar of one degree of freedom is considered. Each oscillator is provided with a control input in order to guarantee steady-state oscillations. This requirement of having a control input to sustain the oscillations can be linked to Huygens’ pendulum clocks, where each pendulum is equipped with an escapement mechanism, which provides an impulsive force to the pendulum in order to keep the clocks running. Then, it is shown that the synchronized motion in the oscillators is independent of the kind of controller used to maintain the oscillations. Rather, the coupling bar, i.e. Huygens’ coupling is considered as the key element in the occurrence of synchronization. In particular, it is shown that the mass of the coupling bar determines the eventual synchronized behaviour in the oscillators, namely in-phase and anti-phase synchronization. The Poincaré method is used in order to determine the existence and stability of these synchronous motions. This is feasible since in the system there exists a natural small parameter, namely the coupling strength, which value is determined by the mass of the coupling bar. Next, the synchronization problem is addressed from a control point of view. First, the synchronization problem of two chaotic oscillators with Huygens’ coupling is discussed. It is shown that by driving the coupling bar with an external periodic excitation, it is possible to trigger the onset of chaos in the oscillators. The mass of the coupling bar is considered as the bifurcation parameter. When the oscillators are in a chaotic state, the synchronization phenomenon will not occur naturally. Consequently, it is demonstrated that by using a master-slave configuration or a mutual synchronization scheme, it is possible to achieve (controlled) synchronization. Secondly, the effect of time delay in the synchronized motion of oscillators with Huygens’ coupling is investigated. In this case, the wooden bar, is replaced by a representative dynamical system. This dynamical system generates a suitable control input for the oscillators such that in closed loop the system resembles a pair of oscillators with Huygens’ coupling. Under this approach, the oscillators do not need to be at the same location and moreover, the dynamical system generating the control input should be implemented separately, using for instance a computer. Consequently, the possibility of having communication time-delays (either in the oscillators or in the applied control input) comes into play. Then, the onset of in-phase and anti-phase synchronization in the coupled/controlled oscillators is studied as a function of the coupling strength and the time delay. In addition to computer simulations, the (natural and controlled) synchronized motion of the oscillators is validated by means of experiments. These experiments are performed in an experimental platform consisting of an elastically supported (controllable) rigid bar (in Huygens’ example the wooden bar) and two (controllable) mass-spring-damper oscillators (the pendulum clocks in Huygens’ case). A key feature of this platform is that its dynamical behaviour can be adjusted. This is possible due to the fact that the oscillators and the coupling bar can be actuated independently, then by using feedback, specific desirable oscillators’ dynamics are enforced and likewise the behaviour of the coupling bar is modified. This feature is exploited in order to experimentally study synchronous behaviour in a wide variety of dynamical systems. Another question considered in this thesis is related to the modeling of the real Huygens experiment. The models used in the first part of this thesis and the ones reported in the literature are simplifications of the real model: the coupling bar has been considered as a rigid body of one degree of freedom. However, in the real Huygens experiment, the bar to which the clocks are attached is indeed an infinite dimensional system for which a rigorous study of the in-phase and antiphase synchronized motion of the two pendula is, as far as is known, still never addressed in the literature. The second part of the thesis focuses on this. A Finite Element Modelling technique is used in order to derive a model consisting of a (finite) set of ordinary differential equations. Numerical results illustrating all the possible stationary solutions of the "true" infinite dimensional Huygens problem are included. In summary, the results contained in the thesis in fact reveal that the synchronized motion observed by Huygens extends beyond pendulum clocks

Control of chaos in nonlinear circuits and systems

Nonlinear circuits and systems, such as electronic circuits (Chapter 5), power converters (Chapter 6), human brains (Chapter 7), phase lock loops (Chapter 8), sigma delta modulators (Chapter 9), etc, are found almost everywhere. Understanding nonlinear behaviours as well as control of these circuits and systems are important for real practical engineering applications. Control theories for linear circuits and systems are well developed and almost complete. However, different nonlinear circuits and systems could exhibit very different behaviours. Hence, it is difficult to unify a general control theory for general nonlinear circuits and systems. Up to now, control theories for nonlinear circuits and systems are still very limited. The objective of this book is to review the state of the art chaos control methods for some common nonlinear circuits and systems, such as those listed in the above, and stimulate further research and development in chaos control for nonlinear circuits and systems. This book consists of three parts. The first part of the book consists of reviews on general chaos control methods. In particular, a time-delayed approach written by H. Huang and G. Feng is reviewed in Chapter 1. A master slave synchronization problem for chaotic Lur’e systems is considered. A delay independent and delay dependent synchronization criteria are derived based on the H performance. The design of the time delayed feedback controller can be accomplished by means of the feasibility of linear matrix inequalities. In Chapter 2, a fuzzy model based approach written by H.K. Lam and F.H.F. Leung is reviewed. The synchronization of chaotic systems subject to parameter uncertainties is considered. A chaotic system is first represented by the fuzzy model. A switching controller is then employed to synchronize the systems. The stability conditions in terms of linear matrix inequalities are derived based on the Lyapunov stability theory. The tracking performance and parameter design of the controller are formulated as a generalized eigenvalue minimization problem which is solved numerically via some convex programming techniques. In Chapter 3, a sliding mode control approach written by Y. Feng and X. Yu is reviewed. Three kinds of sliding mode control methods, traditional sliding mode control, terminal sliding mode control and non-singular terminal sliding mode control, are employed for the control of a chaotic system to realize two different control objectives, namely to force the system states to converge to zero or to track desired trajectories. Observer based chaos synchronizations for chaotic systems with single nonlinearity and multi-nonlinearities are also presented. In Chapter 4, an optimal control approach written by C.Z. Wu, C.M. Liu, K.L. Teo and Q.X. Shao is reviewed. Systems with nonparametric regression with jump points are considered. The rough locations of all the possible jump points are identified using existing kernel methods. A smooth spline function is used to approximate each segment of the regression function. A time scaling transformation is derived so as to map the undecided jump points to fixed points. The approximation problem is formulated as an optimization problem and solved via existing optimization tools. The second part of the book consists of reviews on general chaos controls for continuous-time systems. In particular, chaos controls for Chua’s circuits written by L.A.B. Tôrres, L.A. Aguirre, R.M. Palhares and E.M.A.M. Mendes are discussed in Chapter 5. An inductorless Chua’s circuit realization is presented, as well as some practical issues, such as data analysis, mathematical modelling and dynamical characterization, are discussed. The tradeoff among the control objective, the control energy and the model complexity is derived. In Chapter 6, chaos controls for pulse width modulation current mode single phase H-bridge inverters written by B. Robert, M. Feki and H.H.C. Iu are discussed. A time delayed feedback controller is used in conjunction with the proportional controller in its simple form as well as in its extended form to stabilize the desired periodic orbit for larger values of the proportional controller gain. This method is very robust and easy to implement. In Chapter 7, chaos controls for epileptiform bursting in the brain written by M.W. Slutzky, P. Cvitanovic and D.J. Mogul are discussed. Chaos analysis and chaos control algorithms for manipulating the seizure like behaviour in a brain slice model are discussed. The techniques provide a nonlinear control pathway for terminating or potentially preventing epileptic seizures in the whole brain. The third part of the book consists of reviews on general chaos controls for discrete-time systems. In particular, chaos controls for phase lock loops written by A.M. Harb and B.A. Harb are discussed in Chapter 8. A nonlinear controller based on the theory of backstepping is designed so that the phase lock loops will not be out of lock. Also, the phase lock loops will not exhibit Hopf bifurcation and chaotic behaviours. In Chapter 9, chaos controls for sigma delta modulators written by B.W.K. Ling, C.Y.F. Ho and J.D. Reiss are discussed. A fuzzy impulsive control approach is employed for the control of the sigma delta modulators. The local stability criterion and the condition for the occurrence of limit cycle behaviours are derived. Based on the derived conditions, a fuzzy impulsive control law is formulated so that the occurrence of the limit cycle behaviours, the effect of the audio clicks and the distance between the state vectors and an invariant set are minimized supposing that the invariant set is nonempty. The state vectors can be bounded within any arbitrary nonempty region no matter what the input step size, the initial condition and the filter parameters are. The editors are much indebted to the editor of the World Scientific Series on Nonlinear Science, Prof. Leon Chua, and to Senior Editor Miss Lakshmi Narayan for their help and congenial processing of the edition

Phase synchronization of coupled bursting neurons and the generalized Kuramoto model

Bursting neurons fire rapid sequences of action potential spikes followed by a quiescent period. The basic dynamical mechanism of bursting is the slow currents that modulate a fast spiking activity caused by rapid ionic currents. Minimal models of bursting neurons must include both effects. We considered one of these models and its relation with a generalized Kuramoto model, thanks to the definition of a geometrical phase for bursting and a corresponding frequency. We considered neuronal networks with different connection topologies and investigated the transition from a non-synchronized to a partially phase-synchronized state as the coupling strength is varied. The numerically determined critical coupling strength value for this transition to occur is compared with theoretical results valid for the generalized Kuramoto model.Comment: 31 pages, 5 figure

Identical synchronization of a new chaotic system via nonlinear control and linear active control techniques: A comparative analysis

Most of the synchronization techniques belong to the master-slave (drive-response) system configurations in which the two chaotic systems are coupled in such a manner that the performance of the second (slave /response) system is influenced by the first (drive/master) system and the first system is not disturbed by the exertion of the second (slave / response) system.In this research paper, the synchronization problem of two widely used techniques, the Linear Active Control and Nonlinear Control Algorithms have been studied to achieve chaos synchronization of a new chaotic system. In this study, using the Linear Active Control and the Nonlinear Control algorithms and based on the Lyapunov Stability Theory, it has been shown that the two techniques have excellent transient performance and that analytically as well as graphically, the synchronization is asymptotically globally stable.Numerical simulations are furnished to show the efficiency and effectiveness of the two methods