Synchronization of chaoticsystems is a process where two (or many) chaoticsystems eventually progress identically for different initial conditions in all future states. This means that the dynamical state of one of the system is completely dictated by the dynamical state of the other system . Chaos Synchronization between twochaoticsystems is one of the most primary procedures in complex systems’ control and has wide potential applications in different fields [2-6]. After a pioneering work on chaos synchronization , synchronization of chaotic dynamical systems has received a great interest among researchers in nonlinear sciences for more than two decades . Until now, diverse techniques have been proposed and applied successfully to synchronize two identical (or nearly identical) as well as nonidentical chaoticsystems [8-13]. Notable among those, the Nonlinearcontrolalgorithm [7, 9] is one of the effectual techniques for synchronizing twochaoticsystems . Nonlinearcontrol techniques take the advantage of the given nonlinear system dynamics to produce high- performance designs. No Lyapunov exponents or gain matrix
Control of complex irregular dynamics has evolved as one of the central issues in applied nonlinear science during the last decade. Nowadays the notion of chaos control involving stabilization of unstable periodic or stationary states in nonlinear dynamic systems has been extended to a much wider class of problems. Since the discovery of chaos synchronization introduced in (Carroll, 1990), there have been tremendous interests in studying the synchronization of chaoticsystems. Recently, much research on the fuzzy model- based designs to stability and synchronization for chaoticsystems have been carried out based on Takagi–Sugeno (T–S) fuzzy models (Park et al., 2002; Lian et al., 2001). In (Yan- Wu Wang et al., 2003), a fuzzy model-based designs for Chen’s chaotic stability and synchronization have been proposed. Based on the fuzzy hyper chaotic models, simpler fuzzy controllers have been designed for synchronizing hyper chaoticsystems in (Hongbin Zhang, 2005). In this work, utilizing BELBIC model introduced in (Ali Reza Mehrabian et al., 2006; Saeed Jafarzadeh et al., 2008), we will design an intelligent controller for synchronization of two new 3D
Fractional calculus has a large history in mathematics over 300-years-old. However, its applications to engineering are a recent focus of interest [1-7]. Many systems are known to display fractional order dynamics, such as viscoelastic systems , dielectric polarization , and electrode-electrolyte polarization . Also, secure communications and information encryption based on fractional-order systems are a field of research [11- 14]. There are three approaches to solve fractional-order chaoticsystems: frequency-domain method , Adomian decomposition method (ADM) , and Adams-Bashforth-Moulton algorithm [17, 18]. However, Tavazoei et al.  reported that the frequency-domain method is not always reliable in detecting chaos behavior in nonlinearsystems . In this paper, discretization is performed by Adams-Bashforth-Moulton decomposition method.
Chaotic system is a nonlinear deterministic system with complex and unpredictable behavior. Chaotic behavior as is known to all, is a prevalent phenomenon which can be appeared in nonlinearsystems. It has been also seen in a variety of real system in laboratory such as electrical circuits, chemical reactions and fluid dynamics and so forth . Based on chaos theory, the prominent features of chaoticsystems are that the highly sensitivity to initial conditions. Chaos synchronization is a phenomenon that may happen when two, or more, dissipative chaoticsystems are coupled. Moreover, Synchronizationcontrol is one of the important research area in chaos theory and it is simply means that things occur at the same time. The main problem related to the synchronization
Abstract—In this paper, anti-phase synchronization of two non-identical autonomous chaoticsystems is presented. These twosystems are neither diffeomorphic nor topologically equivalent, but possessed chaotic properties that ease synchronization and antisynchronization based on different control strategies. The Yu-Wang possessed a cross- product quadratic term and a nonlinear hyperbolic term in its algebraic structure while the Burke-Shaw has twononlinear terms which adds complexity to the system's dynamic evolutions. Nonlinearactive controllers were designed to regulate the two exponentially divergent chaotic trajectories of the coupled system to achieve anti-phase synchronization in finite times, while Lyapunov stability theory was employed to test for local and global convergence of the error dynamics to the origin. The results of the various numerical simulations via MATLAB software demonstrates the effectiveness of the coupling scheme and the applicability of the antisynchronized signals in modelling and design of electrical and communications systems that are critical to secure communication and electrical power outage minimization.
On another research frontier, adaptive control method  is an eﬀective way to es- timate the unknown parameters due to its advantages on witnessed rapid and impres- sive developments leading to global stability and tracking results for nonlinearsystems. It has been successfully applied to synchronize chaoticsystems with unknown parame- ters, and many important results have been presented. For example, Park studied adap- tive synchronization of a uniﬁed chaoticsystems with an unknown parameter [, ]. Zhang et al. proposed the adaptive controllers and adaptive laws to synchronize two dif- ferent chaoticsystems with unknown parameters . In , the adaptive complete syn- chronization between chaoticsystems with fully uncertain parameters were realized. Li et al. gave a deeply research on adaptive impulsive synchronization for fractional-order chaoticsystems with unknown parameters . In , adaptive synchronization of two diﬀerent chaoticsystems was addressed by considering the time varying unknown param- eters. Adaptive added-order and reduced-order anti-synchronization of chaoticsystems were investigated in [, ], respectively. He et al. made a thorough inquiry about syn- chronization of hyperchaotic systems with multiple unknown parameters . Zhao et al. presented a discussion of chaos synchronization between the coupled systems on net- work with unknown parameters based on adaptive control method . Liu developed adaptive anti-synchronization of chaotic complex nonlinearsystems with unknown pa- rameters . Wu and Yang achieved the adaptive synchronization of coupled nonidenti- cal chaoticsystems with complex variables and stochastic perturbations . However, all of these works only deal with the synchronization problems between twochaoticsystems with unknown parameters. Up to now, no related results have been established for the synchronization of multiple chaoticsystems with unknown parameters, which is another motivation of this paper.
Synchronization between twochaoticsystems occurs when the trajectory of one of the system asymptotically follows the trajectory of another system due to coupling or due to forcing. In this research paper, the synchronization problem between two identical Li and identical Lorenz systems and nonidentical Li and Lorenz ChaoticSystems have been addressed. In this study, the synchronization is performed through a nonlinear controller based on Lyapuonov Stability Theory to stabilize the error dynamics. It has been shown that the proposed strategies have excellent transient performances using less control effort with fast transient speed and has shown analytically as well as graphically that synchronization is asymptotically globally stable. Numerical simulations are carried out to verify and support the analytical results of the proposed methodology by using mathematica 9.
In this work, a novelnonlinear modeling technique in NANC is proposed to overcome the drawbacks of NLFXLMS algorithm. Sequentially, the proposed model is used to develop a controller algorithmbased on Tangential hyperbolic function (THF). The work is restricted to single input, single output (SISO) ANC system. The feedforward strategy is used to control the noise at the observer . All the transfer function and filters are assumed to be linear except the loudspeaker which is represented by a memory less saturation nonlinearity. The work involves designing and simulating the proposed modeling technique. At the control stage, an alternative THF-NLFXLMS algorithm is proposed and compared with NLFXLMS and FXLMS when ANC system deals with loudspeaker nonlinearity. Figure1.1, illustrates the research scope which is covered in this argumentation.
Recently, other techniques, such as DNA encoding, have been employed to encipher images. In this approach, the pixels of plain image are substituted for DNA sequences. Jain and Rajpal  have proposed an image encryption algorithmbased on DNA encoding, in which the corresponding DNA sequence to each pixel is determined according to its binary presentation. In , Liu and Wang have merged DNA encoding and Logistic map. In this work, each pixel of the plain image is substituted for another pixel, where the coordinates of the latter is determined using the generated numbers by the chaotic map. Next, the pixels are substituted for the equivalent DNA sequences. The proposed algorithms in [15-16] have also combined DNA encoding and chaotic map. The main disadvantage of DNA encoding is its high time complexity. Moreover, in some DNA-based algorithms the data is enlarged. Therefore, this approach is not appropriate to encrypt images.
Chaoticsystems have been invoked as details for, or as casual appreciably to clarification of, real-global behaviors. Several examples are epileptic seizure, heart traumatic inflammation, neural technique, chemical reactions, weather, business manipulate processes. Apart from irregular overall performance of real-world systems, chaos is as well invoked to make clear form such as the real trajectories exhibited in a specified state space or the sojourn times of trajectories in exacting areas of state space [3, 4]. The nature of scientific details in the literature on chaos is carefully under-discussed to put it gently. In 1963 Lorenz proposed a mathematical model of chaos . Rikitake also introduced different kinds of chaoticsystems[3, 6, 7], R¨ossler, Shimizu-Morioka , Chua , Rucklidge , Sprott , Chen . So,in literature many chaotic and hyper-chaoticsystems have appeared. Lately, there has been great involvement in chaotic studies on chaoticsystems and their applications in secure communications, data encryption, etc.
has been studied and to control these chaoticsystems the backstepping control schemes have been employed. In recent year this method has importance in hydraulic servo system , backstepping Decision and fractional derivative equation . Among other applications recursive procedure has importance in to demonstrate to control a third-order phase-locked loops.With the design of the controller the backstepping method is effective to choice of lyapunov exponent. Through the transmission of the signal the trajectory of slave system approaches asymptotically to the trajectory of master system which is input system so that the error dynamics converges to zero. When several single oscillators are coupled together then a complicated system is obtained. For the study of these types of oscillators complex variables are used which are more convenient. Based on Lyapunov function for determination of the controllers the backsteeping technique is used and also for synchronize two identical chaotic system. In this paper, between twochaoticsystems for achieving the global synchronization we design backstepping control method. This presentation is divided in sections: In Section II, formulation of the problem is introduced.in III. Design for chaos synchronization and methodology is presented. SectionIV, deals with numerical simulation results. Section V, presented finally the simulation results.
By constructing two scaling matrices, i.e., a function matrix (t) and a constant matrix W which is not equal to the identity matrix, a kind of W – (t) synchronization between fractional-order and integer-order chaotic (hyper-chaotic) systems with diﬀerent dimensions is investigated in this paper. Based on the fractional-order Lyapunov direct method, a controller is designed to drive the synchronization error convergence to zero asymptotically. Finally, four numerical examples are presented to illustrate the eﬀectiveness of the proposed method.
Although there have been a lot of literature works studying the synchronization problem of chaoticsystems, we ﬁnd the following deﬁciencies: () Most of them require the nonlin- ear parts in the chaoticsystems to satisfy the Lipschitz condition. () Each paper usually only could solve a single synchronization problem, which leads to the lack of a uniﬁed method to solve all the synchronization problems for the same master-slave model. Moti- vated by these factors, we aim in this paper to ﬁnd a kind of eﬀective method to deal with all the synchronization problems for a general master-salve chaotic system. The contribu- tions of this paper are as follows: (i) The lag projective synchronization which can include synchronization, projective synchronization, anti-synchronization and lag synchroniza- tion at the same time is investigated. (ii) The considered master-slave system model is diﬀerent from the systems in the literature, in which the nonlinearities only need to sat- isfy a bounded condition. Moreover, the state equations of the master system and the slave system are non-identical. (iii) The presented results are very concise and it is easy to adjust the synchronization rate by the control gains.
Actually, this open problem, although significant for complete chaos synchronization, is very difficult and cannot admit the optimization solution . For example, in , rigorous criteria are presented to guarantee isochronal synchronization motion, but the criteria are so complicated that specific numerical calculation is nec- essary for particular examples in practice.
In this paper, we use adaptive control method to derive new results for the global chaos synchronization of identical uncertain Li systems (2009), identical uncertain T systems (2008) and non-identical Li and uncertain T systems. In adaptive synchronization of identical chaoticsystems, the parameters of the master and slave systems are unknown and we devise feedback control laws using estimates of the system parameters. In adaptive synchronization of non-identical chaoticsystems, the parameters of the master system are known, but the parameters of the slave systems are unknown and we devise feedback control laws using the estimates of the parameters of the slave system. Our adaptive synchronization results derived in this paper for uncertain Li and T systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the adaptive control method is very effective and convenient to synchronize identical and non-identical Li and T chaoticsystems. Numerical simulations are given to demonstrate the effectiveness of the proposed adaptive synchronization schemes for the global chaos synchronization of the uncertain chaoticsystems addressed in this paper.
(4.1) where X(t), Y(t), a, r are complex functions and parameters, Z(t) is a real function, σ , b are real parameters and the asterisk (*) denote complex conjugate. Thus, in real quantities (3.17) is a five- dimensional dynamical system. It occurs in the theory of nonlinear baroclinic instability in the atmosphere, a problem of great meteorological significance, and also in the laser physics (Rauh et al., 1996).
In 1963, Lorenz  found the first classical chaotic attractor in a three-dimensional autonomous system derived form a simplified model of earth atmospheric convection system. As the first chaotic model, the Lor- enz system has become a paradigm of chaos research. Mathematicians, physicists and engineers from various fields have thoroughly studied the essence of chaos, characteristics of chaoticsystems, bifurcations, routes to chaos, and many other related topics. There are also some chaoticsystems of great significance that are clo- sely related to the Lorenz system but not topologically equivalent to it, such as the Rössler system , the Chen system  and the Lü system . Recently, Liu et al.  proposed a system of three-dimensional autonomous differential equations with only two quad- ratic terms, which is described as follows:
Recently, it has been shown that discontinuous transitions can take place in networks of periodic oscillators , called explosive synchronization (ES). As one kind of abrupt dynamical transitions in nonlinearly coupled systems, it has attracted widespread attention from the systems science community -. Traditionally, the master stability function is used to study the continuous change of systems and to focus on the synchronizability of networks, rather than synchronization processes. However, explosive synchronization is commonly observed in heterogeneous networks. Gómez-Gardeñes et al .  proposed that ES could occur in the networked Kuramoto oscillators and the following two conditions are satisfied: 1) a scale-free network structure and 2) the existence of a positive correlation between the natural frequency of an oscillator and its de- gree. There are many systems in the world that are not Kuramoto systems, but chaoticsystems. A large system is said to undergo a phase transition when one or more of its properties change abruptly after a slight change in a controlling variable. If the transitions are discontinuous or abrupt, they are called a first-order. Conversely, when the transitions are continuous or smooth, they are second-order. Generally, there are two main factors to influence phase transi- tions of complex networks as following: 1) the topological structure of the net- work and 2) the dynamics of the system. More recently, there are many studies of ES based on Kuramoto and little attention is paid to other chaotic dynamics in   . Zhao  studied explosive synchronization of complex net- works with different chaotic oscillators and indicated that explosive synchroni- zation only takes place in the coupled Lorenz systems. However, Zhao only considers the process from incoherence to synchrony, ignoring the process from synchrony to incoherence. Generally speaking, explosive synchronization can be said to happen in complex networks when the following conditions are satisfied: 1) the emergence of the first-order transition and 2) the hysteresis curve appears in the process from synchrony to incoherence.
Some comments regarding the above assumptions are in order. Concerning Assump- tion 1, the Lipschitz constant k is often required to be known for the control design purpose. In fact, it is often difficult to obtain a precise value of k in some practical sys- tems, hence the Lipschitz constant is often selected to be larger, which will induce the control gain to be higher, and the obtained results would be conservative. Assumption 2 is reasonable for the boundness of the chaotic attractor in state space and the interaction of all trajectories inside the attractor. Note also that the constant φ m is only introduced to
combination of current and past error states, which is a modiﬁcation of the normal impulsive one. Some global exponential stability criteria are derived for the error system by utilizing the stability analysis of impulsive diﬀerential equations and diﬀerential inequalities and, moreover, the exponential convergence rate can be speciﬁed. An illustrative example is given to show the eﬀectiveness of the modiﬁed impulsive control scheme.