Chaos is an important phenomenon, happens vastly in both natural and man-made systems. Lorenz  faced to the first chaotic attractor in 1963. In continue, a lot of researches were achieved on chaotic systems [2-11]. A new 3D chaoticsystem (T-system) from the Lorenz system was derived in [12,13]. Over the last two decades, chaos control and synchronization have been absorbed increasingly attentions due to their wide applications in many fields [14–25]. Active control , backstepping  and adaptive control  are three different methods for synchronization of Tsystem. Active control  and backstepping  methods are selected when system parameters are known, and adaptive control  method is applied when system parameters are unknown. GBM [26,27], a new method to optimize backstepping method, controls chaos in nonlinear systems better than backstepping design.
In the Nature, and also in our day life, almost everything is connected to everything. In other words, many systems are in some way coupled to others. The free behavior of a system may change drastically if it is coupled to another (or to others) even if it is an identical system. There are different models of couplings that try to reproduce the variety of interferences that two systems may cause to each other.
This paper presents fuzzy model-based designs for synchronization of new chaoticsystem. The T–S fuzzy models for new chaotic systems are exactly derived. Then utilizing an intelligent controller which based on brain emotional learning (BELBIC), this fuzzy chaoticsystem is synchronized. Numerical simulation results are presented to show the effectiveness of the proposed method.
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 chaotic systems. 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.
Different Schemes of using chaotic systems in designing Chaotic ciphers have been developed since the origin of Chaos in Cryptography. Message- embedded is the latest Scheme of third generation also known as generation of chaotic ciphers. Floriane Anstet et. al compared two encryption schemes, the standard Stream cipher and message embedded cryptosystem. The comparison is based on two main aspects. The first aspect deals with the synchronization of the time-varying keys at the transmission and reception side respectively. The second aspect focuses on the cryptanalysis of the encryption algorithms. The cryptanalysis is concerned with the system parameter retrieving . It has been shown that the identifiable parameters may be good candidates to play the role of the static key against brute force attack. A fundamental conclusion is that the usual cryptosystems encountered in the literature involving only polynomial nonlinearities are weak against algebraic attacks.
Abstract — The chaotic systems play a critical role in a wide range of communication systems, form both economical and technical. Recently, a new fractional-order chaoticsystem was proposed and dynamical analysis was investigated. Here, we focus on synchronization of the drive and response systems, which can have significant impact on the economic side of the problem. Furthermore, stability of mentioned system is discussed based on Lyapunov theorem. It is shown that Lyapunov theorem can be extended to the systems which have terms with orders higher than 2. Numerical simulations prove our claims from the objective evaluation.
The problem on chaos synchronization for a class of chaoticsystem is addressed. Based on impulsive control theory and by constructing a novel Lyapunov functional, new impulsive synchronization strategies are presented and possess more practical application value. Finally some typical numerical simulation examples are included to demonstrate the effectiveness of the theoretical results.
receiver. The transmitter is composed of a chaoticsystem, a sampler, an A/D module and a D/A module. The receiver has a chaoticsystem, a sampler, an A/D module, a D/A module and an error correcting module. The system is running as follows: Step 1, Turn on k1, chaotic states of the transmitter are sampled and transmitted to the receiver as synchronization pulses. Turn off k1 after the chaoticsystem in the receiver is synchronized with the chaoticsystem in the transmitter. Step 2, turn on k2, k3 and k4, chaotic states of the transmitter are sampled and then quantized by the A/D module. The error correcting signals which gained from the digitized chaotic states are then sent to the receiver. The D/A module convert the digitized chaotic states to synchronization pulses. In the receiver side, chaotic states are sampled and then quantized by the A/D module. The digitized chaotic states which modified by the error correcting signals are then converted to synchronization pulses by the D/A module.
Generally, one can classify the main problems in chaos control into three cases: sta- bilization , chaotification, and synchronization. The stabilization problem of the unstable periodic solution (orbit) arises in the suppression of noise and vibrations of various constructions, elimination of harmonics in the communication systems, elec- tronic devices, and so on. These problems are distinguished for the fact that the con- trolled plant is strongly oscillatory, that is, the eigenvalues of the matrix of the linearized system are close to the imaginary axis. The harmful vibrations can be either regular (quasiperiodic) or chaotic. The problems of suppressing the chaotic oscillations by reducing them to the regular oscillations or suppressing them completely can be formalized as stabilization techniques. The second class includes the control problems of excitation or generation of chaotic oscillations. These problems are also called the chaotification or anticontrol.
Abstract: This paper is concerned with the adaptive impulsive synchronization for a class of delay fractional-order chaoticsystem. Firstly, according to the impulsive differential equations theory and the adaptive control theory, the adaptive impulsive controller and the parametric update law are designed, respectively. Secondly, by constructing the suitable response system, the original fractional-order error system can be converted into the integral-order one. Finally, based on the Lyapunov stability theory and the generalized Barbalat’s lemma, some new sufficient conditions are derived to guarantee the asymptotic stability of synchronization error system.
scheme for anti-synchronization of two identi- cal hyperchaotic complex T-systems, with three unknown parameters. We use the result of anti-synchronization for secure communications via the chaotic masking method. By apply- ing this process, we can achieve chaos anti- synchronization of master and slave systems, identify the unknown parameters, and mask and unmask the message signals simultaneously. In secure communication, a parameter and a state of master system are used for masking the informa- tion, such that a chaotic signal is added to the in- formation. The anti-synchronized states and es- timation of unknown parameters of the slave sys- tem are used to unmask the information. Based on the Barbalate’s lemma and Hamilton-Jacobi- Bellman(HJB) technique, an optimal adaptive sliding-mod controller with parameters estima- tion rules is designed to anti-synchronize complex chaoticT-systems asymptotically.
Although hybrid synchronization (both anti-synchroni- zation and synchronization co-exist) has an important application in information processing , only a few researchers study about it. G. Li used a single variable to control the hybrid synchronization of coupled Chen sys- tem . Zhang used a linear feedback control method and an adaptive feedback control method to guarantee the hybrid synchronization in general Lorenz system .
The main contributions of this paper are three-fold: () An eﬀective modiﬁed impulsive controller is designed for the global exponential synchronization of coupled chaotic sys- tems. () Due to the additional integral term of the errors corresponding to each impulse, equipped with the deﬁnitions and results, we establish a uniform comparison system for this case and derive a suﬃcient condition in this paper. () Global exponential synchro- nization of the chaotic systems with the proposed impulsive controller can be simultane- ously realized. In other words, by adding the summation term in the error dynamics, one could achieve the same eﬀect by increasing the impulse distance and reducing the control cost.
As time goes on, more and more researchers began to realize the important role of the synchronization time. To attain a high convergence speed, many eﬀective methods have been introduced and ﬁnite-time control is one of them. Finite-time synchroniza- tion means the optimality in convergence time. Much research work has been done on chaos synchronization based on ﬁnite time (see for instance [–] and the references
In this letter we proposed a new synchronization method for fractional-order chaotic sys- tems based on a simple Lyapunov function. We also proposed some suﬃcient conditions of synchronization for the fractional-order chaotic systems. The proposed method is simple, universal and theoretically rigorous. Furthermore, we have implemented and veriﬁed our method for other fractional-order chaotic systems [, –], namely the Lü chaotic sys- tem, the fractional-order Newton-Leipnik system, the Rössler system, ﬁnancial systems, etc. The numerical simulation results also indicate that the proposed controller can eﬀec- tively make the fractional-order chaotic systems synchronized, and the proposed method provides a theoretical basis for the applications of synchronization method in fractional- order dynamic systems. In future work as regards this topic, we will consider whether the proposed synchronization method can be extended to control other complex chaotic sys- tems, such as the networked fractional chaotic systems [, ] and fractional multi-scroll chaotic systems .
In real world, because of switching phenomenon or sudden noise, many real systems have been found to be subject to instantaneous perturbations and abrupt changes at cer- tain instants. That is, these systems cannot be controlled by continuous control and endure continuous disturbance. Therefore, impulsive control, as a typical discontinuous control scheme, has been widely adopted to design proper controllers for achieving synchroniza- tion or stability [–]. Based on the Lyapunov function method, the Razumikhin tech- nique, or the comparison principle, many valuable results have been obtained, and syn- chronization criteria have been derived. For a given neural network, we can estimate the largest impulsive interval from the derived criteria by ﬁxing impulsive gains and calculat- ing some system constants, for example, Lipschitz or Lipschitz-like constants with respect to neuron activation functions, and vice versa. As we know, diﬀerent neural networks usu- ally have totally diﬀerent system parameters and activation functions, which means that the impulsive controllers with ﬁxed impulsive gains and intervals are not uniﬁed. In other words, the system parameters have more restrictions on the choice of impulsive gains and
Abstract. In this article, a new synchronization scheme is presented by combining the concept of reduced-order synchronization with multi-switching synchronization schemes. The presented scheme, reduced-order multi- switching hybrid synchronization, is notable addition to the earlier multi-switching schemes providing enhanced security in applications of secure communication. Based on the Lyapunov stability theory, the active control method is used to design the controllers and derive sufficient condition for achieving reduced-order multi-switching hybrid synchronization between a new hyperchaotic system taken as drive system and Qi chaoticsystem serving as response system. Numerical simulations are performed in MATLAB using the Runge-Kutta method to verify the effectiveness of the proposed method. The results show the utility and suitability of the active control method for achieving the reduced-order multi-switching hybrid synchronization among dynamical chaotic systems. Keywords: chaos synchronization; reduced order synchronization; multi-switching synchronization; hybrid syn- chronization; active control method.
Synchronization of nonlinear systems, particularly chaotic systems, has attracted the attention of many research- ers  . Many control techniques have been devised for chaos synchronization -. In practice, control systems frequently present time delays due to i) finite time necessary for sensing state information, ii) finite time needed for information processing and transmission and iii) finite time necessary for the control actuator to re- spond to a given command. It is worth noting that the general idea of synchronization of chaotic systems with coupling delay seems to follow the idea of simple stabilization of a slave chaoticsystem in the delayed trajec- tory of its master. As such, given a master system x ( ) t , a slave system y ( ) t and coupling delay τ , it is un- derstood that the system achieve complete synchronization when y ( ) ( t − x t − τ ) = 0 as t → ∞ , as assumed in . This form of synchronization is referred to in the literature as achronal synchronization .
To achieve generalized synchronization control of fractional order chaoticsystem, the key is to use response output signal of the drive system to influence the dynamic behavior of response system, making the dynamic behavior of the response system not sensitive to initial values. The most common and direct method is to calculate the Lyapunov exponent (LEs), but in practice, the more widely used approach is the method of auxiliary system . At first, the approach is to construct an auxiliary system, which is a complete copy of response system, and then an appropriate controller including linear drive output signal is selected, which makes the response output of response system and that of auxiliary system achieve generalized synchronization outputting signal in different initial conditions. The following formula can be expressed as :
We wish, now, to choose an appropriate function u t ( ) in order to solve the problem stated in (1). If we re- call one of the main properties of feedback K, namely, the feedback linearization, we can try to achieve the con- trol by linearizing the systems involved via a combined feedback plus feedforward action. Under assumption 1 we can trivially prove that: