nonlinear dynamical systems. This procedure strongly relies upon proper hierarchical prob- lem decomposition as shown in Fig.32, which particularizes for the well-known double- scroll attractor. The first step in the methodology is to identify the set of equations describ- ing the dynamics. This corresponds to the behavioral level at the top of the hierarchy. The obtained description maps down to the block level, which defines a network synthesis archi- tecture for the problem. At the block level, the different operators, or functional building blocks, required for physical realization, as well as their interconnection, are clearly identi- fied. Each of these blocks must be subsequently mapped down to a collection of intercon- nected circuit elements, thus defining a circuit level. Two different sublevels can be identified; one containing only idealized elements (for instance VCCS’s), and another where these idealized elements are realized using available circuit primitives of the technol- ogy. Fig.32 illustrates both sublevels. Observe that the circuit level infers choosing the physical nature of the variables which support information flow (usually voltages, currents or both). Bottom level in the VLSI design hierarchy define the layout phase, where circuit primitives are codified into geometrical objects required for processing and fabrication.
Kerne´vez 1986; see http://sourceforge.net/projects/auto2000/). These packages reduce the identiﬁcation, location and tracking of equilibrium and periodic solution branches and other bifurcation problems to the continuation of implicitly deﬁned curves, using predictor–corrector methods. For simplicity, consider autonomous dynamical systems and one-parameter bifurcation diagrams that contain equilibria, periodic solutions and Hopf and homoclinic bifurcations. Once a point on a particular branch is located, a continuation package follows the locus of either equilibrium or periodic states on the branch. While following an equilibrium branch, tracking the real parts (and signs) of the eigenvalues gives the stability of the branch that is exchanged between stable and unstable as the real part passes through zero. If a single complex conjugate pair of eigenvalues crosses the imaginary axis, small amplitude (stable or unstable) oscillations emerge at a Hopf bifurcation. Once the bifurcation point is found, continuation can be switched from the equilibrium to the periodic branch. The stability of periodic oscillations is followed using Floquet multipliers.
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 chaotic systems, bifurcations, routes to chaos, and many other related topics. There are also some chaotic systems 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:
This paper is organized as follows: In section 2, the dynamical properties of chemical reactor model including Lyapunov exponents, bifurcations and stability of equilibria, will be discussed. We using sliding mode control method for chaos synchronization of two identical chemical reactors model in section 3. By using the Barbalate's lemma, proved that the error system is asymptotically stable in origin. In section 4, numerical simulations are computed to check the analytical expressions. Our concluding remarks are presented in section 5.
 -. In this case, somewhat counter intuitively, chaotic units synchronize without any relative time delay, although the transmitted signal is received with a large time lag. In most of the rigorous results based on the Lyapunov-Krasovskii stability or Lyapunov-Razumikhin stability, the proposed scheme is very specific, but also the added controller is sometimes too big to be physically practical. One practical scheme is the linear feed- back. However, in such a technique it is very difficult to find the suitable feedback constant, and thus numerical calculation has to be used, e.g., the calculation of the conditional Lyapunov exponents. Due to numerical calcu- lation, such a scheme is not regular since it can be applied only to particular models. More unfortunately, it has been reported that the negativity of the conditional Lyapunov exponents is not a sufficient condition for com- plete chaos synchronization, see . Therefore, the synchronization based on these numerical schemes cannot be strict (i.e., high-qualitative), and is generally not robust against the effect of noise. Especially, in these schemes a very weak noise or a small parameter mismatch can trigger the desynchronization problem due to a sequence of bifurcations .
Chaotic systems have been studied for a long time and nowadays many examples of electronic implementations using discrete devices can be found in the literature. However, the big challenge remains the generation of multi-scroll attractors and their design using CMOS integratedcircuit (IC) technology to develop real life applications. For instance, the control and synchronization of chaotic systems were first proposed no more than three decades ago [1–7], from which some practical developments have impacted areas such as high-performance circuit design (e.g., delta-sigma modulators and power converters), liquid mixing, chemical reactions, biological systems (e.g., sensing signals produced in the human brain, heart or other organs), power electronics, secure communication systems, etc. [8–13]. That way, this new and challenging line for research and development is becoming highly inter-disciplinary, involving systems and control engineers, theoretical and experimental physicist, applied mathematicians, physiologists and, above all, IC design specialists.
the simulation realistic. Band-limited white noise block of Simulink is used to generate the noise. The result of the synchronization is given in Fig. 2 and 3. Fig. 2 and 3 show that master-slave synchronization is achieved after applying the control. Comparative simulation results show that the proposed controller perform better than the EHG based control scheme. The convergence time of the proposed controller is better than that EHG based controller. Steady- state error of the proposed controller is significantly small than that of EHG based controller. One problem of high-gain observer is that it amplifies the measurement noise. This is also evident here as shown in Fig. 3. These noise amplified estimated states deteriorates the performance of the controller.
Programmable NoC is a circuit switched architecture that simplifies the system design by providing flexible networking approach. It is possible to design various network architectures each with its own choice of system routing. The nodes can be dynamically inserted and removed, provided the FPGA supports it. It consists of sub nets, which are in turn connected to router and a bunch of network nodes as shown in Fig II ( and ]. This type of architecture was chosen on account of its ability to place modules that communicate very frequently in the same subnet, thus allowing more effective system communication. A simple handshaking mechanism is used to establish dedicated connections between nodes, to exchange data and to remove connections.
Synchronization has been widely observed in many natural, social and technological systems, and has attracted much attention during the last two decades [ 1 – 12 ]. In previous works, complete synchronization of identical chaotic oscillators has been a special focus due to the availability of analytic framework of the master-stability function (MSF) in determining the critical coupling strength at which synchronization arises and, more generally, the regions in the parameter space where stable synchronization may emerge [ 7 , 13 , 14 ]. In general, the MSF reduces the synchronization analysis to the interplay between two aspects: the dynamics of a single oscillator and the coupling topology characterized by the eigenvalues of the coupling matrix. With knowledge of the dynamics of the individual nodes and the nature of the pairwise coupling to infer the MSF, for any coupled system one can thus predict whether synchronization can occur based on the eigenvalues of the coupling matrix and the coupling strength, without the need of actual simulations. For example, a common class of MSFs have the property that, in the plot of the MSF versus some generalized coupling parameter, there exists a ﬁnite interval in which the MSF assumes negative values. For the normalized coupling strength either below or above this interval, the system cannot be physically synchronized. In this case, synchronization is determined by the largest and the smallest nontrivial eigenvalues of the coupling matrix, and their ratio is called eigenratio, which has been used extensively in the study of the synchronizability of complex networks of various types of topology [ 15 – 20 ]. A general observation is that a smaller eigenratio is beneﬁcial to synchronization and thus is more desirable in designing synchronous networks [ 16 ]. The effect of different coupling schemes among the dynamical variables
Coupled oscillatory systems are very common both in nature and in technology. Some- times the coupling results in synchronization between the oscillations of the two systems - an effect first observed between a pair pendulum clocks by Huygens in 1665, while lying ill in bed. In 1990, Pecora and Carroll (PC) proposed a successful method  to syn- chronize two identical chaotic systems starting from different initial conditions. The idea of synchronization is to use the output of a master system to control a slave system so that the response of the latter follows the output of the master system asymptotically. Since then, chaos synchronization has been developed extensively as an important topic in its own right nonlinear science. It has been investigated by scientists working in fields as different as secure communications, optical, chemical, physical, and biological systems and neural networks. The synchronization of chaos has been reported in systems whose models are identical, similar, and with mismatched-parameters (see [2-6] for reviews). The synchronization of chaos can be induced even in strictly different oscillators [7-9] and systems of different order [10,11]. Nevertheless, given the diversity of phenomena that have been found in chaotic systems , there is not yet a clear and exact meaning of the synchronization in chaos. Many distinct kinds of synchronized systems are to be found in nature. Examples where we can expect to model synchronization between similar oscilla- tors include nephron-nephron synchronization  and the mechanical systems . On the other hand, nonidentical systems can be found in the synaptic communication where some neurons with different dynamic models can behave in a synchronous manner [15,16]. Another example is the synchronization that occurs between heart and lung, where one can observe that circulatory and respiratory systems synchronize , even through they are expected to be described by different models. Such facts add to the complexity of the meaning of the chaoticsynchronization. In the nonlinear science, the definition of the synchronous behavior means that the trajectories of two or more dynamical systems evolve, in some sense, close to one and other along their trajectories.
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.
We introduced a phase oscillator model of glacial cycles and analyzed the bifurcations of the model for the ideal two-frequency quasiperiodic forcing and for the astronomical forcing. It was shown that SNAs appear through nonsmooth saddle-node bifurcations of tori in the model. Based on the results for the phase oscillator model, we conjecture that the bifurcations from quasiperiodic attractors to SNAs found in oscillator models of glacial cycles  are also nonsmooth saddle-node bifurcations. The regime diagram in Fig. 6(b) indicates that mode-locking is likely to occur for the 41 kyr glacial cycles but not likely for the 100 kyr glacial cycles. The sequence of mode-locked 41 kyr cycles is robust to small parameter changes. However, the sequence of 100 kyr glacial cycles can be sensitive to parameter changes when the system has an SNA.
This paper has shown that identication of chaotic or hyper-chaotic systems can be done based on the synchronization of two identical systems. Two sys- tems are synchronized by applying one state feedback controller. Adaptation laws used to nd unknown parameters came from the Lyapunov stability theorem. By applying fewer degrees of dierentiation in some of the adaptation laws (usually parameters with more ripples), less uctuations in convergence of the param- eter occur, as the results have shown in the numerical simulations. Finally, a discussion about the analytical proof of the stability of the controlled system using the fractional adaptation law is presented.
Chaos systems [1-3] have complex dynamical behaviors that possess some special features such as being extremely sensitive to tiny variations of initial conditions, and having bounded trajectories with a positive leading Lyapunov exponent and so on. Chaos control [4, 5]and chaos synchronization are two main problems of chaotic system research. In recent years, chaos synchronization has been attracted increasingly attentions due to their potential applications in the fields of secure communications.
Chaos synchronization has powerful application in chemical reactions, power converters, biological systems, information processing, secure communications, etc. During the last decade, many techniques for handling chaos synchronization have been studied [1-11] . In , through separation between linear and nonlinear terms of the chaotic system, an adaptive synchronous controller for a general class of non-autonomous chaotic systems is proposed. In , based on delayed feedback control and intermittent linear state delayed feedback control, the synchronization problem on non-autonomous chaotic systems is discussed. In , an adaptive controller with parameter update laws is designed to realize the projective synchronization of two different chaotic systems. In , based on the time-domain approach, the tracking synchronization control method is proposed for the uncertain Genesio-Tesi chaotic systems with dead zone nonlinearity. In , the projective synchronization problem for different fractional order chaotic systems is investigated. In , some adaptive control schemes are developed to anti-synchronize two chaotic complex systems. However, in practical control systems,
Based on Lyapunov stability theory, constructive schemes are presented in this paper to research the generalized syn- chronization and the inverse generalized synchronization be- tween different dimensional spatial chaotic dynamical sys- tems. Besides, the new synchronization criterions are estab- lished in the form of simple algebraic conditions which are very convenient to be verified.
Since Pecora and Carroll introduced a method to synchronize two identical chaotic systems with different initial conditions, various synchronization schemes have been studied by many researchers, such as OGY method , observer-based control, adaptive control, feedback control, backstepping control[6,7], and so on. All of these methods have adopted the continuous chaoticsynchronization scheme. To increase the efficiency of bandwidth usage, impulsive chaoticsynchronization [8,9] has been proposed. In synchronization process, the control signals are transmitted from driving system to driven system only at discrete time instants. Thus reduces the amount of information transmitted between the two systems. Various theoretical and experimental results of impulsive chaoticsynchronization can be found in[8-14]. Further studies on how to reduce the synchronization information are implemented by researchers. Basically in three ways, enlarge the fixed impulse intervals, time- varying impulse intervals to enlarge the average impulse
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 chaotic system 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.