Abstract. In this paper, we present new schemes to synchronize different dimensional chaotic and hyperchaoticsystems. Based on coexistence of generalizedsynchronization (GS) and inversegeneralizedsynchronization (IGS), a new type of hybrid chaos synchronization is constructed. Using Lyapunov stability theory and stability theory of linear continuous-time systems, some sufficient conditions are derived to prove the coexistence of generalizedsynchronization and inversegeneralizedsynchronizationbetween 3D master chaotic system and 4D slave hyperchaotic system. Finally, two numerical examples are illustrated with the aim to show the effectiveness of the approaches developed herein.
In recent years, fractional calculus has been noticed for having superior characteris- tics over conventional calculus in the modeling dynamics of natural phenomena [7–13]. The recent development in fractional calculus has been focused on dynamical systems in fractional sense, for example the nonlinear fractional-order systems with disturbances . These systems are characterized by the fact that the order of the derivative is a non- integer number. In particular, it has been shown that may also have complex dynamics such as chaos and bifurcation [15–17]. Some eﬀorts have been recently made devoted to the synchronization of fractional-order chaotic system [18, 19]. It is worth noting that, dif- ferently from integer-order systems, most of the approaches for fractional-order systems are related to the synchronization of identical systems rather than non-identical systems . On the other hand, referring to function-based hybrid synchronization schemes, only few schemes have been proposed to date; see [21–23]. Similar considerations hold for the coexistence of diﬀerent synchronization types in fractional-order systems, given that very few attempts have been made. For example, in  the coexistence of some synchroniza- tion types has been illustrated, including the inversegeneralizedsynchronization and the Q-S synchronization. However, no function-based hybrid synchronization schemes have been analyzed in .
More recently, many investigations were devoted to the chaotic behavior, chaotic con- trol, and synchronization of fractional-order dynamical systems. For example, it has been shown that Chua’s circuit with an order as low as . can produce a chaotic attractor . In , it was shown that nonautonomous Duﬃng systems with an order less than . can still behave in a chaotic manner. In , chaotic behavior of the fractional-order ‘jerk’ model, in which chaotic attractors can be obtained with a system of the order as low as ., was stud- ied. Bifurcations and chaos in the fractional-order simpliﬁed Lorenz system , chaotic behavior and its control in the fractional-order Chen system  were reported. In , chaotic and hyperchaotic behaviors in fractional-order Rössler equations were studied. Chaotic dynamics and synchronization of fractional-order Arneodo systems , the Lü system , and a uniﬁed system , synchronization of fractional-order hyperchaotic modiﬁed systems  were also reported.
In this paper, we have presented a new approach to rigorously study the coexistence of some synchronization types between fractional-order chaoticsystems characterized by diﬀerent dimensions and diﬀerent orders. The paper has shown that identical synchro- nization (IS), antiphase synchronization (AS), and inverse full state hybrid projective syn- chronization (IFSHPS) coexist when synchronizing a three-dimensional master system with a fourth-dimensional slave system. It has been shown that the approach presents the remarkable feature of being both rigorous and applicable to a wide class of commensu- rate and incommensurate systems of diﬀerent dimensions and orders. All the numerical examples reported through the paper have clearly highlighted the capability of the pro- posed approach in successfully achieving the co-existence of IS, AS, and IFSHPS betweenchaotic and hyperchaoticsystems of diﬀerent dimensions for both commensurate and in- commensurate fractional-order systems. These examples of coexistence have included the chaotic commensurate three-dimensional Rössler system of order 2.7, the hyperchaotic commensurate four-dimensional Chen system of order 3.84, the chaotic incommensurate three-dimensional Lü system of order 2.955, and the hyperchaotic incommensurate four- dimensional Lorenz system of order 3.86. Finally, we would stress that the topic related to the coexistence of synchronization types in fractional-order systems is almost unex- plored in the literature. We feel that the additional features introduced by the conceived approach, related to both the number of coexisting synchronization types and the capa- bility to synchronize chaotic dynamics with hyperchaotic ones, provides a deeper insight into the synchronization phenomena betweensystems described by fractional diﬀerential equations.
Recently, the study of dynamics of fractional-order chaoticsystems has received interest of many researchers. Yu and Li in  used Laplace transformation theory and variational itera- tion method to study R¨ossler system, Wu, Lu and Shen discussed the synchronization of a new fractional- order hyperchaotic system via active control , Wang, Yu and Diao in  stud- ied the hybrid projective synchronizationbetween fractional-order chaoticsystems of different dimensions, Sahab and Ziabari  analyzed the chaos between two different hyperchaoticsystems by generalized backstepping method, S.T. Mohammad and H. Mohammad in  pro- posed a controller based on active sliding mode theory to synchronize the chaotic fractional order systems, Zhang and Lu introduced a new type of hybrid synchronization called full state hybrid lag projective synchronization and applied it to the R¨ossler system and the hyperchaotic Lorenz system to verify their results numerically . A. Ouannas in  studied the Q-S syn- chronization of chaotic dynamical systems in continuous-time, Boutefnouchet, Taghvafard and Erjaee in their paper  discussed the phase synchronization in coupled chaoticsystems.
On the other hand, with the development of complex systems, synchronization of chaotic complex systems has gained a great deal of attentions. Some synchronization schemes of chaotic real systems were extended to the complex space, such as complete synchronization , anti-synchronization [, ], projective synchronization , etc. Recently, many authors have studied some new kinds of synchronization for complex dy- namical systems, for example, complex complete synchronization , complex projective synchronization , complex modiﬁed projective synchronization [, ], and so forth.
Chaos is an omnipresent phenomenon. Scientists who understand its existence have been struggling to control chaos to our benefit. There is a great need to control the chaoticsystems as chaos theory plays an important role in industrial applications particularly in chemical reac- tions, biological systems, information processing and secure communications [1-3]. Many scientists who are interested in this field have struggled to achieve the syn- chronization or anti-synchronization of different hyper- chaoticsystems. Therefore due to its complexity and applications, a wide variety of approaches have been proposed for the synchronization or anti-synchronization of hyperchaoticsystems. The types of synchronization used so far include generalized active control [4-8], non- linear control [9,10], and adaptive control [11-19].
Recently impulsive control theory and its application in chaos synchronization have become a research hotspot. For instance, based on impulsive control strategies, the reduced- order observer for the synchronization of generalized Lorenz chaoticsystems is built in , the adaptive modified function projective synchronization of multiple time-delayed chaotic Rossler systems is discussed in , the hybrid synchronization of L¨u hyperchaotic system with disturbances is investigated in , and the robust synchronization of perturbed Chen’s fractional-order chaoticsystems is studied in . However, these results are just about one kind of chaoticsystems, which limits their applied scope. Hence how to design the impulsive strategy to realize the synchronization suitable for more chaoticsystems activates our research.
Based on Lyapunov stability theory, constructive schemes are presented in this paper to research the generalized syn- chronization and the inversegeneralizedsynchronization 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.
Chaos refers to the high sensitivity of a dynamical system to small changes in the ini- tial condition. Discrete chaoticsystems have been around for a while. Perhaps the most commonly studied and applied discrete chaotic system is the Hénon map, which was intro- duced in 1976  as a discretization of the Poincaré section of the famous continuous-time Lorenz system. Soon after, numerous other maps were proposed and became of interest to researchers in the ﬁelds of communications and control including the Lozi system  and the 2-component ﬂow model . Generalizations of these 2-component systems to higher dimensions were proposed at a later stage including, for instance, the generalized Hénon map  and the Stefanski system . This type of chaoticsystems has found applications in many ﬁelds including engineering and science [14–17].
Explosive synchronization (ES), as one kind of abrupt dynamical transitions in nonlinearly coupled systems, has become a hot spot of modern complex networks. At present, many results of ES are based on the networked Kura- moto oscillators and little attention has been paid to the influence of chaotic dynamics on synchronization transitions. Here, the unified chaoticsystems (Lorenz, Lü and Chen) and Rössler systems are studied to report evidence of an explosive synchronization of chaoticsystems with different topological network structures. The results show that ES is clearly observed in coupled Lorenz systems. However, the continuous transitions take place in the coupled Chen and Lü systems, even though a big shock exits during the syn- chronization process. In addition, the coupled Rössler systems will keep syn- chronous once the entire network is completely synchronized, although the coupling strength is reduced. Finally, we give some explanations from the dynamical features of the unified chaoticsystems and the periodic orbit of the Rössler systems.
and where ǫ, δ > 0, are small parameters. In this case, the underlying (ODE) part of system (1) induces an asymptotic evolution to a unique limit cycle for the trajectories diferent from (0, 0). We showed some results on asymptotic behaviour and synchronization for the network. Here, we will generalize some of these results. Let us consider a network of coupled reaction diffusion systems of the following generalized FHN type,
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 chaoticsystems [2-11]. A new 3D chaotic system (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 T system. 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.
Synchronization of chaoticsystems has become an active research area because of its po- tential applications in diﬀerent industrial areas [–]. Communication security scheme is one of the hottest ﬁelds based on chaos synchronization. In this secure communication scheme, the message signals are injected to a chaotic carrier in the transmitter and then are masked or encrypted. The resulting masked signals are transmitted across a public channel to the receiver. To recover the message in the receiver, the synchronization be- tween the chaoticsystems at the transmitter and receiver ends is required. Since Pecora and Carroll  originally proposed the synchronization of the drive and response systems with diﬀerent initial states in , many synchronization techniques such as coupling control , adaptive control , feedback control , fuzzy control , observer-based control , etc. have been developed in the literature.
presented the adaptive synchronization of fractional Lorenz systems using a re- duced number of control signals and parameters . Kajbaf et al. used sliding mode controller to obtain chaoticsystems . Wang et al. proposed a new feedback synchronization criterion based on the largest Lyapunov exponent . However, most synchronization criterions were obtained under ideal cir- cumstances. If parameters perturbation and external disturbance exist, this kind of criterions will take no effect. According to this practical problem, some solu- tions have been presented. For examples, Jiang et al. proposed a LMI criterion  for chaotic feedback synchronization. Although the simulations showed that it is robust to a random noise with zero mean, but no rigorous mathematical proof was provided and we can’t determine if their method is effective for other kinds of noise. In Ref. , parameters perturbation was involved in their scheme. The theoretical proof and numerical simulations were given in their work, but external disturbance didn’t receive attention, which made their method unila- teral.
 -. 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 .
chaoticsystems (Sara Dadras and Hamid Reza Momeni, 2009). Simulation results depicts that this proposed controller can synchronize these chaoticsystems. The rest of the paper is organized as follows. In Section 2, the Brain Emotional Learning Based Intelligent Controller (BELBIC) is described. In Section 3, the T–S fuzzy models will be presented for a new 3D chaotic system (Sara Dadras and Hamid Reza Momeni, 2009). In Section 4, synchronizationbetween two new fuzzy chaotic system is achieved by BELBIC is described. Finally, Section 5 provides, conclusion of this work.
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.
In the phenomenon of hybrid synchronization of chaoticsystems, coexistence of complete synchronization and anti-synchronization occurs. Sometimes because of this coexistence this type of synchronization is also referred to as mixed synchronization . Complete synchro- nization implies that the differences of state variables of synchronized systems with different initial values converge to zero. Anti-synchronization is a phenomenon in which the state vari- ables of synchronized systems with different initial values have the same absolute values but of opposite signs. Consequently, the sum of two signals is expected to converge to zero when anti-synchronization occurs. The coexistence of complete and anti-synchronization turns out to be very useful in applications of secure communication. Many successful studies have been reported on hybrid synchronization of chaoticsystems [34–36].
In this paper, we derive new results for the anti-synchronization of identical Pan chaoticsystems (2010), identical Liu chaoticsystems (2004) and non-identical Pan and Liu chaoticsystems by active nonlinear control. The stability results for the anti-synchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active nonlinear control method is effective and convenient to achieve anti-synchronization of the chaoticsystems addressed in this paper. Numerical simulations are shown to illustrate the effectiveness of the anti-synchronization results derived in this paper for identical and non-identical Pan and Liu chaoticsystems.