Abstract. In this paper, we present new schemes to synchronize different dimensional **chaotic** and **hyperchaotic** **systems**. Based on **coexistence** of **generalized** **synchronization** (GS) and **inverse** **generalized** **synchronization** (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 **generalized** **synchronization** and **inverse** **generalized** **synchronization** **between** 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.

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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 [14]. 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** [20]. 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 [24] the **coexistence** of some synchroniza- tion types has been illustrated, including the **inverse** **generalized** **synchronization** and the Q-S **synchronization**. However, no function-based hybrid **synchronization** schemes have been analyzed in [25].

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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.

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In this paper, we have presented a new approach to rigorously study the **coexistence** of some **synchronization** types **between** fractional-order **chaotic** **systems** 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 **between** **chaotic** and **hyperchaotic** **systems** 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 **between** **systems** described by fractional diﬀerential equations.

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Recently, the study of dynamics of fractional-order **chaotic** **systems** has received interest of many researchers. Yu and Li in [20] 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 [21], Wang, Yu and Diao in [22] stud- ied the hybrid projective **synchronization** **between** fractional-order **chaotic** **systems** of different dimensions, Sahab and Ziabari [23] analyzed the chaos **between** two different **hyperchaotic** **systems** by **generalized** backstepping method, S.T. Mohammad and H. Mohammad in [24] 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 [25]. A. Ouannas in [26] studied the Q-S syn- chronization of **chaotic** dynamical **systems** in continuous-time, Boutefnouchet, Taghvafard and Erjaee in their paper [27] discussed the phase **synchronization** in coupled **chaotic** **systems**.

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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.

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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 **chaotic** **systems** 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- **chaotic** **systems**. Therefore due to its complexity and applications, a wide variety of approaches have been proposed for the **synchronization** or anti-**synchronization** of **hyperchaotic** **systems**. The types of **synchronization** used so far include **generalized** active control [4-8], non- linear control [9,10], and adaptive control [11-19].

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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 **chaotic** **systems** is built in [16], the adaptive modified function projective **synchronization** of multiple time-delayed **chaotic** Rossler **systems** is discussed in [17], the hybrid **synchronization** of L¨u **hyperchaotic** system with disturbances is investigated in [18], and the robust **synchronization** of perturbed Chen’s fractional-order **chaotic** **systems** is studied in [19]. However, these results are just about one kind of **chaotic** **systems**, which limits their applied scope. Hence how to design the impulsive strategy to realize the **synchronization** suitable for more **chaotic** **systems** activates our research.

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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.

Chaos refers to the high sensitivity of a dynamical system to small changes in the ini- tial condition. Discrete **chaotic** **systems** 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 [9] 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 [10] and the 2-component ﬂow model [11]. Generalizations of these 2-component **systems** to higher dimensions were proposed at a later stage including, for instance, the **generalized** Hénon map [12] and the Stefanski system [13]. This type of **chaotic** **systems** has found applications in many ﬁelds including engineering and science [14–17].

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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 **chaotic** **systems** (Lorenz, Lü and Chen) and Rössler **systems** are studied to report evidence of an explosive **synchronization** of **chaotic** **systems** 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 **chaotic** **systems** and the periodic orbit of the Rössler **systems**.

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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,

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Chaos is an important phenomenon, happens vastly in both natural and man-made **systems**. Lorenz [1] faced to the first **chaotic** attractor in 1963. In continue, a lot of researches were achieved on **chaotic** **systems** [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 [28], backstepping [28] and adaptive control [28] are three different methods for **synchronization** of T system. Active control [28] and backstepping [28] methods are selected when system parameters are known, and adaptive control [28] 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.

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presented the adaptive **synchronization** of fractional Lorenz **systems** using a re- duced number of control signals and parameters [13]. Kajbaf et al. used sliding mode controller to obtain **chaotic** **systems** [14]. Wang et al. proposed a new feedback **synchronization** criterion based on the largest Lyapunov exponent [15]. 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 [16] 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. [17], 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.

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[11] [14]-[16]. 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 [17]. 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 [18].

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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 **chaotic** **systems**, 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 **chaotic** **systems**, 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 **chaotic** **systems**. Numerical simulations are given to demonstrate the effectiveness of the proposed adaptive **synchronization** schemes for the global chaos **synchronization** of the uncertain **chaotic** **systems** addressed in this paper.

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In the phenomenon of hybrid **synchronization** of **chaotic** **systems**, **coexistence** of complete **synchronization** and anti-**synchronization** occurs. Sometimes because of this **coexistence** this type of **synchronization** is also referred to as mixed **synchronization** [33]. 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 **chaotic** **systems** [34–36].

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In this paper, we derive new results for the anti-**synchronization** of identical Pan **chaotic** **systems** (2010), identical Liu **chaotic** **systems** (2004) and non-identical Pan and Liu **chaotic** **systems** 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 **chaotic** **systems** 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 **chaotic** **systems**.

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