- Research
- Open Access
Coexistence of identical synchronization, antiphase synchronization and inverse full state hybrid projective synchronization in different dimensional fractional-order chaotic systems
- Adel Ouannas^{1},
- Xiong Wang^{2},
- Viet-Thanh Pham^{3}Email author,
- Giuseppe Grassi^{4} and
- Toufik Ziar^{5}
https://doi.org/10.1186/s13662-018-1485-2
© The Author(s) 2018
- Received: 29 September 2017
- Accepted: 12 January 2018
- Published: 23 January 2018
Abstract
The topic related to the coexistence of different synchronization types between fractional-order chaotic systems is almost unexplored in the literature. Referring to commensurate and incommensurate fractional systems, this paper presents a new approach to rigorously study the coexistence of some synchronization types between nonidentical systems characterized by different dimensions and different orders. In particular, the paper shows that identical synchronization (IS), antiphase synchronization (AS), and inverse full state hybrid projective synchronization (IFSHPS) coexist when synchronizing a three-dimensional master system with a fourth-dimensional slave system. The approach, which can be applied to a wide class of chaotic/hyperchaotic fractional-order systems in the master-slave configuration, is based on two new theorems involving the fractional Lyapunov method and stability theory of linear fractional systems. Two examples are provided to highlight the capability of the conceived method. In particular, referring to commensurate systems, the coexistence of IS, AS, and IFSHPS is successfully achieved between the chaotic three-dimensional Rössler system of order 2.7 and the hyperchaotic four-dimensional Chen system of order 3.84. Finally, referring to incommensurate systems, the coexistence of IS, AS, and IFSHPS is successfully achieved between the chaotic three-dimensional Lü system of order 2.955 and the hyperchaotic four-dimensional Lorenz system of order 3.86.
Keywords
- coexistence of synchronization types
- fractional-order systems
- chaos synchronization
- commensurate and incommensurate systems
- nonidentical systems
MSC
- 34H10
- 26A33
- 34A08
1 Introduction
By starting from the milestone by Pecora and Carroll [1], over the last years, great efforts have been devoted to the study of chaos synchronization in dynamical systems described by integer-order differential equations and difference equations [2]. Given two systems in the master-slave configuration, the objective in chaos synchronization is to make the response system variables synchronized in time with the corresponding drive system variables. At the beginning, Pecora and Carroll introduced the concept of complete (identical) synchronization (IS), but, year after year, different types of synchronization have been proposed in the literature, for continuous- and discrete-time systems [3–10]. For example, in projective synchronization (PS) the response system variables are scaled replicas of the drive system variables [11]. When the scaling factor is ‘−1’, antiphase synchronization (AS) is obtained. On the other hand, when the scaling factor is different for each drive system variable, full state hybrid projective synchronization (FSHPS) is achieved [12]. Recently, in [13, 14] a synchronization scheme has been presented, where each drive system state synchronizes with a linear combination of response system states. Since drive system states and response system states have been inverted in [13, 14] with respect to the FSHPS, the scheme has been called inverse full state hybrid projective synchronization (IFSHPS). Additionally, recent papers have investigated the coexistence of several synchronization types when synchronizing two chaotic systems both in integer-order differential systems and discrete-time systems [15–17]. In particular, the approach developed in [15] has illustrated a rigorous study to prove the coexistence of some synchronization types between discrete-time chaotic (hyperchaotic) systems.
Besides integer-order systems, attention has been recently focused on systems described by fractional-order differential equations [18, 19]. Researches in the literature have shown that fractional-order systems, as generalizations of well-known integer-order systems, are characterized by chaotic dynamics [20, 21]. These systems include the fractional Lorenz system, the fractional Chua system, the fractional Rössler system, the fractional Chen system, and the fractional Lü system [22–26]. Specifically, researches have shown that chaos is achievable when the system order is less than 3, whereas hyperchaos can be obtained when the system order is less than 4. Referring to synchronization, studies have shown that chaotic fractional-order systems can also be synchronized [27]. However, differently from integer-order systems, few synchronization types have been introduced for fractional-order systems. Moreover, most of the approaches are related to the synchronization of identical fractional-order systems. Very few methods for synchronizing nonidentical fractional-order chaotic systems have been illustrated [28–30]. Additionally, the topic related to the coexistence of different synchronization types between fractional-order systems is almost unexplored [31, 32].
Based on these considerations, this paper presents a new approach to rigorously study the coexistence of some synchronization types between fractional-order systems characterized by different dimensions and different orders. In particular, the paper shows that identical synchronization (IS), antiphase synchronization (AS), and inverse full state hybrid projective synchronization (IFSHPS) coexist in the synchronization of a three-dimensional master system with a four-dimensional slave system. The approach presents the remarkable feature of being both rigorous and applicable to a wide class of commensurate and incommensurate fractional-order systems of different dimensions and different orders. It is worth noting that the proposed approach is more general than those illustrated in [31, 32] because, among other things, it guarantees the coexistence of three different synchronization types (instead of only two types, as in [31, 32]). This increased complexity provides a deeper insight into the synchronization phenomena between systems described by fractional differential equations.
The paper is organized as follows. In Section 2, the basic notions on fractional derivatives and on the stability of fractional systems are given. In Section 3, the meaning of coexistence of IS, AS, and IFSHPS in fractional-order systems is illustrated. In Section 4, by exploiting fractional Lyapunov stability theory the coexistence of IS, AS, and IFSHPS between two commensurate fractional-order systems of different dimensions is proved. Additionally, by using the stability theory of fractional linear systems, in Section 5, the coexistence of IS, AS, and IFSHPS between two incommensurate fractional-order systems of different dimensions is illustrated. Referring to numerical examples, at first the coexistence of IS, AS, and IFSHPS between the chaotic fractional-order commensurate Rössler system and the hyperchaotic fractional-order commensurate Chen system is successfully achieved. Note that the Rössler system is a three-dimensional system of order 2.7, whereas the Chen system is a four-dimensional system of order 3.84. An additional numerical example illustrates the coexistence of IS, AS and IFSHPS between the chaotic fractional-order the incommensurate Lü system and the hyperchaotic fractional-order incommensurate Lorenz system. Note that the Lü system is a three-dimensional system of order 2.955, whereas the Lorenz system is a four-dimensional system of order 3.86. All the numerical examples prove the capability of the proposed approach in successfully achieving the coexistence of IS, AS, and IFSHPS between chaotic and hyperchaotic systems of different dimensions for both commensurate and incommensurate fractional-order systems. Finally, in Section 6, the advantages and the novelty of the conceived approach with respect to those existing in the literature are discussed in detail.
2 Basic concepts
In the following, some basic concepts on fractional derivatives and on the stability properties of fractional linear/nonlinear systems are briefly illustrated.
Definition 1
([33])
Now we state a well-known theorem on the stability of fractional linear systems.
Theorem 1
([37])
Now we report a theorem on fractional nonlinear systems.
Theorem 2
Starting from previous theorem, we can give the following lemma.
Lemma 1
([40])
3 Problem formulation
Based on the master-slave synchronizing system described by (7)-(8), we can give the following definition of coexistence of different synchronization types.
Definition 2
4 Coexistence of IS, AS, and IFSHPS for commensurate systems
In this section, we focus ourselves on the coexistence of IS, AS, and IFSHPS for commensurate chaotic (hyperchaotic) systems. First, we prove a new theorem. Afterward, we illustrate in detail an application involving the chaotic fractional-order Rössler system (as a master system) and the hyperchaotic fractional-order Chen system (as a slave system).
Theorem 3
Proof
5 Coexistence of IS, AS, and IFSHPS for incommensurate systems
In this section, we analyze the coexistence of IS, AS, and IFSHPS for incommensurate chaotic (hyperchaotic) systems. A new theorem, tailored for synchronizing incommensurate systems, is first proved. Successively, an example involving the chaotic fractional-order Lü system (as a master system) and the hyperchaotic fractional-order Lorenz system (as a slave system) is described in detail.
Theorem 4
Proof
6 Discussion
The aim of this section is to highlight the novelty introduced by the present approach. First of all, it is worth analyzing the results achieved so far in the literature on the same topic, that is, the coexistence of some synchronization types in different dimensional fractional-order chaotic systems. This topic is almost unexplored in the literature [31, 32]. For example, in [31] the authors have presented a hybrid synchronization method for fractional-order systems, which is a combination of only two synchronization types, that is, generalized synchronization and inverse generalized synchronization. Our approach is more general than that in [31], since it guarantees the coexistence of three different synchronization types, that is, identical synchronization (IS), antiphase synchronization (AS), and inverse full state hybrid projective synchronization (IFSHPS). Another method for fractional-order systems has been developed in [32], where a combination of full state hybrid projective synchronization (FSHPS) and inverse full state hybrid projective synchronization (IFSHPS) has been presented. However, the method in [32] can only be applied to incommensurate systems, besides the fact that it allows the coexistence of only two synchronization types. Our approach is more general than that in [32], since it can be applied to both commensurate and incommensurate systems (see Theorems 3 and 4, respectively), besides the fact that it guarantees the coexistence of three different synchronization types (instead of only two, as in [32]).
Based on previous considerations, it should be clear that the method proposed herein provides a contribution to the topic related to the coexistence of some fractional synchronization types, since it guarantees the coexistence of three different synchronization types for both commensurate and incommensurate fractional-order systems of different dimensions and different orders. This increased complexity related to both the number of synchronization types and the capability to synchronize chaotic dynamics with hyperchaotic ones and provides a deeper insight into the synchronization phenomena between systems described by fractional differential equations.
7 Conclusions
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 synchronization (IS), antiphase synchronization (AS), and inverse full state hybrid projective synchronization (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 commensurate and incommensurate systems of different dimensions and orders. All the numerical examples reported through the paper have clearly highlighted the capability of the proposed approach in successfully achieving the co-existence of IS, AS, and IFSHPS between chaotic and hyperchaotic systems of different dimensions for both commensurate and incommensurate 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 unexplored 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 capability to synchronize chaotic dynamics with hyperchaotic ones, provides a deeper insight into the synchronization phenomena between systems described by fractional differential equations.
Declarations
Acknowledgements
The authors acknowledge Prof. GuanRong Chen, Department of Electronic Engineering, City University of Hong Kong, for suggesting many helpful references.
Funding
The author Xiong Wang was supported by the National Natural Science Foundation of China (No. 61601306) and Shenzhen Overseas High Level Talent Peacock Project Fund (No. 20150215145C).
Authors’ contributions
AO and VTP suggested the model. AO, XW, VTP, GG, and TZ helped in results interpretation and manuscript evaluation. XW, GG, and TZ helped to evaluate, revise, and edit the manuscript. XW, GG, and TZ supervised the development of work. AO and VTP drafted the article. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Carroll, TL, Pecora, LM: Synchronizing chaotic circuits. IEEE Trans. Circuits Syst. 38(4), 453-456 (1991) View ArticleMATHGoogle Scholar
- Wen, G, Grassi, G, Feng, Z, Liu, X: Special issue on advances in nonlinear dynamics and control. J. Franklin Inst. 352(8), 2985-2986 (2015) MathSciNetView ArticleGoogle Scholar
- Wang, XF, Chen, GR: Synchronization in small-world dynamical networks. Int. J. Bifurc. Chaos 12, 187-192 (2002) View ArticleGoogle Scholar
- Cao, JD, Li, HX, Ho, DWC: Synchronization criteria of Lur’s systems with time-delay feedback control. Chaos Solitons Fractals 23, 1285-1298 (2005) MathSciNetView ArticleMATHGoogle Scholar
- He, WL, Qian, F, Cao, JD, Han, QL: Impulsive synchronization of two nonidentical chaotic systems with time-varying delay. Phys. Lett. A 375, 498-504 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Pourdehi, S, Karimaghaee, P, Karimipour, D: Adaptive controller design for lag-synchronization of two non-identical time-delayed chaotic systems with unknown parameters. Phys. Lett. A 375, 1769-1778 (2011) View ArticleMATHGoogle Scholar
- Rehan, M: Synchronization and anti-synchronization of chaotic oscillators under input saturation. Appl. Math. Model. 37, 6829-6837 (2013) MathSciNetView ArticleGoogle Scholar
- Wu, A: Hyperchaos synchronization of memristor oscillator system via combination scheme. Adv. Differ. Equ. 2014, 86 (2014) MathSciNetView ArticleGoogle Scholar
- Pecora, LM, Carroll, TL: Synchronization of chaotic circuits. Chaos 25, 097611 (2015) View ArticleMATHGoogle Scholar
- Abrams, DM, Pecora, LM, Motter, AE: Introduction to focus issue - patterns of network synchronization. Chaos 26, 094601 (2016) View ArticleGoogle Scholar
- Manieri, R, Rehacek, J: Projective synchronization in three-dimensional chaotic systems. Phys. Rev. Lett. 82(15), 3042-3045 (1999) View ArticleGoogle Scholar
- Hu, M, Xu, Z, Zhang, R: Full state hybrid projective synchronization in continuous-time chaotic (hyperchaotic) systems. Commun. Nonlinear Sci. Numer. Simul. 13(2), 456-464 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Ouannas, A, Grassi, G: Inverse full state hybrid projective synchronization for chaotic maps with different dimensions. Chin. Phys. B 25(9), 090503 (2016) View ArticleGoogle Scholar
- Ouannas, A, Azar, AT, Ziar, T: On inverse full state hybrid function projective synchronization for continuous-time chaotic dynamical systems with arbitrary dimensions. Differ. Equ. Dyn. Syst. (2017). https://doi.org/10.1007/s12591-017-0362-x Google Scholar
- Ouannas, A, Grassi, G: A new approach to study the coexistence of some synchronization types between chaotic maps with different dimensions. Nonlinear Dyn. 86(2), 1319-1328 (2016) View ArticleMATHGoogle Scholar
- Ouannas, A: Co-existence of various synchronization-types in hyperchaotic maps. Nonlinear Dyn. Syst. Theory 16(3), 312-321 (2016) MathSciNetMATHGoogle Scholar
- Ouannas, A, Azar, AT, Vaidyanathan, S: New hybrid synchronization schemes based on coexistence of various types of synchronization between master-slave hyperchaotic systems. Int. J. Comput. Appl. Technol. 55(2), 112-120 (2017) View ArticleGoogle Scholar
- Pinto, CMA: Strange dynamics in a fractional derivative of complex-order network of chaotic oscillators. Int. J. Bifurc. Chaos 25(1), 1550003 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Ugurlu, E, Baleanu, D, Tas, K: Regular fractional differential equations in the Sobolev space. Fract. Calc. Appl. Anal. (2017). https://doi.org/10.1515/fca-2017-0041 MathSciNetMATHGoogle Scholar
- Cafagna, D, Grassi, G: On the simplest fractional-order memristor-based chaotic system. Nonlinear Dyn. 70(2), 1185-1197 (2012) MathSciNetView ArticleGoogle Scholar
- Cafagna, D, Grassi, G: Fractional-order systems without equilibria: the first example of hyperchaos and its application to synchronization. Chin. Phys. B 24(8), 080502 (2015) View ArticleGoogle Scholar
- Grigorenko, I, Grigorenko, E: Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91, 034101 (2003) View ArticleGoogle Scholar
- Li, CP, Deng, WH, Xu, D: Chaos synchronization of the Chua system with a fractional order. Physica A 360, 171-185 (2006) MathSciNetView ArticleGoogle Scholar
- Li, C, Chen, G: Chaos and hyperchaos in fractional order Rössler equations. Physica A 341, 55-61 (2004) MathSciNetView ArticleGoogle Scholar
- Lu, JG, Chen, G: A note on the fractional-order Chen system. Chaos Solitons Fractals 27, 685-688 (2006) View ArticleMATHGoogle Scholar
- Deng, WH, Li, CP: Chaos synchronization of the fractional Lü system. Physica A 353, 61-72 (2005) View ArticleGoogle Scholar
- Wu, GC, Baleanu, D, Huang, LL: Chaos synchronization of the fractional Rucklidge system based on new Adomian polynomials. J. Appl. Nonlinear Dyn. (2017). https://doi.org/10.5890/JAND.2017.09.006 MathSciNetGoogle Scholar
- Ouannas, A, Al-sawalha, MM, Ziar, T: Fractional chaos synchronization schemes for different dimensional systems with non-identical fractional-orders via two scaling matrices. Optik 127(20), 8410-8418 (2016) View ArticleGoogle Scholar
- Ouannas, A, Grassi, G, Ziar, T, Odibat, Z: On a function projective synchronization scheme for non-identical fractional-order chaotic (hyperchaotic) systems with different dimensions and orders. Optik 136, 513-523 (2017) View ArticleGoogle Scholar
- Ouannas, A, Abdelmalek, S, Bendoukha, S: Coexistence of some chaos synchronization types in fractional-order differential equations. Electron. J. Differ. Equ. 2017, 128 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Ouannas, A, Azar, AT, Vaidyanathan, S: A robust method for new fractional hybrid chaos synchronization. Math. Methods Appl. Sci. 40(5), 1804-1812 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Ouannas, A, Azar, AT, Vaidyanathan, S: A new fractional hybrid chaos synchronization. Int. J. Model. Identif. Control 27(4), 314-323 (2017) View ArticleMATHGoogle Scholar
- Caputo, M: Linear models of dissipation whose q is almost frequency independent. Geophys. J. R. Astron. Soc. 13, 529-539 (1967) MathSciNetView ArticleGoogle Scholar
- Oldham, KB, Spanier, J: The Fractional Calculus. Academic Press, New York (1974) MATHGoogle Scholar
- Podlubny, I: Fractional Differential Equations. Academic Press, New York (1999) MATHGoogle Scholar
- Samko, SG, Klibas, AA, Marichev, OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Amsterdam (1993) Google Scholar
- Matignon, D: Stability results of fractional differential equations with applications to control processing. IMACS, IEEE-SMC, Lille, France (1996) Google Scholar
- Gorenko, R, Mainardi, F: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A, Mainardi, F (eds.) Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York (1997) Google Scholar
- Li, Y, Chen, Y, Podlubny, I: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59, 1810-1821 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Chen, D, Zhang, R, Liu, X, Ma, X: Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks. Commun. Nonlinear Sci. Numer. Simul. 19, 4105-4121 (2014) MathSciNetView ArticleGoogle Scholar
- Li, C, Chen, G: Chaos and hyperchaos in fractional order Rössler equations. Physica A 341, 55-61 (2004) MathSciNetView ArticleGoogle Scholar
- Li, T-Z, Wang, Y, Luo, M-K: Control of fractional chaotic and hyperchaotic systems based on a fractional order controller. Chin. Phys. B 23(8), 080501 (2014) View ArticleGoogle Scholar
- Deng, W, Li, C, Lü, J: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn. 48, 409-416 (2007) MathSciNetView ArticleMATHGoogle Scholar