- Research
- Open Access
A fractional-order form of a system with stable equilibria and its synchronization
- Xiong Wang^{1},
- Adel Ouannas^{2},
- Viet-Thanh Pham^{3}Email author and
- Hamid Reza Abdolmohammadi^{4}
https://doi.org/10.1186/s13662-018-1479-0
© The Author(s) 2018
- Received: 26 September 2017
- Accepted: 7 January 2018
- Published: 16 January 2018
Abstract
There has been an increasing interest in studying fractional-order chaotic systems and their synchronization. In this paper, the fractional-order form of a system with stable equilibrium is introduced. It is interesting that such a three-dimensional fractional system can exhibit chaotic attractors. Full-state hybrid projective synchronization scheme and inverse full-state hybrid projective synchronization scheme have been designed to synchronize the three-dimensional fractional system with different four-dimensional fractional systems. Numerical examples have verified the proposed synchronization schemes.
Keywords
- chaos
- equilibrium
- hidden attractor
- fractional order
- synchronization
MSC
- 34H10
- 26A33
- 34A08
1 Introduction
There has been a dramatic increase in studying chaos and systems with chaotic behavior in the past decades [1–3]. Applications of chaos have been witnessed in various areas ranging from path planning generator [4], secure communications [5–7], audio encryption scheme [8], image encryption [9–11], to truly random number generator [12, 13]. Many three-dimensional (3D) autonomous chaotic systems have been found and reported in the literature [14]. It has previously been observed that common 3D autonomous chaotic systems, such as Lorenz system [15], Chen system [16], Lü system [17], or Yang’s system [18, 19], have one saddle and two unstable saddle-foci. However, recent evidence suggests that chaos can be observed in 3D autonomous systems with stable equilibria [20, 21].
Several attempts have been made to investigate chaotic systems with stable equilibria. Yang and Chen proposed a chaotic system with one saddle and two stable node-foci [20]. In spite of the fact that the Yang-Chen system connected the original Lorenz system and the original Chen system, it was not diffeomorphic with the original Lorenz and Chen systems. Yang et al. found an unusual Lorenz-like chaotic system with two stable node-foci [21]. By using the center manifold theory and normal form method, Wei investigated delayed feedback on such a chaotic system with two stable node-foci [22]. A six-term system with stable equilibria was presented in [23]. Interestingly, the six-term system with stable equilibria exhibited a double-scroll chaotic attractor [23]. In addition, the generalized Sprott C system with only two stable equilibria was introduced in [24, 25]. It is worth noting that systems with stable equilibria are systems with ‘hidden attractors’ [26–29]. Hidden attractors have received considerable attention recently because of their roles in theoretical and practical problems [30–38].
Synchronization schemes used for fractional-order forms of chaotic systems with hidden attractors
System | Dimension | Synchronization scheme |
---|---|---|
[39] | 4 | adaptive sliding mode |
[40] | 4 | observer-based method |
[41] | 4 | one-way coupling |
[42] | 4 | feedback controller |
3 | observer-based method | |
[45] | 3 | unidirectional coupling |
[46] | 3 | generalized projective synchronization |
This work | 3 | full-state hybrid projective synchronization and inverse full-state hybrid projective synchronization |
The aim of this study is to examine the fractional-order form of a 3D system with stable equilibria and its full-state hybrid projective synchronization schemes. The model of the fractional system and its chaotic behavior are presented in Section 2. Different types of full-state hybrid projective synchronization schemes are investigated in Section 3. Section 4 presents results and discussions. Finally, the concluding remarks are drawn in the last section.
2 Fractional-order form of the system with stable equilibria
3 Synchronization schemes of the 3D fractional-order chaotic system
Among all types of chaos synchronization schemes, full-state hybrid projective synchronization (FSHPS) is one of the most noticeable types. It has been widely used in the synchronization of fractional chaotic (hyperchaotic) systems [68]. In this type of synchronization, each slave system state achieves synchronization with linear combination of master system states. Recently, an interesting scheme has been introduced [69], where each master system state synchronizes with a linear constant combination of slave system states. Since master system states and slave system states have been inverted with respect to the FSHPS, the new scheme has been called the inverse full-state hybrid projective synchronization (IFSHPS). Obviously, the problem of IFSHPS is more difficult than the problem of FSHPS. Studying the inverse problems of synchronization which produce new types of chaos synchronization is an attractive research topic. Based on fractional-order Lyapunov approach, this section first analyzes the FSHPS between the 3D fractional chaotic system (5) and a 4D fractional chaotic system with an infinite number of equilibrium points. Successively, the IFSHPS is proved between the 3D fractional chaotic system (5) and a 4D fractional chaotic system without equilibrium points.
3.1 Fractional-order Lyapunov method
Lemma 1
Lemma 2
([71])
3.2 Full-state hybrid projective synchronization (FSHPS) and inverse full-state hybrid projective synchronization (IFSHPS)
In the next, we present the definitions of FSHPS and IFSHPS for the master system (8) and the slave system (9).
Definition 1
Definition 2
3.3 FSHPS between the 3D fractional chaotic system and 4D fractional chaotic system with an infinite number of equilibrium points
Theorem 1
Proof
3.4 IFSHPS between the 3D fractional system and the 4D fractional chaotic system without equilibrium points
Theorem 2
Proof
4 Results and discussion
Figure 1 displays the chaotic attractor of three-dimensional autonomous system (1). Interestingly, system (1) has two stable equilibrium points, and attractors in this system are ‘hidden attractors’ because the basin of attraction for a hidden attractor is not connected with any unstable fixed point [27–29]. Recently, a hidden attractor has been discovered in different systems such as Chua’s system [27], model of drilling system [32], extended Rikitake system [36], and DC/DC converter [38]. The identification of hidden attractors in practical applications is important to avoid the sudden change to undesired behavior [29]. Previous research has established that derivatives are important in the field of mathematical modeling [51–54]. Fractional-order system (5) involving derivative orders is a generalization of autonomous system (1). Fractional system (5) exhibits chaos for the incommensurate orders \(q_{1}=0.98\), and \(q_{2}=q_{3}=0.99\) as shown in Figure 2. In addition, we have found that fractional system (5) also can generate chaotic behavior for commensurate orders.
Researchers have shown an increased interest in control and synchronization of fractional-order systems [55–57, 68–70]. We have studied the synchronization of fractional systems with different orders. Figure 3 illustrates the FSHPS between the considered 3D fractional chaotic system (5) and the 4D fractional chaotic system with an infinite number of equilibrium points. Figure 4 indicates the IFSHPS between the introduced 3D fractional system and the 4D fractional chaotic system without equilibrium. The complexity of proposed synchronization schemes can be used in secure communication and chaotic encryption schemes. In our future works, we will use the recent stability results of fractional systems [55–57] for discrete fractional systems and their synchronization.
5 Conclusions
In this work, the fractional-order form of a 3D chaotic system with two stable equilibrium points has been studied. Remarkably, the fractional system can display chaotic behavior. Moreover, we have studied two types of synchronization for such a 3D fractional system: full-state hybrid projective synchronization and inverse full-state hybrid projective synchronization. By using the proposed synchronization schemes, we have obtained the synchronization between the 3D fractional-order system and the 4D fractional-order system with an infinite number of equilibrium points as well as the 4D fractional-order system without equilibrium. Further studies regarding practical applications of this fractional system will be carried out in our next works.
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
XW and VTP suggested the model, helped in results interpretation and manuscript evaluation. AO and HRA helped to evaluate, revise, and edit the manuscript. XW and HRA 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
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