Synchronization of fractionalorder and integerorder chaotic (hyperchaotic) systems with different dimensions
 Xiaoyan Yang^{1},
 Heng Liu^{2}Email authorView ORCID ID profile and
 Shenggang Li^{1}
https://doi.org/10.1186/s1366201713994
© The Author(s) 2017
Received: 6 July 2017
Accepted: 12 October 2017
Published: 24 October 2017
Abstract
By constructing two scaling matrices, i.e., a function matrix \(\Lambda (t)\) and a constant matrix W which is not equal to the identity matrix, a kind of \(W\Lambda(t)\) synchronization between fractionalorder and integerorder chaotic (hyperchaotic) systems with different dimensions is investigated in this paper. Based on the fractionalorder Lyapunov direct method, a controller is designed to drive the synchronization error convergence to zero asymptotically. Finally, four numerical examples are presented to illustrate the effectiveness of the proposed method.
Keywords
1 Introduction
The fractional calculus theory, which is a generalization of the traditional integerorder calculus, can date back to 300 years ago. However, until recent 10 years, it has attracted increasing attention due to its popular use in the scientific fields and the engineeringoriented fields. Compared with the integer calculus, the fractional one can explain and handle many challenging problems more adequately and effectively [1–5].
Chaos synchronization is the dynamical process which means making two or more oscillators keep the same rhythms under a weak interaction [6]. Since Pecora and Carroll [7] proposed a pioneering method to synchronize two identical chaotic systems, synchronization of fractionalorder chaotic dynamical systems has gained a lot of popularity for its potential applications in secure communication and cryptography, telecommunication, signal and control processing, chaos synchronization [8–14]. Several types of synchronization techniques and methods, such as adaptive control, sliding mode control [15, 16], complete synchronization, projective synchronization (PS), and function projective synchronization (FPS) [17–19], have been proposed for fractionalorder dynamical systems. Among those existing synchronization methods, FPS, which has been introduced by Chen and Li [20, 21], was widely employed for synchronizing chaotic systems. Some scaling function matrices, which can be given with one’s need, are used in FPS. In fact, the scaling function matrix usually exhibits flexibility and unpredictability. By using error feedback control scheme, FPS of complex dynamical networks with or without external disturbances was discussed in [22]. Ref. [23] investigated adaptive switched modified FPS between two complex nonlinear hyperchaotic systems with unknown parameters. Ref. [24] discussed modified function projective combination synchronization of hyperchaotic systems.
It should be pointed out that in the above mentioned literature, synchronization of fractionalorder or integerorder chaotic systems was mainly discussed. Synchronization between fractionalorder and integerorder chaotic systems is widely perceived as contributing to generating hybrid chaotic transient signals, which are quite difficult to be decrypted in communication. Up to now, only a few works have been given to investigate this problem, for instance, by using the stability theory of fractionalorder linear system, Ref. [25] investigated modified general functional projective synchronization between a class of integerorder and fractionalorder chaotic systems. Ref. [26] discussed the dual projective synchronization between integerorder and fractionalorder chaotic systems (one can refer to [27–29] for more details). Actually, some dynamical systems usually have nonidentical dimensions. However, papers which have discussed the synchronization between fractionalorder and integerorder chaotic (hyperchaotic) systems with different dimensions are not common. Ref. [30] investigated adaptive generalized function matrix projective lag synchronization between fractionalorder and integerorder complex networks with delayed coupling and different dimensions. However, the controller in [30] has a very complicated form. Note that two scaling matrices (a function matrix and a nonunit constant matrix), which are more general than other scaling factors in FPS, have not been used to discuss the synchronization between fractionalorder and integerorder chaotic (hyperchaotic) systems with different dimensions. Besides, it is well known that the quadratic Lyapunov functions provide an important tool for stability analysis in the integerorder nonlinear systems. Therefore, how to use quadratic Lyapunov functions in the stability analysis of fractionalorder systems is meaningful.

Based on the Lyapunov direct method, the synchronization of fractionalorder and integerorder chaotic (hyperchaotic) systems with different dimensions is discussed by using a constant matrix and a function matrix.

With respect to different systems with nonidentical dimensions, different controllers are constructed to achieve \(W\Lambda(t)\) synchronization.
The rest of this paper is arranged as follows. In Section 2, some necessary theories and the mathematical models of fractionalorder and integerorder systems are given. The problem of \(W\Lambda(t)\) synchronization of fractionalorder and integerorder chaotic (hyperchaotic) systems is investigated in Section 3. In Section 4, the corresponding numerical simulations are presented to demonstrate the effectiveness of the main results. Finally, the conclusions are given in Section 5.
2 Preliminaries
2.1 Some related theories
Some necessary lemmas and properties of the Caputo fractional derivative operator are listed below. For convenience, we always assume that \(0 < \alpha<1\) in the rest of our paper.
Property 1
([31])
Theorem 1
([32])
Lemma 1
([33])
Lemma 2
([34])
2.2 Problem description
In this section, two cases will be considered.
Remark 1
Generally speaking, dimension l is an integer satisfying \(0< l\leq\max (m, n)\). For the convenience of our discussions, we will consider the conditions that \(l=m\) or \(l=n\).
Definition 1
Assumption 1
Assume that the scaling matrices W and \(\Lambda(t)=(\Lambda_{ks}(t)) \in R^{l\times m}\) are bounded, \(\Lambda_{ks}(t)\) are continuously differentiable and bounded functions or constants, and the derivatives of \(\Lambda_{ks}(t)\) (\(k=1, \ldots, l\); \(s=1, \ldots, m\)) are bounded.
Remark 2
 (1)
When \(l=n\), \(W\neq I\).
 (2)
When \(l=n\), \(W=I \in R^{n\times n}\), and \(\Lambda(t)=(\Lambda _{ks}(t)) \in R^{n\times m}\), our method is simplified to be FPS.
 (3)
When \(l=n\), \(W=I \in R^{n\times n}\), and \(\Lambda(t)=C \in R^{n\times m}\) is a nonzero constant matrix, our method is simplified to be PS.
 (4)
When \(m=n\), \(W=I \), \(\Lambda(t)=(\Lambda_{ks}(t)) \in R^{n\times n}\), our method is simplified to be FPS of chaotic systems with the same dimensions.
 (5)
When \(m=n\), \(W=I \), \(\Lambda(t)=I \in R^{n\times n}\), our method is simplified to be complete synchronization.
 (6)
When \(m=n\), \(W=I \), \(\Lambda(t)=\operatorname{diag}(d, \ldots, d) \in R^{n\times n}\) is a nonzero constant matrix, our method is simplified to be PS of chaotic systems with the same dimensions.
 (7)
When \(m=n\), \(W=I \), \(\Lambda(t)=I \in R^{n\times n}\), our method is simplified to be antiphase synchronization.
 (8)
Our method provides multiple selections. Both the drive system and the response system are related to the dimension of \(e(t)\), that is to say, \(e(t) \in R^{n} \) and \(e(t) \in R^{m} \) can be achieved simultaneously, and this will be shown in our simulation part. Therefore, for some complex dynamical systems, we can choose the smaller dimension to get better reduction results.
Remark 3
It follows from Remark 2 that the proposed synchronization method is more general than other kinds of scaling synchronization, and our results are also effective for synchronization between fractionalorder and integerorder chaotic or hyperchaotic systems with the same dimensions.
3 Synchronization controller design and stability analysis
In this section, we will construct the synchronization controllers with different dimensions.
3.1 Synchronization under case 1
3.1.1 \(l=n\)
Theorem 2
Proof
3.1.2 \(l=m\)
To proceed, the following assumption is needed.
Assumption 2
The controller component \(U_{i}(t)\) of controller \(U(t)\) is 0 for \(i=m+1, \ldots, n\).
3.2 Synchronization under case 2
3.2.1 \(l=n\)
3.2.2 \(l=m\)
Assumption 3
The control component \(H_{i}(t)\) of controller \(H(t)\) is 0 for \(i=m+1, \ldots, n\).
Theorem 5
The driveresponse system (9) will be synchronized in m dimension under Assumption 3 and the control matrix \(L_{2}\) if we design the controller as (23) and (25).
Remark 4
Since the proofs of Theorem 3, Theorem 4, and Theorem 5 are similar to that of Theorem 2, the processes will be omitted here.
Remark 5
Specially, to simplify calculations, the above feedback gain matrices \(Q_{1}\), \(Q_{2}\), \(L_{1}\), and \(L_{2}\) can be chosen such that their corresponding matrices \(P_{1}\), \(P_{2}\), \(T_{1}\), and \(T_{2} \) are diagonally positive definite.
Remark 6
For the above cases, we know that the asymptotical stability of the synchronization error systems is mainly decided by the above feedback gain matrices \(Q_{1}\), \(Q_{2}\), \(L_{1}\), and \(L_{2}\). The scaling matrices W and \(\Lambda(t)\) have no effect on the selection of these feedback gain matrices; consequently, if Definition 1 and Assumption 1 are satisfied, the corresponding positive definite matrices \(P_{1}\), \(P_{2}\), \(T_{1}\), and \(T_{2}\) will not change with the scaling matrices W and \(\Lambda(t)\). Therefore, according to certain chaotic (hyperchaotic) systems, one can focus on optimizing the construction of the controller \(U(t)\) to build the scaling matrices W and \(\Lambda (t)\). It should be pointed out that the continuously bounded functions \(\sin(t)\) and \(\cos(t)\) will display more excellent properties than other functions in the process of control. Based on Definition 1 and Assumption 1, for the purpose of getting better control performance, we usually employ functions \(\sin(t)\) and \(\cos(t)\) to construct the scaling function matrix \(\Lambda(t)\).
4 Numerical simulation
In this section, four numerical examples are presented to verify the effectiveness of our results.
4.1 Synchronization between integerorder Chen system and fractionalorder hyperchaotic Chen system
4.2 Synchronization between integerorder Rössler system and fractionalorder hyperchaotic Lorenz system
4.3 Synchronization between fractionalorder Rössler system and integerorder hyperchaotic Chen system
4.4 Synchronization between fractionalorder Lü system and integerorder hyperchaotic Lorenz system
5 Conclusions
In this paper, a kind of control approach about the synchronization of fractionalorder and integerorder chaotic (hyperchaotic) systems with different dimensions is proposed. To get new results, more simplified control schemes were designed by using two scaling matrices, and a quadratic Lyapunov function is used in the stability analysis of the synchronization error system. Finally, numerical simulations about the stabilization and synchronization problems of chaotic and hyperchaotic dynamical systems are used to testify the validity and usefulness of the proposed method.
Declarations
Acknowledgements
The authors are very grateful to the editor(s) and reviewers for giving valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant No. 11771263) and the Natural Science Foundation of Anhui Province of China (Grant No. 1508085QA16).
Authors’ contributions
All authors contributed equally. 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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