- Research
- Open Access
Projective synchronization of fractional-order delayed neural networks based on the comparison principle
- Weiwei Zhang^{1}Email authorView ORCID ID profile,
- Jinde Cao^{2, 3},
- Ranchao Wu^{4},
- Ahmed Alsaedi^{5} and
- Fuad E. Alsaadi^{6}
https://doi.org/10.1186/s13662-018-1530-1
© The Author(s) 2018
- Received: 12 October 2017
- Accepted: 15 February 2018
- Published: 27 February 2018
Abstract
This paper considers projective synchronization of fractional-order delayed neural networks. Sufficient conditions for projective synchronization of master–slave systems are achieved by constructing a Lyapunov function, employing a fractional inequality and the comparison principle of linear fractional equation with delay. The corresponding numerical simulations demonstrate the feasibility of the theoretical result.
Keywords
- Fractional order
- Comparison principle
- Projective synchronization
- Time-delay
1 Introduction
Neural networks have attracted great attention due to their wide applications, including the signal processing, parallel computation, optimization, and artificial intelligence. The dynamical behaviors of neural networks have been widely studied, particularly synchronization, which is one of the most important topics and therefore has been given much attention [1–6]. However, the majority of existing results considered modeling integer-order neural networks.
It is well known that fractional calculus is the generalization of integer-order calculus to arbitrary order. Compared to classical integer-order models, fractional-order calculus offers an excellent instrument for the description of memory and hereditary properties of dynamical processes. The existence of infinite memory can help fractional-order models better describe the system’s dynamical behaviors as illustrated in [7–23]. Taking these factors into consideration, fractional calculus was introduced to neural networks forming fractional-order neural networks, and some interesting results on synchronization were demonstrated [24–29]. Among all kinds of synchronization, projective synchronization, in which the master and slave systems are synchronized up to a scaling factor, is an important concept in both theoretical and practical manners. Recently, some results with respect to projective synchronization of fractional-order neural networks were considered [30–32]. In [30], projective synchronization for fractional neural networks was studied. Through the employment of a fractional-order differential inequality, the projective synchronization of fractional-order memristor-based neural networks was shown in [31]. By using an LMI-based approach, the global Mittag–Leffler projective synchronization for fractional-order neural networks was investigated in [32].
However, time delay, which is unavoidable in biological, engineering systems, and neural networks, was not taken into account in most of the previous works. To the best of our knowledge, projective synchronization of fractional-order neural networks was previously investigated at the presence of time delay through the use of Laplace transform [33], and no special Lyapunov functions were derived for synchronization analysis. In this paper, new methods are introduced to investigate the projective synchronization of fractional-order delayed neural networks. The study includes constructing a Lyapunov function, applying a fractional inequality and the comparison principle of linear fractional equation with delay, and obtaining new sufficient conditions.
The rest of this article is organized as follows. In Sect. 2, some definitions and lemmas are introduced, and the model description is given. In Sect. 3, the projective synchronization schemes are presented, and sufficient conditions for projective synchronization are obtained. Numerical simulations are presented in Sect. 4. Conclusions are drawn in Sect. 5.
2 Preliminaries and model description
It has to be noted that Riemann–Liouville fractional derivative and Caputo fractional derivative are the most commonly used among all the definitions of fractional-order integrals and derivatives. Due to the advantages of the Caputo fractional derivative, it is adopted in this work.
Definition 1
([7])
Definition 2
([7])
For generalities, the following definition, assumption, and lemmas are presented.
Definition 3
If there exists a nonzero constant β such that, for any two solutions \(x(t)\) and \(y(t)\) of systems (1) and (3) with different initial values, one can get \(\lim_{t\rightarrow\infty} \Vert y(t)-\beta x(t) \Vert =0\), then the master system (1) and the slave system (3) can achieve globally asymptotically projective synchronization, where \(\Vert \cdot \Vert \) denotes the Euclidean norm of a vector.
Assumption 1
Lemma 1
([32])
When \(P=E\) is an identity matrix, then \(\frac{1}{2}D^{\alpha }[x^{T}(t)x(t)]\leq x^{T}(t)D^{\alpha}x(t)\).
Lemma 2
([34])
Lemma 3
([34])
3 Projective synchronization
In this section, master–slave projective synchronization of delayed fractional-order neural networks is discussed. The aim is to design a suitable controller to achieve the projective synchronization between the slave system and the master system.
Let \(e_{i}(t)=y_{i}(t)-\beta x_{i}(t)\) (\(i=1, 2, \ldots, n\)) be the synchronization errors.
Remark 1
The control function \(u_{i}(t)\) is a hybrid control, \(v_{i}(t)\) is an open loop control, and \(w_{i}(t)\) is a linear control.
Theorem 1
Proof
Remark 2
Remark 3
Remark 4
The control function \(u_{i}(t)\) is a hybrid control, \(v_{i}(t)\) is an open loop control, and \(w_{i}(t)\) is an adaptive feedback control.
Remark 5
Let \(d_{i}(0)\geq0\), then \(d_{i}(t)=d_{i}(0)+I^{\alpha}(\gamma_{i} \Vert y_{i}(t)-\beta x_{i}(t) \Vert ^{2})\geq d_{i}(0)\). So it is easy to get \(d_{i}(t)\geq0\).
Theorem 2
Proof
Remark 6
Remark 7
Remark 8
By using an LMI-based approach, Wu et al. investigated global Mittag–Leffler projective synchronization for fractional-order neural networks [32], but without considering delay.
Remark 9
In [33], by using the Laplace transform, the hybrid projective synchronization of fractional-order memristor-based neural networks with time delays was discussed, but the theoretical synchronization results are poor and the sufficient conditions are complex. For comparison purposes, in this paper, the projective synchronization of fractional-order delayed neural networks is studied by constructing a Lyapunov function, with the employment of a fractional inequality and the comparison principle of linear fractional equation with delay. The results are simpler and more theoretical.
4 Numerical simulations
Remark 10
In simulations, the projective coefficient β is a nonzero constant, which is selected arbitrarily.
5 Conclusions
In this paper, the projective synchronization of delayed fractional-order neural networks is investigated. In order to obtain general results, an effective controller is designed, a fractional inequality and the comparison principle of linear fractional equation with delay are implemented, and some sufficient conditions are given to ensure that the master–slave systems are able to obtain projective synchronization. Numerical simulations are used to show the effectiveness of the method proposed.
Declarations
Acknowledgements
This study is supported by the National Natural Science Foundation of China (No. 11571016), the Natural Science Foundation of Anhui Province (No. 1608085MA14), and the Natural Science Foundation of the Higher Education Institutions of Anhui Province (No. KJ2015A152).
Authors’ contributions
All authors contributed equally to the manuscript. All the 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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