- Research
- Open Access
Synchronization results for a class of fractional-order spatiotemporal partial differential systems based on fractional Lyapunov approach
- Adel Ouannas^{1},
- Xiong Wang^{2},
- Viet-Thanh Pham^{3}Email author,
- Giuseppe Grassi^{4} and
- Van Van Huynh^{5}
- Received: 4 December 2018
- Accepted: 7 April 2019
- Published: 17 April 2019
Abstract
In this paper, the problem of synchronization of a class of spatiotemporal fractional-order partial differential systems is studied. Subject to homogeneous Neumann boundary conditions and using fractional Lyapunov approach, nonlinear and linear control schemes have been proposed to synchronize coupled general fractional reaction–diffusion systems. As a numerical application, we investigate complete synchronization behaviors of coupled fractional Lengyel–Epstein systems.
Keywords
- Fractional calculus
- Synchronization
- Fractional Lyapunov approach
- Reaction–diffusion systems
- Lengyel–Epstein systems
MSC
- 35Axx
- 34A08
- 34D06
1 Introduction
The phenomenon of synchronization has attracted the interest of many researchers from various fields due to its potential applications in nonlinear sciences [1]. Synchronization is the process of controlling the output of a dynamical slave system in order to force its variables to match those of a corresponding master system in time [2]. Various kinds of control schemes have been introduced in the past to synchronize dynamical systems such as complete (anti-) synchronization [3], lag synchronization [4], function projective synchronization [5], generalized synchronization [6], and Q-S synchronization [7]. Recently, the topic of synchronization between dynamical systems described by fractional-order differential equations started to attract increasing attention [8–11].
Most of the research efforts have been devoted to the study of synchronization problems in low-dimensional nonlinear dynamical systems. Synchronizing high-dimensional systems in which state variables depend on not only the time but also the spatial position remains a challenge. These high-dimensional systems are generally modeled in spatial-temporal domain by partial differential systems. Recently, the search for synchronization has moved to high-dimensional nonlinear dynamical systems [12–15]. Over the last years, some studies have investigated synchronization of spatially extended systems demonstrating spatiotemporal chaos such as the works presented in [16–18]. Reaction–diffusion systems have shown important roles in modeling various spatiotemporal patterns that arise in chemical and biological systems [19, 20]. Reaction–diffusion systems can describe a wide class of rhythmic spatiotemporal patterns observed in chemical and biological systems, such as circulating pulses on a ring, oscillating spots, target waves, and rotating spirals. Synchronization dynamics of reaction–diffusion systems has been studied in [21, 22] using phase reduction theory. It has been shown that reaction–diffusion systems can exhibit synchronization in a similar way to low-dimensional oscillators. The effect of time-delay autosynchronization on uniform oscillations in a reaction–diffusion system has been presented in [23]. Furthermore, generalized synchronization [24], an approach based on semi-group theory [25, 26], functional spaces approach [27], the backstepping synchronization approach [28], the graph-theoretic synchronization approach [29], biological signal transmission using synchronous control [30], pinning impulsive synchronization [31], impulsive type synchronization [32], and hybrid adaptive synchronization strategy [33] for coupled reaction–diffusion systems have been introduced. To the best of our knowledge, the study of synchronization behaviors for fractional-order reaction–diffusion systems remains to this day a new and mostly unexplored field. This has motivated us to examine the phenomenon and develop suitable synchronization control laws.
This work presents a novel contribution to the topic of synchronization in some class of fractional-order spatiotemporal partial differential systems. The main aim of the present paper is to study the problem of complete synchronization in coupled fractional reaction–diffusion systems. By using fractional Lyapunov approach, nonlinear and linear control laws have been proposed to realize complete synchronization for general fractional reaction–diffusion systems. Synchronization behaviors of coupled fractional-order Lengyel–Epstein systems are obtained to demonstrate the effectiveness and feasibility of the proposed control techniques. The remainder of this paper is organized as follows: Sect. 2 illustrates some basic concepts on fractional calculus. In Sect. 3, we present two different synchronization schemes that cover two cases: nonlinear scheme and linear scheme. Finally, in order to show the applicability of the developed schemes, Sect. 4 considers the synchronization of coupled fractional Lengyel–Epstein systems. Concluding remarks are given in Sect. 5.
2 Basic concepts
Before delving into the main results and proofs of the study at hand, it is important to list some key definitions and results that will be useful at later stages.
Definition 1
Definition 2
Theorem 1
([36])
Lemma 1
([37])
\(\forall t>0\): \(\frac{1}{2}D_{t}^{p} ( X ^{T}(t)X(t) ) \leq X^{T}(t)D_{t}^{p} ( X(t) ) \).
3 Main results
3.1 Nonlinear control law
In this subsection, we outline the problem of controlling the coupled master and slave systems given in Eqs. (4) and (5) using nonlinear controllers.
Theorem 2
Proof
3.2 Linear control law
Theorem 3
Proof
4 Numerical applications
4.1 Nonlinear case
4.2 Linear case
5 Discussion and conclusion
The paper investigates, based on the fractional Lyapunov approach and using the master–slave concept, the synchronization control for a class of fractional spatiotemporal partial differential systems. First, a spatial-time coupling protocol for the synchronization is suggested, then novel control methods that include nonlinear and linear controllers are proposed to realize complete synchronization between coupled fractional-order reaction–diffusion systems. In both cases, the proposed control schemes stabilize the synchronization error states where the zero solution of the error system becomes globally asymptotically stable.
Suitable sufficient conditions for achieving synchronization of coupled fractional Lengyel–Epstein systems via suitable nonlinear and linear controllers applied to the slave are derived. As a result, from the performed numerical simulations, using the Matlab function “q-Homotopy Analysis Transform algorithm”, we can observe that the addition of the designed nonlinear and linear controllers to the controlled fractional Lengyel–Epstein system updates the coupled systems dynamics such that the system states become synchronized. Comparing the numerical simulations shown in Figs. 3, 4, 5, and 6, we can easily observe that the linear control scheme realizes synchronization faster than the nonlinear case. Also, the nonlinear control scheme requires the removal of nonlinear terms from the slave system, which may increase the cost of the controllers. So, the cost of the controllers in the nonlinear case is higher than that in the linear case.
The study confirms that the problem of complete synchronization in coupled high-dimensional fractional-order spatiotemporal systems can be realized using nonlinear and linear controllers. Also, we can easily see that the research results obtained in this paper can be extended to many other types of fractional spatiotemporal systems with reaction–diffusion terms.
Declarations
Acknowledgements
The authors acknowledge Prof. GuanRong Chen, Department of Electronic Engineering, City University of Hong Kong for suggesting many helpful references.
Availability of data and materials
Not applicable.
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, XW, and GG suggested the model, helped in result interpretation and manuscript evaluation. VTP, GG, and VVH helped to evaluate, revise, and edit the manuscript. XW, GG, and VVH 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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