- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 191063, 10 pages

http://dx.doi.org/10.1155/2012/191063

## Projective Synchronization of *N*-Dimensional Chaotic Fractional-Order Systems via Linear State Error Feedback Control

^{1}Nonlinear Dynamics and Chaos Group, School of Management, Tianjin University, Tianjin 300072, China^{2}Center for Applied Mathematics, School of Economics and Management, Shandong University of Science and Technology, Qingdao 266510, China

Received 4 April 2012; Accepted 16 June 2012

Academic Editor: Her-Terng Yau

Copyright © 2012 Baogui Xin and Tong Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Based on linear feedback control technique, a projective synchronization scheme of *N*-dimensional chaotic fractional-order systems is proposed, which consists of master and slave fractional-order financial systems coupled by linear state error variables. It is shown that the slave system can be projectively synchronized with the master system constructed by state transformation. Based on the stability theory of linear fractional order systems, a suitable controller for achieving synchronization is designed. The given scheme is applied to achieve projective synchronization of chaotic fractional-order financial systems. Numerical simulations are given to verify the effectiveness of the proposed projective synchronization scheme.

#### 1. Introduction

The fractional calculus, as a very old mathematical topic, has been in existence for more than 300 years [1], but it has not been widely used in the science and engineering for many years, because its geometrical or physical interpretation has been not widely accepted [2, 3]. However, due to the long memory advantage, in the recent past, the fractional calculus has been widely applied to diffusion processes [4], Sprott chaotic systems [5], happiness and love [6], economics and finances [7, 8], and so on.

Chaos synchronization has been widely investigated in science and engineering such as humanistic community [9], physical science [10], and secure communications [11]. The chaos projective synchronization was first reported by Mainieri and Rehacek [12]. This type of projective synchronization is interesting due to its property of proportionally diminished or enlarged synchronizing responses, but the early work was limited to a certain kind of nonlinear systems with partly linear properties. Chaos projective synchronization has been an active research topic in nonlinear science until Wen and Xu [13, 14] proposed an observer-based control method and showed “no special limitation” to nonlinear systems themselves to achieve this type of chaos synchronization. Wen and coauthors tried to explore the potential applications of projective synchronization to noise reduction in mechanical engineering [15, 16] or design bifurcation solutions based on the property of projective synchronization [17]. Synchronization of fractional-order chaotic systems was first presented by Deng and Li [18]. There has been an increasing interest in fractional-order chaos synchronization during the last few years because of its potentials in both theory and applications [19]. Peng et al. [20] proposed the generalized projective synchronization scheme of fractional order chaotic systems via a transmitted signal. Shao [21] proposed a method to achieve general projective synchronization of two fractional order Rossler systems. Odibat et al. [22] studied synchronization of 3-dimensional chaotic fractional-order systems via linear control. The advantage of the linear feedback controller is that it is robust and linear, and moreover, it is easier to be designed and implemented for chaos synchronization than standard PID feedback controller, sliding mode controller, nonlinear feedback controller, and so on [23–25].

Huang and Li [26] reported an integer order financial model as follows: where is the interest rate, is the investment demand, is the price index, is the saving amount, is the cost per investment, is the demand elasticity of commercial markets, and all three constants .

Chen [7] considered the generalization of system (1.1) for the fractional-order model which takes the following form: where the th-order fractional derivative is given by the following Caputo definition, .

*Definition 1.1 (see [27]). *The th-order fractional derivative of function with respect to and the terminal value 0 is written as
where is an integer and satisfies .

*Remark 1.2. *If , the system (1.2) degenerates into the system (1.1).

The remainder of this paper is organized as follows. In Section 2, a projective synchronization scheme of *n*-dimensional chaotic fractional-order systems is proposed. In Section 3, a projective synchronization scheme of chaotic fractional-order financial systems is studied. In Section 4, the Adams-Bashforth-Moulton predictor-corrector scheme of a fractional-order system is described. Numerical simulations are given in Section 5 to show the effectiveness of the proposed synchronization scheme. Finally, the paper is concluded in Section 6.

#### 2. A Projective Synchronization Scheme of -Dimensional Chaotic Fractional-Order Systems

*Definition 2.1. *The projective synchronization discussed in this paper is defined as two relative chaotic dynamical systems can be synchronous with a desired scaling factor.

Consider a fractional-order chaotic system as follows:
where is an *n*-dimensional state vector of the system, is an linear constant matrix, is an linear constant matrix, is a continuous nonlinear vector function.

For the given system (2.1), one can construct the following new system
where is an *n*-dimensional state vector of the system, is a continuous nonlinear vector function, are linear constant matrix, is a desired scaling factor, is a linear state error feedback controller.

The synchronization error between the master system (2.1) and the slave system (2.2) is defined as
The linear state error feedback controller is defined as
where is an linear constant matrix.

Then the error system can be written as
where is an linear constant matrix. Obviously the orginal point is the equilibrium point of system (2.5).

According to the stability criterion of linear fractional-order dynamical system, one can directly obtain the following theorem.

Theorem 2.2. *If is an upper or lower triangular matrix and all eigenvalues satisfy , then the equilibrium point of synchronization error is asymptotically stable and , that is, the master system (2.1) and the slave system (2.2) achieve projective synchronization.*

*Remark 2.3. * If and , the above synchronization scheme is similar to the synchronization scheme in [28].

*Remark 2.4. * If and , the above synchronization scheme degenerates into the synchronization scheme proposed by Odibat et al. [22].

*Remark 2.5. * If , the above synchronization scheme degenerates into the synchronization scheme proposed by Xin et al. [29].

#### 3. A Projective Synchronization Scheme of Chaotic Fractional-Order Financial Systems

In order to investigate the synchronization behaviors of two chaotic fractional-order financial systems, one can set a master-slave configuration with a master system given by the fractional-order financial systems (1.2) and with a slave system (denoted by the subscript ) as follows: where have the same meanings as of system (1.2), is a desired scaling factor, are linear state error feedback controllers.

Proposition 3.1. * The drive system (1.2) and the slave system (3.1) will approach global synchronization for any initial condition if anyone of the following control laws holds:
**
where , and .*

*Proof. *The synchronization errors between the master system (1.2) and the slave system (3.1) are defined as , , . Subtracting (1.2) from (3.1) yields the error system as follows:
For the first control law in Proposition 3.1, substituting (3.2) into the error system (3.4), the system (3.4) can be rewritten as follows:
which has one equilibrium point at . Its Jacobian matrix evaluated at equilibrium point is given by
which is a lower triangular matrix and its eigenvalues satisfy .

For the second control law in Proposition 3.1, substituting (3.3) into the error system (3.4), the system (3.4) can be rewritten as follows:
which has one equilibrium point at . Its Jacobian matrix evaluated at equilibrium point is given by
which is an upper triangular matrix and its eigenvalues satisfy .

It follows from Theorem 2.2 that system (3.4) is asymptotically stable, that is, the master system (1.2) and the slave system (3.1) are synchronized finally.

The Proposition 3.1 is proved.

#### 4. Numerical Method for Solving System (1.2)

An improved Adams-Bashforth-Moulton predictor-corrector scheme [30] can be employed to solve fractional-order ordinary differential equations. The improved predictor-corrector scheme of system (1.2) can be described as follows.

With the initial value (), , system (1.2) is equivalent to the Volterra integral equations as follows:

Consider the uniform grid for some integers and , system (4.1) can be approximated to the following difference equations: where Errors of the above method are where .

#### 5. Numerical Simulations

Based on the Adams-Bashforth-Moulton predictor-corrector scheme, one can let the master system (1.2) and the slave system (3.1) with parameters , , , , , , , initial values , , , , , . The following numerical simulations are carried out to illustrate the main results.

From the first control law of Proposition 3.1, the linear controllers have the following form: , , . The chaotic attractors of the master system (1.2) and the slave system (3.1) are shown in Figure 1(a). Synchronization errors between systems (1.2) and (3.1) are shown in Figure 1(b). Time evolutions of , , , , and are shown in Figures 1(c)–1(e), respectively. From Figures 1(a)–1(e), it is clear that the projective synchronization is achieved for all these values.

From the second control law of Proposition 3.1, the linear controllers have the following form: , , . The chaotic attractors of the master system (1.2) and the slave system (3.1) are shown in Figure 2(a). Synchronization errors between systems (1.2) and (3.1) are shown in Figure 2(b). Time evolutions of , , , , , and are shown in Figures 2(c)–2(e), respectively. From Figures 2(a)–2(e), it is clear that the projective synchronization is achieved for all these values.

#### 6. Conclusions

In this paper, we propose a projective synchronization scheme of *n*-dimensional chaotic fractional-order systems via line error feedback control, and apply the scheme to achieve synchronization of the chaotic fractional-order financial systems. Numerical simulations validate the main results of this work.

#### Acknowledgment

This work was supported in part by Excellent Young Scientist Foundation of Shandong Province (Grant no. BS2011SF018), National Social Science Foundation of China (Grant no. 12BJY103), Humanities and Social Sciences Foundation of the Ministry of Education of China (Grant no. 11YJCZH200), and Research Project of “SUST Spring Bud” (Grant no. 2010AZZ067).

#### References

- B. Ross,
*A Brief History and Exposition of the Fundamental Theory of Fractional Calculus*, vol. 457 of*Lecture Notes in Mathematics*, Springer, New York, NY, USA, 1975. View at Zentralblatt MATH - R. Hilfer,
*Fractional Calculus in Physics*, World Scientific, Singapore, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. E. Gutiérrez, J. M. Rosário, and J. MacHado, “Fractional order calculus: basic concepts and engineering applications,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 375858, 19 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - X. Jiang, M. Xu, and H. Qi, “The fractional diffusion model with an absorption term and modified Fick's law for non-local transport processes,”
*Nonlinear Analysis. Real World Applications*, vol. 11, no. 1, pp. 262–269, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. H. Erjaee and M. Alnasr, “Phase synchronization in coupled Sprott chaotic systems presented by fractional differential equations,”
*Discrete Dynamics in Nature and Society*, vol. 2009, Article ID 753746, 10 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Song, S. Xu, and J. Yang, “Dynamical models of happiness with fractional order,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 15, no. 3, pp. 616–628, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. C. Chen, “Nonlinear dynamics and chaos in a fractional-order financial system,”
*Chaos, Solitons & Fractals*, vol. 36, no. 5, pp. 1305–1314, 2008. View at Publisher · View at Google Scholar · View at Scopus - B. G. Xin, T. Chen, and Y. Q. Liu, “Complexity evolvement of a chaotic fractional-order financial system,”
*Acta Physica Sinica*, vol. 60, no. 4, Article ID 048901, 2011. View at Google Scholar · View at Scopus - L. Song and J. Yang, “Chaos control and synchronization of dynamical model of happiness with fractional order,” in
*Proceedings of the 4th IEEE Conference on Industrial Electronics and Applications (ICIEA '09)*, pp. 919–924, May 2009. View at Publisher · View at Google Scholar · View at Scopus - B. Ratajska-Gadomska and W. Gadomski, “On control of chaos and synchronization in the vibronic laser,”
*Optics Express*, vol. 17, no. 16, pp. 14166–14171, 2009. View at Publisher · View at Google Scholar · View at Scopus - J. Cai, M. Lin, and Z. Yuan, “Secure communication using practical synchronization between two different chaotic systems with uncertainties,”
*Mathematical & Computational Applications*, vol. 15, no. 2, pp. 166–175, 2010. View at Google Scholar · View at Zentralblatt MATH - R. Mainieri and J. Rehacek, “Projective synchronization in three-dimensional chaotic systems,”
*Physical Review Letters*, vol. 82, no. 15, pp. 3042–3045, 1999. View at Publisher · View at Google Scholar · View at Scopus - G. L. Wen and D. Xu, “Observer-based control for full-state projective synchronization of a general class of chaotic maps in any dimension,”
*Physics Letters A*, vol. 333, no. 5-6, pp. 420–425, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - G. Wen and D. Xu, “Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems,”
*Chaos, Solitons & Fractals*, vol. 26, no. 1, pp. 71–77, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - G. Wen, Y. Lu, Z. Zhang, C. Ma, H. Yin, and Z. Cui, “Line spectra reduction and vibration isolation via modified projective synchronization for acoustic stealth of submarines,”
*Journal of Sound and Vibration*, vol. 324, no. 3-5, pp. 954–961, 2009. View at Publisher · View at Google Scholar · View at Scopus - G. Wen, S. Yao, Z. Zhang, et al., “Vibration control for active seat suspension system based on projective chaos synchronisation,”
*International Journal of Vehicle Design*, vol. 58, no. 1, pp. 1–14, 2012. View at Publisher · View at Google Scholar - G. Wen, “Designing Hopf limit circle to dynamical systems via modified projective synchronization,”
*Nonlinear Dynamics*, vol. 63, no. 3, pp. 387–393, 2011. View at Publisher · View at Google Scholar - W. H. Deng and C. P. Li, “Chaos synchronization of the fractional Lü system,”
*Physica A*, vol. 353, no. 1–4, pp. 61–72, 2005. View at Publisher · View at Google Scholar · View at Scopus - W. Deng, “Generalized synchronization in fractional order systems,”
*Physical Review E*, vol. 75, no. 5, Article ID 056201, 2007. View at Publisher · View at Google Scholar · View at Scopus - G. Peng, Y. Jiang, and F. Chen, “Generalized projective synchronization of fractional order chaotic systems,”
*Physica A*, vol. 387, no. 14, pp. 3738–3746, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - S. Shao, “Controlling general projective synchronization of fractional order Rossler systems,”
*Chaos, Solitons & Fractals*, vol. 39, no. 4, pp. 1572–1577, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Z. M. Odibat, N. Corson, M. A. Aziz-Alaoui, and C. Bertelle, “Synchronization of chaotic fractional-order systems via linear control,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 20, no. 1, pp. 81–97, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Xin, T. Chen, and Y. Liu, “Synchronization of chaotic fractional-order WINDMI systems via linear state error feedback control,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 859685, 10 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Wei, “Synchronization of coupled nonidentical fractional-order hyperchaotic systems,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 430724, 9 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. M. El-Dessoky and E. Saleh, “Generalized projective synchronization for different hyperchaotic dynamical systems,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 437156, 19 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Huang and H. Li,
*Theory and Method of the Nonlinear Economics*, Sichuan University Press, Chengdu, China, 1993. - I. Pdlubny,
*Fractional Differential Equations*, Academic Press, New York, NY, USA, 1999. - L. Chen, Y. Chai, and R. Wu, “Control and synchronization of fractional-order financial system based on linear control,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 958393, 21 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Xin, T. Chen, and Y. Liu, “Projective synchronization of chaotic fractional-order energy resources demand-supply systems via linear control,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 11, pp. 4479–4486, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Deng, “Numerical algorithm for the time fractional Fokker-Planck equation,”
*Journal of Computational Physics*, vol. 227, no. 2, pp. 1510–1522, 2007. View at Publisher · View at Google Scholar