Abstract and Applied Analysis

Volume 2014, Article ID 675840, 10 pages

http://dx.doi.org/10.1155/2014/675840

## A Novel Scheme Adaptive Hybrid Dislocated Synchronization for Two Identical and Different Memristor Chaotic Oscillator Systems with Uncertain Parameters

^{1}College of Automation, Huazhong University of Science and Technology, Hubei 430074, China^{2}School of Science, Hubei University of Technology, Wuhan 430068, China^{3}School of Information Science and Engineering, Central South University, Changsha 410012, China

Received 12 July 2013; Revised 11 December 2013; Accepted 15 December 2013; Published 27 January 2014

Academic Editor: Chengjian Zhang

Copyright © 2014 Jie Chen et al. 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

The drive system can synchronize with the response system by the scaling factor in the traditional projective synchronization. This paper proposes a novel adaptive hybrid dislocated synchronization with uncertain parameters scheme for chaos synchronization using the Lyapunov stability theory. The drive system is synchronized by the sum of hybrid dislocated state variables for the response system. By designing effective hybrid dislocated adaptive controller and hybrid dislocated adaptive law of the parameters estimation, we investigate the synchronization of two identical memristor chaotic oscillator systems and two different memristor chaotic oscillator systems with uncertain parameters. Finally, the numerical simulation examples are provided to show the effectiveness of our method.

#### 1. Introduction

A chaotic system has complex dynamical behaviors that possess some special features such as being extremely sensitive to tiny variations of initial conditions and having bounded trajectories in the phase space with a positive leading Lyapunov exponent and so on. Pecora and Carroll [1] have realized chaos synchronization in 1990, many types of synchronization methods have been investigated in the past 10 years. Based on the Lyapunov stability theory, some kinds of synchronization have been intensively studied and a lot of theoretical results have been obtained, such as complete synchronization [1], partial synchronization [2], anti-synchronization [3], generalized synchronization [4–7], phase synchronization [8], anti-phase synchronization [9], lag synchronization [10], projective synchronization [11–14], time scale synchronization [15], combination synchronization [16–19], and compound synchronization [20]. In this period, several theoretical methods have been developed to realize chaos synchronization such as OGY method [21], feedback control method [22–24], active control method [25], backstepping method [26], adaptive control method [27], sliding mode control method [28], impulsive control method [29, 30], coupling control method [31], and observer control method [32], and so on.

In recent years, projective synchronization received many attractions, which characterizes that the state vectors of synchronized systems become proportional with a scaling factor. Mainieri and Rehacek have studied projective synchronization in coupled partially linear chaotic systems such as the Lorenz system [11]. Projective synchronization in general class of chaotic systems including nonpartially-linear chaotic system has been achieved with nonlinear observer control [33]. More recently, a new synchronization method referred to as “modified projective synchronization” has been proposed in [34, 35], where the chaotic systems can synchronize up to a constant scaling matrix. The above projective synchronization is confined to three-dimensional chaotic systems, the projective synchronization problem for a class of four-dimensional chaotic systems is concerned [36, 37].

The above methods realize the projective synchronization whose scaling factor is a constant or a function for the corresponding state variable. Xu et al. has realized the general hybrid projective dislocated synchronization between two chaotic nonlinear systems, which includes complete dislocated synchronization, dislocated antisynchronization, and projective dislocated synchronization as its special items [38]. The transmitted signals are such complex and unpredictable that they may have stronger antiattack ability and antitranslated capability than that transmitted by the usual transmission model. In our paper, the drive system is synchronized by the sum of hybrid dislocated state variables for the response system. What is more, the memristor chaotic oscillator system is a new four-dimensional (4D) autonomous chaotic system and can produce novel rich and complex dynamic activities, which is different from the traditional chaotic systems for the unique memory of the memristor initial state. Motivated by the existing works, we focus on not only the identification of parameters but also the novel adaptive hybrid dislocated control synchronization.

In this paper, the problem of chaos synchronization to memristor chaotic oscillator system with uncertain parameters is considered. At first, we give a general scheme description for synchronization with uncertain parameters between two identical and two different chaotic systems. Then the chaos synchronization of the systems is proved by the Lyapunov stability theory. Finally, the numerical simulation examples are given to show the effectiveness of our method.

#### 2. Chaos Synchronization

The section will discuss hybrid dislocated adaptive method to achieve synchronization for memristor chaotic oscillator systems with uncertain parameters. Synchronization between two identical chaotic systems and two different chaotic systems are considered, respectively.

##### 2.1. Chaos Synchronization between Two Identical Chaotic Systems

A drive system is given by and the corresponding response system is written by where and are state vectors, and are two continuous functions, the estimated parameter vector is for the parameter vector , and is the control law to be constructed.

*Definition 1. *If there exist non-zero constants , the following condition
can hold true, such that the drive system (1) and the corresponding response system (2) have realized hybrid projective complete dislocated synchronization with uncertain parameters.

Specifically, the error dynamics system of general hybrid projective complete dislocated synchronization with uncertain parameters is given by Lyapunov function is designed as follows: where and are the positive definite constant matrices. The time derivative of along the trajectories of (5) is written by

If there exists suitable feedback control law and constants to make hold true, based on the Lyapunov stability theorem, the drive system (1) and the response system (2) have completed general hybrid projective complete dislocated synchronization with uncertain parameters.

*Remark 2. *If , the rest , then hybrid projective complete dislocated synchronization will become into complete dislocated synchronization. If , the rest , then hybrid projective complete dislocated synchronization will become into dislocated anti-synchronization. If (, is constant, , ), the rest , then hybrid projective complete dislocated synchronization will become into projective dislocated synchronization.

###### 2.1.1. Main Results

Memristor chaotic oscillator system [39] is described by where , , , and are state variables of the drive system (7), and , , , and are the real constants of the drive system (7). is a piecewise linear function of the form: where is state variable. is given as the following expression:

Actually, system (7) shows chaotic when , , , and as shown in Figure 1. Two identical memristor chaotic oscillator systems are given, where the drive system with the variable drives the response system having identical equations denoted the variable . The drive system is (7), the response system is described by where , , , and are parameters of the response system (10) which needs to be estimated , , , and are the control laws to assure that two chaotic systems can be synchronized.

Let where are constants, and The control law is given as follows:

The update rules for unknown parameters , , , and are given as follows:

Theorem 3. *The drive system (7) and response system (10) can complete the hybrid projective complete dislocated synchronization for any initial conditions and via the control law (15) and the update law (16).*

*Proof. *The error dynamics can be gained as follows:
Substituting (12), (13), and (15) into (17), we get
Substituting (7), (10), and (14) into (18), it is easy to gain the error dynamics as follows:
The following Lyapunov candidate is chosen by
where
Then the differential of the Lyapunov function along the trajectory of error system (19) is
Since is negative semidefinite, we cannot immediately obtain that the origin of error system (19) is asymptotically stable. In fact, as , then and . From the error system (10), we get , . So we have
where . Thus and . According to the Barbalat’s lemma, we have and . Therefore, the response system (10) synchronizes the drive system (7) by the controller (15). This completes the proof.

*Remark 4. *A chaotic system with unknown parameters cannot be known in advance, but it has a given geometric topology structure. The control laws , , , and are too complex to realize the synchronization control; the reasonable design of , , , and are the key factors to a successful method.

###### 2.1.2. Simulation and Results

In the numerical simulations, the fourth-order Runge-Kutta method is applied to solve the systems with time step size 0.001. It is assumed that the initial condition, and are employed. Parameters are designed as , , , , , , , , , , , . Synchronization of the systems (7) and (10) by hybrid dislocated adaptive control law (15) and (16) with the initial estimated parameters , , , and are displayed in Figures 2 and 3. Figure 2 displays synchronization errors of systems (7) and (10). Figure 3 shows that the estimated values of the unknown parameters , , , and that can converge to , , , and .

*Remark 5. *In the simulation, parameters are chosen to make the following condition hold true.

##### 2.2. Chaos Synchronization between Two Different Systems

A drive system is described by and the corresponding response system is defined as follows: where and are state vectors, and are two continuous functions, the estimated parameter vectors of the vectors and are and , and is a control law to be designed.

*Definition 6. *For the drive system (24) and response system (25), let the vector error state be
where , , , , and .

Suppose

Two kinds of cases (I) and (II) are discussed for the further research.

*Case I (**)*. The order of the drive system is not lower than that of the response system. When , , , , , and , such that the system (24) and system (25) are general hybrid projective complete dislocated synchronization with uncertain parameters.

*Case II (**)*. The order of the drive system is lower than that of the response system. When , , , , , and , such that the system (24) and system (25) are general hybrid projective complete dislocated synchronization with uncertain parameters.

Then there exists suitable feedback control law and , , so as to

In the following, we will give a principle to find suitable feedback control law such that the two chaotic systems are hybrid projective complete dislocated synchronization with uncertain parameters. Construct a dynamical Lyapunov function: where , , and are the positive definite constant matrices. The time derivative of along the trajectories of (29) is

A reasonable control law is designed such that is negative definite. Then based on the Lyapunov’s function method, the general hybrid projective complete dislocated synchronization with uncertain parameters of chaotic systems (24) and (25) is realized by the given designed feedback control law .

###### 2.2.1. Main Results

Here, an example is given to show the effectiveness of above method. The memristor chaotic oscillator system described by (7) drives the other memristor chaotic oscillator system [40]: where the memristor chaotic oscillator system exhibits a chaotic attractor at parameters , , , and as Figure 4.

Let where are real constants.

Let the control law be as follows:

Let

The update laws for unknown parameters , , , , , , , and are given as follows:

Theorem 7. *The drive system (7) and the response system (31) can realize the hybrid projective complete dislocated synchronization for any initial conditions and by the control law (33) and the update laws (37).*

*Proof. *It is easy to see that the error dynamics can be obtained as follows:
Substituting (7) and (31) into (38), we have
Substituting (33), (34), and (35) into (39), we get
Substituting (7), (31), and (36) into (40), it is easy to gain the error dynamics as follows:
The following Lyapunov candidate is chosen:
where

Then, the differential of the Lyapunov function along the trajectory of error system (32) is gained by
Since is negative semidefinite, we cannot immediately obtain that the origin of error system (32) is asymptotically stable. In fact, as , then , , and . From the error system (32), we have , , and . Since , then we have
where . Thus , , and . According to the Barbalat’s lemma, we have , and . Therefore, the response system (31) synchronizes the drive system (7) by the control law (33). This completes the proof.

###### 2.2.2. Simulation and Results

In the numerical simulations, the fourth-order Runge-Kutta method is also used to solve the systems with time step size 0.001. The initial condition, and , are employed. Parameters are chosen as , , , , , , , , , , , , , , , and . Synchronization of the systems (7) and (31) the control law (33) and the update laws (37) with the initial estimated parameters , , , and and , , , and , are shown in Figures 5–7. Figure 5 displays synchronization errors of systems (7) and (31). Figures 6 and 7 show that the estimated values of the unknown parameters , , , and and , , , and of the unknown parameters converge to , , , and and , , , and , respectively.

*Remark 8. *In the simulations, are chosen to make hold true.

#### 3. Conclusion

In this paper, based on adaptive synchronization and general hybrid projective dislocated synchronization, we propose a novel hybrid dislocated adaptive synchronization scheme for asymptotic chaos synchronization using the Lyapunov stability theory. Complete dislocated synchronization, dislocated anti-synchronization, projective dislocated synchronization, and parameter identification are considered as its special items. In this way, we investigate the synchronization between two identical memristor chaotic oscillator systems and two different memristor chaotic oscillator systems with four uncertain parameters. Finally, two numerical simulation examples are provided to show the effectiveness and correctness of our method.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work was supported in part by the National Natural Science Foundation of China under Grants 61170031, 61272114, 61370039, and 61373041.

#### References

- L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,”
*Physical Review Letters*, vol. 64, no. 8, pp. 821–824, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z.-M. Ge and Y.-S. Chen, “Synchronization of unidirectional coupled chaotic systems via partial stability,”
*Chaos, Solitons and Fractals*, vol. 21, no. 1, pp. 101–111, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Fu and Z. Li, “Robust adaptive anti-synchronization of two different hyperchaotic systems with external uncertainties,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 1, pp. 395–401, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. He and J. Cao, “Generalized synchronization of chaotic systems: an auxiliary system approach via matrix measure,”
*Chaos*, vol. 19, no. 1, Article ID 013118, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - A. A. Koronovskii, O. I. Moskalenko, and A. E. Hramov, “On the use of chaotic synchronization for secure communication,”
*Physics-Uspekhi*, vol. 52, no. 12, pp. 1213–1238, 2009. View at Publisher · View at Google Scholar · View at Scopus - A. A. Koronovskii, O. I. Moskalenko, and A. E. Hramov, “Hidden data transmission using generalized synchronization in the presence of noise,”
*Technical Physics*, vol. 55, no. 4, pp. 435–441, 2010. View at Publisher · View at Google Scholar · View at Scopus - P. K. Roy, C. Hens, I. Grosu, and S. K. Dana, “Engineering generalized synchronization in chaotic oscillators,”
*Chaos*, vol. 21, no. 1, Article ID 013106, 2011. View at Publisher · View at Google Scholar · View at Scopus - M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,”
*Physical Review Letters*, vol. 76, no. 11, pp. 1804–1807, 1996. View at Publisher · View at Google Scholar · View at Scopus - M.-C. Ho, Y.-C. Hung, and C.-H. Chou, “Phase and anti-phase synchronization of two chaotic systems by using active control,”
*Physics Letters A*, vol. 296, no. 1, pp. 43–48, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - S. K. Bhowmick, P. Pal, P. K. Roy, and S. K. Dana, “Lag synchronization and scaling of chaotic attractor in coupled system,”
*Chaos*, vol. 22, no. 2, Article ID 023151, 2012. View at Google Scholar - 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 - L. Kocarev and U. Parlitz, “Synchronizing spatiotemporal chaos in coupled nonlinear oscillators,”
*Physical Review Letters*, vol. 77, no. 11, pp. 2206–2209, 1996. View at Publisher · View at Google Scholar · View at Scopus - G.-H. Li, “Modified projective synchronization of chaotic system,”
*Chaos, Solitons and Fractals*, vol. 32, no. 5, pp. 1786–1790, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Sun, Y. Shen, G. Zhang, G. Cui, and Y. Wang, “General hybrid projective complete dislocated synchronization with non-derivative and derivative coupling based on parameter identification in several chaotic and hyperchaotic systems,”
*Chinese Physics B*, vol. 22, no. 4, Article ID 040508, 2013. View at Google Scholar - A. E. Hramov and A. A. Koronovskii, “Time scale synchronization of chaotic oscillators,”
*Physica D*, vol. 206, no. 3-4, pp. 252–264, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Runzi, W. Yinglan, and D. Shucheng, “Combination synchronization of three classic chaotic systems using active backstepping design,”
*Chaos*, vol. 21, no. 4, Article ID 043114, 2011. View at Publisher · View at Google Scholar · View at Scopus - R. Luo and Y. Wang, “Active backstepping-based combination synchronization of three different chaotic systems,”
*Advanced Science, Engi- Neering and Medicine*, vol. 4, no. 2, pp. 142–147, 2012. View at Publisher · View at Google Scholar - J. Sun, Y. Shen, G. Zhang, C. Xu, and G. Cui, “Combination-combination synchronization among four identical or different chaotic systems,”
*Nonlinear Dynamics*, vol. 73, no. 3, pp. 1211–1222, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - J. Sun, Y. Shen, X. Wang, and J. Chen, “Finite-time combination-combination synchronization of four different chaotic systems with unknown parameters via sliding mode control,”
*Nonlinear Dynamics*, 2013. View at Publisher · View at Google Scholar - J. Sun, Y. Shen Y Quan, and C. Xu, “Compound synchronization of four memristor chaotic oscillator systems and secure communication,”
*Chaos*, vol. 23, no. 1, Article ID 013140, 2013. View at Google Scholar - E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,”
*Physical Review Letters*, vol. 64, no. 11, pp. 1196–1199, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Li, J. Yu, and Z. Wang, “Delay-range-dependent synchronization criterion for Lur'e systems with delay feedback control,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 5, pp. 1796–1803, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. G. Zhang, T. D. Ma, G.-B. Huang, and Z. L. Wang, “Robust global exponential synchronization of uncertain chaotic delayed neural networks via dual-stage impulsive control,”
*IEEE Transactions on Systems, Man, and Cybernetics B*, vol. 40, no. 3, pp. 831–844, 2010. View at Publisher · View at Google Scholar · View at Scopus - H.-H. Chen, G.-J. Sheu, Y.-L. Lin, and C.-S. Chen, “Chaos synchronization between two different chaotic systems via nonlinear feedback control,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 12, pp. 4393–4401, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. N. Agiza and M. T. Yassen, “Synchronization of Rossler and Chen chaotic dynamical systems using active control,”
*Physics Letters A*, vol. 278, no. 4, pp. 191–197, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Yu and H.-X. Li, “Adaptive hybrid projective synchronization of uncertain chaotic systems based on backstepping design,”
*Nonlinear Analysis: Real World Applications*, vol. 12, no. 1, pp. 388–393, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. G. Zhang, W. Huang, Z. L. Wang, and T. Y. Chai, “Adaptive synchronization between two different chaotic systems with unknown parameters,”
*Physics Letters A*, vol. 350, no. 5-6, pp. 363–366, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - H.-T. Yau, “Design of adaptive sliding mode controller for chaos synchronization with uncertainties,”
*Chaos, Solitons and Fractals*, vol. 22, no. 2, pp. 341–347, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Xie, C. Wen, and Z. Li, “Impulsive control for the stabilization and synchronization of Lorenz systems,”
*Physics Letters A*, vol. 275, no. 1-2, pp. 67–72, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Sun, Y. Shen, and G. Zhang, “Transmission projective synchronization of multi-systems with non-delayed and delayed coupling via impulsive control,”
*Chaos*, vol. 22, no. 4, Article ID 043107, 2012. View at Google Scholar - D. Ghosh, I. Grosu, and S. K. Dana, “Design of coupling for synchronization in time-delayed systems,”
*Chaos*, vol. 22, no. 3, Article ID 033111, 2012. View at Google Scholar - J. Sun and Q. Yin, “Robust fault-tolerant full-order and reduced-order observer synchronization for differential inclusion chaotic systems with unknown disturbances and parameters,”
*Journal of Vibration and Control*, 2013. View at Publisher · View at Google Scholar - G. Wen and D. Xu, “Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems,”
*Chaos, Solitons and Fractals*, vol. 26, no. 1, pp. 71–77, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - J. H. Park, “Adaptive modified projective synchronization of a unified chaotic system with an uncertain parameter,”
*Chaos, Solitons & Fractals*, vol. 34, no. 5, pp. 1152–1159, 2007. View at Google Scholar - J. H. Park, “Adaptive controller design for modified projective synchronization of Genesio-Tesi chaotic system with uncertain parameters,”
*Chaos, Solitons and Fractals*, vol. 34, no. 4, pp. 1154–1159, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. H. Park, “Adaptive synchronization of a four-dimensional chaotic system with uncertain parameters,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 6, no. 3, pp. 305–310, 2005. View at Google Scholar · View at Scopus - J. H. Park, “Adaptive control for modified projective synchronization of a four-dimensional chaotic system with uncertain parameters,”
*Journal of Computational and Applied Mathematics*, vol. 213, no. 1, pp. 288–293, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Y. Xu, W. Zhou, and J.-A. Fang, “Hybrid dislocated control and general hybrid projective dislocated synchronization for the modified Lü chaotic system,”
*Chaos, Solitons and Fractals*, vol. 42, no. 3, pp. 1305–1315, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Wen, Z. Zeng, and T. Huang, “Adaptive synchronization of memristor-based Chua's circuits,”
*Physics Letters A*, vol. 376, no. 44, pp. 2775–2780, 2012. View at Publisher · View at Google Scholar - B. Bao, Z. Liu, and J. Xu, “Transient chaos in smooth memristor oscillator,”
*Chinese Physics B*, vol. 19, no. 3, Article ID 030510, 2010. View at Google Scholar