The problem on chaos synchronization for a class of chaotic system is addressed. Based on impulsive control theory and by constructing a novel Lyapunov functional, new impulsive synchronization strategies are presented and possess more practical application value. Finally some typical numerical simulation examples are included to demonstrate the effectiveness of the theoretical results

Dong, X

Shi, Peng

Wang, B

Discrete Dynamics in Nature and Society

The problem on chaos synchronization for a class of chaotic system is addressed. Based on impulsive control theory and by constructing a novel Lyapunov functional, new impulsive synchronization strategies are presented and possess more practical application value. Finally some typical numerical simulation examples are included to demonstrate the effectiveness of the theoretical results.</jats:p

Bo Wang

Peng Shi

Xiucheng Dong

Crossref

Research ArticleOn the Impulsive Synchronization Control fora Class of Chaotic SystemsBo Wang,1,2 Peng Shi,3,4 and Xiucheng Dong11School of Electrical and Information Engineering, Xihua University, Chengdu 610039, China2School of Mathematics and Computer Science, Yunnan University of Nationalities, Kunming 650031, China3College of Automation, Harbin Engineering University, Harbin, Heilongjiang 150001, China4School of Engineering and Science, Victoria University, Melbourne, VIC 8001, AustraliaCorrespondence should be addressed to Bo Wang; pc wb@163.comReceived 30 September 2014; Revised 10 December 2014; Accepted 10 December 2014; Published 28 December 2014Academic Editor: Hamid R. KarimiCopyright © 2014 Bo Wang et al.This is an open access article distributed under the Creative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.The problem on chaos synchronization for a class of chaotic system is addressed. Based on impulsive control theory and byconstructing a novel Lyapunov functional, new impulsive synchronization strategies are presented and possess more practicalapplication value. Finally some typical numerical simulation examples are included to demonstrate the effectiveness of thetheoretical results.1. IntroductionChaos synchronization can be found in many areas, suchas laser physics and chemical reactor. Up to now manykinds of methods have been developed to realize chaossynchronization, such as feedback control method [1–3],sliding control method [4–6], backstepping control method[7, 8], fuzzy control method [9–12], and adaptive controlmethod [13–15].Recently impulsive control theory and its application inchaos synchronization have become a research hotspot. Forinstance, based on impulsive control strategies, the reduced-order observer for the synchronization of generalized Lorenzchaotic systems is built in [16], the adaptive modifiedfunction projective synchronization of multiple time-delayedchaotic Rossler systems is discussed in [17], the hybridsynchronization of Lu¨ hyperchaotic systemwith disturbancesis investigated in [18], and the robust synchronization ofperturbed Chen’s fractional-order chaotic systems is studiedin [19]. However, these results are just about one kind ofchaotic systems, which limits their applied scope. Hence howto design the impulsive strategy to realize the synchronizationsuitable for more chaotic systems activates our research.In this paper, the following class of chaotic system isintroduced:?̇? = 𝑓 (𝑧) ,𝑧 (𝑡0+) = 𝑧0,(1)where 𝑧(𝑡) ∈ 𝑅𝑛 is the system state vector and𝑓(𝑧) is the non-linear function below second order and can be transformedas follows:?̇? = 𝐶 + 𝐴𝑧 + 𝐵 (𝑧) 𝑧,𝑧 (𝑡0+) = 𝑧0,𝐵𝑖𝑗 (𝑧) =12𝑑 (𝑓𝑖 (𝑧))𝑑𝑧𝑗,(2)where 𝐴 and 𝐶 are linear constant matrices and 𝐵(𝑧) is thelinear matrix about 𝑧.System model (2) includes some chaotic systems, such asLorenz system, Rossler system, Chen system, and Lu¨ system.Hence, the research on system model (2) will possess morepractical application value, compared with those focused onone kind of chaotic system.Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014, Article ID 675721, 5 pageshttp://dx.doi.org/10.1155/2014/6757212 Discrete Dynamics in Nature and SocietyThe remainder of this paper is organized as follows.The model description and preliminaries are proposed inSection 2. Based on impulsive control theory and Lyapunovmethod, we shall try to propose a new and practical impulsivestrategy to realize the synchronization for a class of chaoticsystems in Section 3. Finally, some typical examples will beincluded to show the correctness of the theoretical results inSection 4, and the paper will be concluded in Section 5.2. Model Description and PreliminariesConsider the following main chaotic system:̇𝑦 = 𝐶 + 𝐴𝑦 + 𝐵 (𝑦) 𝑦,𝑦 (𝑡0+) = 𝑦0,𝐵𝑖𝑗(𝑦) =12𝑑 (𝑓𝑖(𝑦))𝑑𝑦𝑗,(3)where 𝑦(𝑡) ∈ 𝑅𝑛 is the state variable of the main system.The slave system is?̇? = 𝐶 + 𝐴𝑥 + 𝐵 (𝑥) 𝑥, 𝑡 ̸= 𝑡𝑘,Δ𝑥 = 𝐵𝑘(𝑥 − 𝑦) , 𝑡 = 𝑡𝑘,𝑥 (𝑡0+) = 𝑥0,𝐵𝑖𝑗(𝑥) =12𝑑 (𝑓𝑖(𝑥))𝑑𝑥𝑗.(4)Define the error state variable as𝑒 (𝑡) = 𝑥 (𝑡) − 𝑦 (𝑡) . (5)We get the following error dynamical system:̇𝑒 = 𝐴𝑒 + 𝐵 (𝑥) 𝑥 − 𝐵 (𝑦) 𝑦, 𝑡 ̸= 𝑡𝑘,Δ𝑒 = 𝐵𝑘𝑒, 𝑡 = 𝑡𝑘,𝑒 (𝑡0+) = 𝑒0.(6)Hence, the problem to be addressed in this paper is to designan impulsive control method such that the tracking errorvariable satisfieslim𝑡→∞‖𝑒 (𝑡)‖ = lim𝑡→∞󵄩󵄩󵄩󵄩𝑥 (𝑡) − 𝑦 (𝑡)󵄩󵄩󵄩󵄩󳨀→ 0. (7)3. Main ResultsIn this section, based on Lyapunov method and impulsivecontrol theory, the following theoretical results are presented.Theorem 1. For the given class of chaotic systems, based on thefollowing impulsive control strategy𝑎 = 𝜆max ((𝑢𝑘− 1)Δ𝑘𝑢𝑘+ Π𝑇+ Π) < 0, (8)whereΠ =1𝑢𝑘(𝐴 + 𝐵 (𝑥) + 𝐵 (𝑦)) ,𝑢𝑘= 𝜆max [(𝐼 + 𝐵𝑘)𝑇(𝐼 + 𝐵𝑘)] ,𝐵𝑘< 0,Δ𝑘= 𝑡𝑘+1− 𝑡𝑘,(9)the slave chaotic system (4) with any initial conditions willsynchronize the master chaotic system (3).Proof. Choose the following Lyapunov functional candidate:𝑉 (𝑡) = 𝑔 (𝑡) 𝑒𝑇(𝑡) 𝑒 (𝑡)(10)with𝑔 (𝑡) =(𝑢𝑘− 1)(𝑡𝑘+1− 𝑡𝑘) 𝑢𝑘(𝑡𝑘− 𝑡) +1𝑢𝑘,𝑡 ∈ [𝑡𝑘, 𝑡𝑘+1) .(11)Considering Δ𝑘= 𝑡𝑘+1− 𝑡𝑘, we have𝑔 (𝑡) =(𝑢𝑘− 1)Δ𝑘𝑢𝑘(𝑡𝑘− 𝑡) +1𝑢𝑘,𝑡 ∈ [𝑡𝑘, 𝑡𝑘+1) .(12)When 𝑡 ∈ (𝑡𝑘, 𝑡𝑘+1), the time derivative of 𝑉(𝑡) along thetrajectories of the error dynamical system is given bẏ𝑉 (𝑡) = ̇𝑔 (𝑡) 𝑒𝑇(𝑡) 𝑒 (𝑡) + 2𝑔 (𝑡) 𝑒𝑇(𝑡) ̇𝑒 (𝑡)=(𝑢𝑘− 1)Δ𝑘𝑢𝑘𝑒𝑇(𝑡) 𝑒 (𝑡) + 2𝑔 (𝑡) 𝑒𝑇(𝑡) ̇𝑒 (𝑡) .(13)For 𝐵(𝑒)𝑥 = 𝐵(𝑥)𝑒, the error dynamical system (6) can betransformed aṡ𝑒 (𝑡) = (𝐴 + 𝐵 (𝑥) + 𝐵 (𝑦)) 𝑒 (𝑡) , 𝑡 ̸= 𝑡𝑘,Δ𝑒 (𝑡) = 𝐵𝑘𝑒 (𝑡) , 𝑡 = 𝑡𝑘,𝑒 (𝑡0+) = 𝑒0.(14)Hencė𝑉 (𝑡) ≤ 𝑒𝑇(𝑡) ((𝑢𝑘− 1)Δ𝑘𝑢𝑘+ 𝑔 (𝑡) (𝐴 + 𝐵 (𝑥) + 𝐵 (𝑦))𝑇+ 𝑔 (𝑡) (𝐴 + 𝐵 (𝑥) + 𝐵 (𝑦)) 𝑒 (𝑡) .(15)It can be concluded thaṫ𝑉 (𝑡) ≤ 𝑒𝑇(𝑡) ((𝑢𝑘− 1)Δ𝑘𝑢𝑘+ Π𝑇+ Π) 𝑒 (𝑡)≤ 𝑎𝑉 (𝑡) ,(16)Discrete Dynamics in Nature and Society 3whereΠ =1𝑢𝑘(𝐴 + 𝐵 (𝑥) + 𝐵 (𝑦)) . (17)Hence(𝑉 (𝑡) 𝑒−𝑎𝑡)󸀠= (̇𝑉 (𝑡) − 𝑎𝑉 (𝑡)) 𝑒−𝑎𝑡≤ 0.(18)Integrating the above inequality from 𝑡𝑘to 𝑡, we have𝑉 (𝑡) 𝑒−𝑎𝑡≤ 𝑉 (𝑡𝑘) 𝑒−𝑎𝑡𝑘. (19)Then the following inequality can be obtained:𝑉 (𝑡) ≤ 𝑉 (𝑡𝑘+) exp (𝑎 (𝑡 − 𝑡𝑘)) . (20)Considering𝑉 (𝑡𝑘+) = 𝑔 (𝑡𝑘+) 𝑒𝑇(𝑡𝑘+) 𝑒 (𝑡𝑘+)= 𝑔 (𝑡𝑘+) [(𝐼 + 𝐵𝑘) 𝑒 (𝑡𝑘−)]𝑇[(𝐼 + 𝐵𝑘) 𝑒 (𝑡𝑘−)]= 𝑉 (𝑡𝑘−) ,(21)we have𝑉 (𝑡) ≤ 𝑉 (𝑡0+) exp (𝑎 (𝑡 − 𝑡0)) . (22)One can obtain‖𝑒 (𝑡)‖ ≤ 𝑒0exp(12𝑎 (𝑡 − 𝑡0)) . (23)Hence, it can be concluded that the error dynamical system(6) is stable. That means that the slave system (4) cansynchronize with the master system (3) based on the givenimpulsive control strategy and this completes the proof.Next, let 𝐵𝑘= 𝑏𝐼 and Δ𝑘= Δ, and based on Theorem 1,we can derive the following theoretical results, and its proofis omitted.Theorem 2. For the given class of chaotic system, based on thefollowing impulsive control strategy𝑎 = 𝜆max ((𝑢 − 1)Δ ⋅ 𝑢+ Π𝑇+ Π) < 0 (24)withΠ =1𝑢(𝐴 + 𝐵 (𝑥) + 𝐵 (𝑦)) ,𝑢 = 𝜆max [(𝐼 + 𝑏𝐼)𝑇(𝐼 + 𝑏𝐼)]𝑏 < 0,(25)the slave chaotic system (4) with any initial conditions willsynchronize with the master chaotic system (3).−10 010 2030 4050−20−1001020−30−20−100102030y1y2y3Figure 1: Dynamic behavior of the chaotic system in case 1.4. Example and SimulationIn this section, we will verify the proposed methodology bygiving two illustrative examples. First consider the unifiedchaotic system as follows:?̇?1= (25𝑟 + 10) (𝑧2− 𝑧1) ,?̇?2= (28 − 35𝑟) 𝑧1− 𝑧1𝑧3+ (29𝑟 − 1) 𝑧2,?̇?3= 𝑧1𝑧2−𝑟 + 83𝑧3.(26)It can be transformed as?̇? = 𝐶 + 𝐴𝑧 + 𝐵 (𝑧) 𝑧 (27)with𝐴 =[[[−25𝑟 − 10 25𝑟 + 10 028 − 35𝑟 29𝑟 − 1 00 0 −𝑟 + 83]]],𝐵 (𝑧) =12[[0 0 0−𝑧30 −𝑧1𝑧2𝑧10]], 𝐶 =[[000]].(28)Unified chaotic system has different dynamic behavior withdifferent parameter 𝑟. For instance, the unified chaotic systemrepresents Lu¨ system when 𝑟 = 0.8, Lorenz system when 0 ≤𝑟 < 0.8, and Chen system when 0.8 < 𝑟 ≤ 1.The numerical simulation is with initial condition 𝑥0=[7, 9, 2]𝑇, 𝑦0= [−1, 3, 5]𝑇, and the simulation step 0.001second. Based onTheorem 2,we choose the impulsive controlparameters Δ = 0.02, 𝑏 = −0.8.Remark 3. Figure 1 depicts the dynamic behavior of thechaotic system in case 1. It can be seen that the state variablemoves in a scope and will never converge to a constant withthe lapse of time. Figure 2 depicts the time response of theerror variable of master-salve system in case 1. It can be seenthat the error variable closes to zero quickly based on thegiven impulse control method, which verifies the correctnessof our theoretical results.4 Discrete Dynamics in Nature and Society−50510−10−5050 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−50510t (s)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2t (s)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2t (s)e 3e 2e 1Figure 2: Time response of the error variable in case 1.Next, consider the following Rossler system?̇?1= − 𝑧2− 𝑧3,?̇?2= 𝑧1+ 0.15𝑧2,?̇?3= 0.2 + 𝑧3(𝑧1− 10) .(29)It can be transformed as follows:?̇? = 𝐶 + 𝐴𝑧 + 𝐵 (𝑧) 𝑧 (30)with𝐴 =[[0 −1 −11 0.15 00 0 −10]], 𝐵 (𝑧) =12[[0 0 00 0 0𝑧30 𝑧1]],𝐶 =[[000.2]].(31)The numerical simulation is with initial condition 𝑥0=[−1, 5, −4]𝑇, 𝑦0= [7, −6, 2]𝑇 and the simulation step 0.001second. Based onTheorem 2,we choose the impulsive controlparameters Δ = 0.02 and 𝑏 = −0.8.Remark 4. Figure 3 depicts the dynamics of the chaoticsystem in case 2. Figure 4 depicts the time response of theerror variable of master-salve system in case 2. It can be seenthat the error variable converges to zero quickly based on theproposed impulse control method.From the above numerical simulations, it can be con-cluded that our impulsive synchronization strategy has morepractical application value compared with those specific toone kind of chaotic system.−15 −10−5 05 1015 20−20−10010200510152025303540y1y2y3Figure 3: Dynamical behavior of the chaotic system in case 2.−10−500510150 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−6−4−20t (s)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2t (s)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2t (s)e 3e 2e 1Figure 4: Time response of the error state variable in case 2.5. ConclusionsThis paper focuses on the chaos synchronization problemand tries to figure out the strategies suitable for more chaoticsystems. Based on the impulsive control technique and Lya-punov stability theory, we have presented the new impulsivesynchronization strategies for a class of chaotic systems.Finally some numerical simulations have been carried out todemonstrate the effectiveness of our theoretical results.Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper.AcknowledgmentsThis work was partially supported by the AustralianResearch Council (DP140102180, LP140100471), the 111Project (B12018), the Open Research Fund of MinistryDiscrete Dynamics in Nature and Society 5of Education (SZjj2011-006), the Chunhui Plan Projectof Ministry of Education (Z2011089, Z2014055), XihuaUniversity Young Scholars Training Program (01201419),the Open Research Subject of Key Laboratory of Signal andInformation Processing of Sichuan Province (szjj2014-018),and the National Natural Science Foundation of China(61174058, 61134001, and 11461062).References[1] A. E. 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Baleanu, “Complete synchronization ofcommensurate fractional order chaotic systems using slidingmode control,”Mechatronics, vol. 23, no. 7, pp. 873–879, 2013.[6] H. R. Karimi, “A slidingmode approach to𝐻∞synchronizationof master-slave time-delay systems with Markovian jumpingparameters and nonlinear uncertainties,” Journal of the FranklinInstitute, vol. 349, no. 4, pp. 1480–1496, 2012.[7] Y. Yu and H.-X. Li, “Adaptive hybrid projective synchronizationof uncertain chaotic systems based on backstepping design,”Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp.388–393, 2011.[8] N. Chen, F. Song, G. Li, X. Sun, and C. Ai, “An adaptivesliding mode backstepping control for the mobile manipulatorwith nonholonomic constraints,” Communications in NonlinearScience andNumerical Simulation, vol. 18, no. 10, pp. 2885–2899,2013.[9] M. Chadli, H. R. Karimi, and P. 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Jasek,“An investigation on evolutionary reconstruction of continuouschaotic systems,”Mathematical andComputerModelling, vol. 57,no. 1-2, pp. 2–15, 2013.[14] U. E. Vincent and R. Guo, “Finite-time synchronization for aclass of chaotic and hyperchaotic systems via adaptive feedbackcontroller,” Physics Letters, Section A: General, Atomic and SolidState Physics, vol. 375, no. 24, pp. 2322–2326, 2011.[15] H. R. Karimi, M. Zapateiro, and N. Luo, “Adaptive synchroniza-tion of master-slave systems with mixed neutral and discretetime-delays and nonlinear perturbations,” Asian Journal ofControl, vol. 14, no. 1, pp. 251–257, 2012.[16] Z. Zhang, H. Shao, Z. Wang, and H. Shen, “Reduced-orderobserver design for the synchronization of the generalizedLorenz chaotic systems,” Applied Mathematics and Computa-tion, vol. 218, no. 14, pp. 7614–7621, 2012.[17] K. S. Sudheer and M. Sabir, “Adaptive modified functionprojective synchronization of multiple time-delayed chaoticRossler system,” Physics Letters A, vol. 375, no. 8, pp. 1176–1178,2011.[18] Q. Li, R. Dong, and B. Lu, “Hybrid synchronization of Lu¨hyperchaotic system with disturbances by sliding mode con-trol,” Procedia Engineering, vol. 15, pp. 23–27, 2011.[19] M. Asheghan, M. Beheshti, and M. Tavazoei, “Robust synchro-nization of perturbed Chen's fractional-order chaotic systems,”Communications in Nonlinear Science and Numerical Simula-tion, vol. 16, pp. 1044–1051, 2011.Submit your manuscripts athttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical Problems in EngineeringHindawi Publishing Corporationhttp://www.hindawi.comDifferential EquationsInternational Journal ofVolume 2014Applied MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical PhysicsAdvances inComplex AnalysisJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014OptimizationJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Operations ResearchAdvances inJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Function SpacesAbstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal of Mathematics and Mathematical SciencesHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AlgebraDiscrete Dynamics in Nature and SocietyHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Decision SciencesAdvances inDiscrete MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Stochastic AnalysisInternational Journal of

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On the Impulsive Synchronization Control for a Class of Chaotic Systems

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On the impulsive synchronization control for a class of chaotic systems

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Published 28 December 2014The problem on chaos synchronization for a class of chaotic system is addressed. Based on impulsive control theory and by constructing a novel Lyapunov functional, new impulsive synchronization strategies are presented and possess more practical application value. Finally some typical numerical simulation examples are included to demonstrate the effectiveness of the theoretical results.Bo Wang, Peng Shi, and Xiucheng Don

Wang, B.

Shi, P.

Dong, X.

Adelaide Research &amp; Scholarship

PUBLISHED VERSION   Bo Wang, Peng Shi, and Xiucheng Dong On the impulsive synchronization control for a class of chaotic systems Discrete Dynamics in Nature and Society, 2014; 2014:675721-1-675721-5 Copyright © 2014 Bo Wang 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. Originally published at: http://doi.org/10.1155/2014/675721                   PERMISSIONS   http://creativecommons.org/licenses/by/3.0/        http://hdl.handle.net/2440/94846  Research ArticleOn the Impulsive Synchronization Control fora Class of Chaotic SystemsBo Wang,1,2 Peng Shi,3,4 and Xiucheng Dong11School of Electrical and Information Engineering, Xihua University, Chengdu 610039, China2School of Mathematics and Computer Science, Yunnan University of Nationalities, Kunming 650031, China3College of Automation, Harbin Engineering University, Harbin, Heilongjiang 150001, China4School of Engineering and Science, Victoria University, Melbourne, VIC 8001, AustraliaCorrespondence should be addressed to Bo Wang; pc wb@163.comReceived 30 September 2014; Revised 10 December 2014; Accepted 10 December 2014; Published 28 December 2014Academic Editor: Hamid R. KarimiCopyright © 2014 Bo Wang et al.This is an open access article distributed under the Creative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.The problem on chaos synchronization for a class of chaotic system is addressed. Based on impulsive control theory and byconstructing a novel Lyapunov functional, new impulsive synchronization strategies are presented and possess more practicalapplication value. Finally some typical numerical simulation examples are included to demonstrate the effectiveness of thetheoretical results.1. IntroductionChaos synchronization can be found in many areas, suchas laser physics and chemical reactor. Up to now manykinds of methods have been developed to realize chaossynchronization, such as feedback control method [1–3],sliding control method [4–6], backstepping control method[7, 8], fuzzy control method [9–12], and adaptive controlmethod [13–15].Recently impulsive control theory and its application inchaos synchronization have become a research hotspot. Forinstance, based on impulsive control strategies, the reduced-order observer for the synchronization of generalized Lorenzchaotic systems is built in [16], the adaptive modifiedfunction projective synchronization of multiple time-delayedchaotic Rossler systems is discussed in [17], the hybridsynchronization of Lu¨ hyperchaotic systemwith disturbancesis investigated in [18], and the robust synchronization ofperturbed Chen’s fractional-order chaotic systems is studiedin [19]. However, these results are just about one kind ofchaotic systems, which limits their applied scope. Hence howto design the impulsive strategy to realize the synchronizationsuitable for more chaotic systems activates our research.In this paper, the following class of chaotic system isintroduced:?̇? = 𝑓 (𝑧) ,𝑧 (𝑡0+) = 𝑧0,(1)where 𝑧(𝑡) ∈ 𝑅𝑛 is the system state vector and𝑓(𝑧) is the non-linear function below second order and can be transformedas follows:?̇? = 𝐶 + 𝐴𝑧 + 𝐵 (𝑧) 𝑧,𝑧 (𝑡0+) = 𝑧0,𝐵𝑖𝑗 (𝑧) =12𝑑 (𝑓𝑖 (𝑧))𝑑𝑧𝑗,(2)where 𝐴 and 𝐶 are linear constant matrices and 𝐵(𝑧) is thelinear matrix about 𝑧.System model (2) includes some chaotic systems, such asLorenz system, Rossler system, Chen system, and Lu¨ system.Hence, the research on system model (2) will possess morepractical application value, compared with those focused onone kind of chaotic system.Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014, Article ID 675721, 5 pageshttp://dx.doi.org/10.1155/2014/6757212 Discrete Dynamics in Nature and SocietyThe remainder of this paper is organized as follows.The model description and preliminaries are proposed inSection 2. Based on impulsive control theory and Lyapunovmethod, we shall try to propose a new and practical impulsivestrategy to realize the synchronization for a class of chaoticsystems in Section 3. Finally, some typical examples will beincluded to show the correctness of the theoretical results inSection 4, and the paper will be concluded in Section 5.2. Model Description and PreliminariesConsider the following main chaotic system:̇𝑦 = 𝐶 + 𝐴𝑦 + 𝐵 (𝑦) 𝑦,𝑦 (𝑡0+) = 𝑦0,𝐵𝑖𝑗(𝑦) =12𝑑 (𝑓𝑖(𝑦))𝑑𝑦𝑗,(3)where 𝑦(𝑡) ∈ 𝑅𝑛 is the state variable of the main system.The slave system is?̇? = 𝐶 + 𝐴𝑥 + 𝐵 (𝑥) 𝑥, 𝑡 ̸= 𝑡𝑘,Δ𝑥 = 𝐵𝑘(𝑥 − 𝑦) , 𝑡 = 𝑡𝑘,𝑥 (𝑡0+) = 𝑥0,𝐵𝑖𝑗(𝑥) =12𝑑 (𝑓𝑖(𝑥))𝑑𝑥𝑗.(4)Define the error state variable as𝑒 (𝑡) = 𝑥 (𝑡) − 𝑦 (𝑡) . (5)We get the following error dynamical system:̇𝑒 = 𝐴𝑒 + 𝐵 (𝑥) 𝑥 − 𝐵 (𝑦) 𝑦, 𝑡 ̸= 𝑡𝑘,Δ𝑒 = 𝐵𝑘𝑒, 𝑡 = 𝑡𝑘,𝑒 (𝑡0+) = 𝑒0.(6)Hence, the problem to be addressed in this paper is to designan impulsive control method such that the tracking errorvariable satisfieslim𝑡→∞‖𝑒 (𝑡)‖ = lim𝑡→∞󵄩󵄩󵄩󵄩𝑥 (𝑡) − 𝑦 (𝑡)󵄩󵄩󵄩󵄩󳨀→ 0. (7)3. Main ResultsIn this section, based on Lyapunov method and impulsivecontrol theory, the following theoretical results are presented.Theorem 1. For the given class of chaotic systems, based on thefollowing impulsive control strategy𝑎 = 𝜆max ((𝑢𝑘− 1)Δ𝑘𝑢𝑘+ Π𝑇+ Π) < 0, (8)whereΠ =1𝑢𝑘(𝐴 + 𝐵 (𝑥) + 𝐵 (𝑦)) ,𝑢𝑘= 𝜆max [(𝐼 + 𝐵𝑘)𝑇(𝐼 + 𝐵𝑘)] ,𝐵𝑘< 0,Δ𝑘= 𝑡𝑘+1− 𝑡𝑘,(9)the slave chaotic system (4) with any initial conditions willsynchronize the master chaotic system (3).Proof. Choose the following Lyapunov functional candidate:𝑉 (𝑡) = 𝑔 (𝑡) 𝑒𝑇(𝑡) 𝑒 (𝑡)(10)with𝑔 (𝑡) =(𝑢𝑘− 1)(𝑡𝑘+1− 𝑡𝑘) 𝑢𝑘(𝑡𝑘− 𝑡) +1𝑢𝑘,𝑡 ∈ [𝑡𝑘, 𝑡𝑘+1) .(11)Considering Δ𝑘= 𝑡𝑘+1− 𝑡𝑘, we have𝑔 (𝑡) =(𝑢𝑘− 1)Δ𝑘𝑢𝑘(𝑡𝑘− 𝑡) +1𝑢𝑘,𝑡 ∈ [𝑡𝑘, 𝑡𝑘+1) .(12)When 𝑡 ∈ (𝑡𝑘, 𝑡𝑘+1), the time derivative of 𝑉(𝑡) along thetrajectories of the error dynamical system is given bẏ𝑉 (𝑡) = ̇𝑔 (𝑡) 𝑒𝑇(𝑡) 𝑒 (𝑡) + 2𝑔 (𝑡) 𝑒𝑇(𝑡) ̇𝑒 (𝑡)=(𝑢𝑘− 1)Δ𝑘𝑢𝑘𝑒𝑇(𝑡) 𝑒 (𝑡) + 2𝑔 (𝑡) 𝑒𝑇(𝑡) ̇𝑒 (𝑡) .(13)For 𝐵(𝑒)𝑥 = 𝐵(𝑥)𝑒, the error dynamical system (6) can betransformed aṡ𝑒 (𝑡) = (𝐴 + 𝐵 (𝑥) + 𝐵 (𝑦)) 𝑒 (𝑡) , 𝑡 ̸= 𝑡𝑘,Δ𝑒 (𝑡) = 𝐵𝑘𝑒 (𝑡) , 𝑡 = 𝑡𝑘,𝑒 (𝑡0+) = 𝑒0.(14)Hencė𝑉 (𝑡) ≤ 𝑒𝑇(𝑡) ((𝑢𝑘− 1)Δ𝑘𝑢𝑘+ 𝑔 (𝑡) (𝐴 + 𝐵 (𝑥) + 𝐵 (𝑦))𝑇+ 𝑔 (𝑡) (𝐴 + 𝐵 (𝑥) + 𝐵 (𝑦)) 𝑒 (𝑡) .(15)It can be concluded thaṫ𝑉 (𝑡) ≤ 𝑒𝑇(𝑡) ((𝑢𝑘− 1)Δ𝑘𝑢𝑘+ Π𝑇+ Π) 𝑒 (𝑡)≤ 𝑎𝑉 (𝑡) ,(16)Discrete Dynamics in Nature and Society 3whereΠ =1𝑢𝑘(𝐴 + 𝐵 (𝑥) + 𝐵 (𝑦)) . (17)Hence(𝑉 (𝑡) 𝑒−𝑎𝑡)󸀠= (̇𝑉 (𝑡) − 𝑎𝑉 (𝑡)) 𝑒−𝑎𝑡≤ 0.(18)Integrating the above inequality from 𝑡𝑘to 𝑡, we have𝑉 (𝑡) 𝑒−𝑎𝑡≤ 𝑉 (𝑡𝑘) 𝑒−𝑎𝑡𝑘. (19)Then the following inequality can be obtained:𝑉 (𝑡) ≤ 𝑉 (𝑡𝑘+) exp (𝑎 (𝑡 − 𝑡𝑘)) . (20)Considering𝑉 (𝑡𝑘+) = 𝑔 (𝑡𝑘+) 𝑒𝑇(𝑡𝑘+) 𝑒 (𝑡𝑘+)= 𝑔 (𝑡𝑘+) [(𝐼 + 𝐵𝑘) 𝑒 (𝑡𝑘−)]𝑇[(𝐼 + 𝐵𝑘) 𝑒 (𝑡𝑘−)]= 𝑉 (𝑡𝑘−) ,(21)we have𝑉 (𝑡) ≤ 𝑉 (𝑡0+) exp (𝑎 (𝑡 − 𝑡0)) . (22)One can obtain‖𝑒 (𝑡)‖ ≤ 𝑒0exp(12𝑎 (𝑡 − 𝑡0)) . (23)Hence, it can be concluded that the error dynamical system(6) is stable. That means that the slave system (4) cansynchronize with the master system (3) based on the givenimpulsive control strategy and this completes the proof.Next, let 𝐵𝑘= 𝑏𝐼 and Δ𝑘= Δ, and based on Theorem 1,we can derive the following theoretical results, and its proofis omitted.Theorem 2. For the given class of chaotic system, based on thefollowing impulsive control strategy𝑎 = 𝜆max ((𝑢 − 1)Δ ⋅ 𝑢+ Π𝑇+ Π) < 0 (24)withΠ =1𝑢(𝐴 + 𝐵 (𝑥) + 𝐵 (𝑦)) ,𝑢 = 𝜆max [(𝐼 + 𝑏𝐼)𝑇(𝐼 + 𝑏𝐼)]𝑏 < 0,(25)the slave chaotic system (4) with any initial conditions willsynchronize with the master chaotic system (3).−10 010 2030 4050−20−1001020−30−20−100102030y1y2y3Figure 1: Dynamic behavior of the chaotic system in case 1.4. Example and SimulationIn this section, we will verify the proposed methodology bygiving two illustrative examples. First consider the unifiedchaotic system as follows:?̇?1= (25𝑟 + 10) (𝑧2− 𝑧1) ,?̇?2= (28 − 35𝑟) 𝑧1− 𝑧1𝑧3+ (29𝑟 − 1) 𝑧2,?̇?3= 𝑧1𝑧2−𝑟 + 83𝑧3.(26)It can be transformed as?̇? = 𝐶 + 𝐴𝑧 + 𝐵 (𝑧) 𝑧 (27)with𝐴 =[[[−25𝑟 − 10 25𝑟 + 10 028 − 35𝑟 29𝑟 − 1 00 0 −𝑟 + 83]]],𝐵 (𝑧) =12[[0 0 0−𝑧30 −𝑧1𝑧2𝑧10]], 𝐶 =[[000]].(28)Unified chaotic system has different dynamic behavior withdifferent parameter 𝑟. For instance, the unified chaotic systemrepresents Lu¨ system when 𝑟 = 0.8, Lorenz system when 0 ≤𝑟 < 0.8, and Chen system when 0.8 < 𝑟 ≤ 1.The numerical simulation is with initial condition 𝑥0=[7, 9, 2]𝑇, 𝑦0= [−1, 3, 5]𝑇, and the simulation step 0.001second. Based onTheorem 2,we choose the impulsive controlparameters Δ = 0.02, 𝑏 = −0.8.Remark 3. Figure 1 depicts the dynamic behavior of thechaotic system in case 1. It can be seen that the state variablemoves in a scope and will never converge to a constant withthe lapse of time. Figure 2 depicts the time response of theerror variable of master-salve system in case 1. It can be seenthat the error variable closes to zero quickly based on thegiven impulse control method, which verifies the correctnessof our theoretical results.4 Discrete Dynamics in Nature and Society−50510−10−5050 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−50510t (s)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2t (s)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2t (s)e 3e 2e 1Figure 2: Time response of the error variable in case 1.Next, consider the following Rossler system?̇?1= − 𝑧2− 𝑧3,?̇?2= 𝑧1+ 0.15𝑧2,?̇?3= 0.2 + 𝑧3(𝑧1− 10) .(29)It can be transformed as follows:?̇? = 𝐶 + 𝐴𝑧 + 𝐵 (𝑧) 𝑧 (30)with𝐴 =[[0 −1 −11 0.15 00 0 −10]], 𝐵 (𝑧) =12[[0 0 00 0 0𝑧30 𝑧1]],𝐶 =[[000.2]].(31)The numerical simulation is with initial condition 𝑥0=[−1, 5, −4]𝑇, 𝑦0= [7, −6, 2]𝑇 and the simulation step 0.001second. Based onTheorem 2,we choose the impulsive controlparameters Δ = 0.02 and 𝑏 = −0.8.Remark 4. Figure 3 depicts the dynamics of the chaoticsystem in case 2. Figure 4 depicts the time response of theerror variable of master-salve system in case 2. It can be seenthat the error variable converges to zero quickly based on theproposed impulse control method.From the above numerical simulations, it can be con-cluded that our impulsive synchronization strategy has morepractical application value compared with those specific toone kind of chaotic system.−15 −10−5 05 1015 20−20−10010200510152025303540y1y2y3Figure 3: Dynamical behavior of the chaotic system in case 2.−10−500510150 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−6−4−20t (s)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2t (s)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2t (s)e 3e 2e 1Figure 4: Time response of the error state variable in case 2.5. ConclusionsThis paper focuses on the chaos synchronization problemand tries to figure out the strategies suitable for more chaoticsystems. Based on the impulsive control technique and Lya-punov stability theory, we have presented the new impulsivesynchronization strategies for a class of chaotic systems.Finally some numerical simulations have been carried out todemonstrate the effectiveness of our theoretical results.Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper.AcknowledgmentsThis work was partially supported by the AustralianResearch Council (DP140102180, LP140100471), the 111Project (B12018), the Open Research Fund of MinistryDiscrete Dynamics in Nature and Society 5of Education (SZjj2011-006), the Chunhui Plan Projectof Ministry of Education (Z2011089, Z2014055), XihuaUniversity Young Scholars Training Program (01201419),the Open Research Subject of Key Laboratory of Signal andInformation Processing of Sichuan Province (szjj2014-018),and the National Natural Science Foundation of China(61174058, 61134001, and 11461062).References[1] A. E. 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On the impulsive synchronization control for a class of chaotic systems

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On the impulsive synchronization control for a class of chaotic systems

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