Generalized Function Projective Lag Synchronization in Fractional-Order Chaotic Systems Yancheng Ma Guoan Wu and Lan Jiang denotes fractional order of drive system Abstract In this paper a new synchronization of fractionalorder chaotic systems called generalized function projective lag synchronization is introduced. his synchronization method is a generalization of function projective synchronization and function projective lag synchronization. Based on stability orem of linear fractional order systems a suitable nonlinear fractional-order controller is designed for synchronization of different structural systems. hree examples are given to verify effectiveness of proposed method. l : Rc Rc indicates continuous function of drive x(t ) ( x (t ) x2 (t ) xc (t )) R c indicates state vector of drive system c denotes dimension of drive is time delay. he fractional-order response system is defined as: D y(t ) f ( y(t )) u( x(t ) y(t ) z(t )) where D signifies fractional-order differential operator. denotes fractional order of response system Index erms Chaos fractional order generalized function projective lag synchronization nonlinear controller. f : Rd Rd I. indicates y (t ) ( y (t ) y2 (t ) continuous d u(t ) (u (t ) u2 (t ) ud (t )) R d is nonlinear controller to be designed later. Choose fractional- order reference system as: Dq z(t ) h( z(t )) system h : Rd Rd denotes continuous function. z (t ) ( z (t ) z2 (t ) zd (t )) R d denotes state vector System (3) is an attractor. Define error state vector as: e(t ) y(t ) C( x(t )) x(t ) z(t ) K ( x ) Q ( x ) where matrix c ( x )} =min{c d} and Q( x ) (c ( x ) max{c d} A. he Definition of GFPLS Choose fractional-order drive system as: or c ( x )) condition is system dimension d c matrix is defined as K ( x ) Q ( x ) () where D signifies fractional-order differential operator. Definition : GFPLS is realized. Assume re exists a u ( x y (t ) z (t )) such nonlinear controller that Manuscript received March 28 26; revised September 2 26. Authors are with uazhong University of Science and echnology China (e-mail: ONYMA989@qq.com). doi:.878/ijiee.26.6.5.642 (4) where C( x(t )) Rd c x x(t ). here exists two conditions one condition is system dimension d c E GFPLS MEOD x(t ) x() t (3) where D q signifies fractional-order differential operator. q indicates fractional order of reference K ( x ) diag{c ( x ) c2 ( x ) D x(t ) l ( x(t )) function. yd (t )) R indicates state vector of response system d denotes dimension of response INRODUCION Since pioneering and meaningful work of Pecora and Carroll [] chaotic synchronization has become a hot topic and been studied by more and more scholars in recent decades. It has many applications in field of physics chemistry biology and ors especially in secure communication. From existing literature re are many different types of synchronization method such as complete synchronization [2]-[5] projective synchronization [6] generalized synchronization [7] [8] robust synchronization [9] function projective synchronization (FPS) [] function projective lag synchronization (FPLS) []. here are many control scheme utilizing in chaotic synchronization such as adaptive control [2]-[4] sliding mode control [5]-[7] fuzzy sliding mode control [8] [9]. In [2] a reference system is added to make synchronization more complex. Inspired by [2] we introduce a new type of synchronization method named generalized function projective lag synchronization (GFPLS) which is a generalization of FPS and FPLS. he organization of this paper is as follows. he GFPLS method is introduced in Section II. Simulation and results are presented in Section III. In Section IV conclusions are proposed. II. (2) lim e(t ). t 299
Remark : system (3) denotes constant zero GFPLS degenerated to FPLS. Remark 2: system (3) denotes constant zero and time delay is zero GFPLS degenerated to FPS. Remark 3: System (3) can be or attractors such as chaos hyperchaos periodic function and quasi period function. in [2] error system (6) asymptotically stabilizes at origin. GFPLS is achieved. he proof is completed. III. SIMULAION AND RESULS In this section three examples presented in [] are utilized to demonstrate effectiveness of GFPLS with same and different dimension. A predictor-corrector method proposed in [22] [23] is used to solve fractionalorder differential equations. B. he Stability Analysis of GFPLS Consider controller as: u D (C ( x ) x ) D ( z ) f (C ( x ) x ) f ( z ) G ( x y (t ) z (t ))e A. GFPLS with Same Dimension d=c Choose fractional-order Rössler system as drive (5) where is compensation vector Rd G ( x y (t ) z (t )) R d d is a polynomial matrix. According to (2) (3) and (5) we obtain fractional-order error system as follows: D e(t ) ( F ( x y(t ) z (t )) G( x y(t ) z(t )))e(t ) D x (t ) x2 (t ) x3 (t ) D x2 (t ) x (t ) x2 (t ) D x3 (t ) 2 x (t ) x3 (t ) 3 x3 (t ) (4) x(t ) x() t (6) =.95 2 3 =.4.2 x() (.5.5.5) where F ( x y(t ) z(t ))e(t ) f ( y(t )) f (C ( x ) x ) f ( z(t )). drive system shows chaotic attractor drawn in Fig.. Proposition : C ( x ) and time delay is given GFPLS is accomplished if re exists G ( x y (t ) z (t )) R d d such that F ( x y(t ) z(t )) G(x y(t ) z(t ))= N ( x y(t ) z(t )) (7) where N [ni j ] R d d i j ni j n j i i =j ni j R Proof: Assume is an arbitrary eigenvalue of matrix F ( x y (t ) z (t )) G ( x y (t ) z (t )) and related nonzero eigenvector denotes. We may get (F ( x y(t ) z(t )) G( x y(t ) z(t ))) Fig.. he phase trajectory of fractional-order Rössler (8) Choose fractional-order Lüsystem as response Multiplying at left of (8) we have ( F ( x y(t ) z (t )) G( x y(t ) z (t ))) D y (t ) ( y2 (t ) y (t )) u ( x y (t ) z (t )) D y2 (t ) y (t ) y3 (t ) 2 y2 (t ) u2 ( x y (t ) z (t )) (9) (5) D y3 (t ) y (t ) y2 (t ) 3 y3 (t ) u3 ( x y (t ) z (t )) where denotes conjugate transpose of a matrix * also indicates an eigenvalue of matrix =.9( 2 3 ) (35 283) y () (4.23.5) response system denotes chaotic attractor shown in Fig. 2. F ( x y(t ) z (t )) G( x y(t ) z(t )) Similarly we have ( F ( x y(t ) z(t )) G( x y(t ) z(t ))) * () According to (9) and () we obtain * = [( F ( x y z ) G ( x y z )) ( F ( x y z ) G ( x y z )) ] / () / where ( N ( x y z ) N ( x y z ) ) R d d. We are able to obtain that denotes a real positive- Fig. 2. he phase trajectory of fractional-order Lü definite diagonal matrix. hus we can have * 2real ( ) Choose fractional-order Chen system as reference D q z (t ) ( z2 (t ) z (t )) (2) D q z2 (t ) ( 3 ) z (t ) z (t ) z3 (t ) 3 z2 (t ) ence we can obtain (6) D z3 (t ) z (t ) z2 (t ) 2 z3 (t ) q arg( ) / 2 / 2 (3) q.95( 2 3 ) (353 28) z () (523) reference system indicates chaotic attractor drawn in Fig. 3. Based on fractional-order stability orem proposed 3
Fig. 3. he phase trajectory of fractional-order Chen he error states are defined as ei (t ) yi (t ) ci ( x ) xi zi (t ) xi xi (t ) i 23. According to (5) and (5) we can get F ( x y(t ) z (t )) (c3 ( x ) x3 z3 (t )) 2 y (t ) y2 (t ) (c ( x ) x z (t )) 3 (c3 ( x ) x3 z3 (t ) ) y2 (t ) G( x y(t ) z(t )) 2 d y (t ) (c ( x ) x z (t )) Fig. 5. he phase trajectory of fractional-order hyperchaotic Lorenz Choose fractional-order Rössler system as response where d> we can obtain where N [ni j ] R d d D y (t ) y2 (t ) y3 (t ) u ( x y (t ) z (t )) i j ni j n j i i =j ni j R D y2 (t ) y (t ) y2 (t ) u2 ( x y (t ) z (t )) i j 23. he GFPLS with same dimension is realized based on proposition. C ( x ) diag x2 35 x3 2.5 4 x3 2 x (8) D y3 (t ) 2 y (t ) y3 (t ) 3 y3 (t ) u3 ( x y (t ) z (t )) Choose fractional-order Lü system as reference.5 d 5 error state curves of GFPLS with same D q z (t ) ( z2 (t ) z (t )) dimension are shown in Fig. 4. D q z2 (t ) z (t ) z3 (t ) 2 z2 (t ) (9) D q z3 (t ) z (t ) z2 (t ) 3 z3 (t ) Based on (4) we can obtain error states e (t ) y (t ) c ( x ) x z (t ) e2 (t ) y2 (t ) c2 ( x ) x2 z2 (t ) e3 (t ) y3 (t ) c3 ( x ) x3 c4 ( x ) x4 z3 (t ) where xi xi (t ) i 23 4. According to (5) and (8) we can have F ( x y(t ) z(t )) c ( x ) x c ( x ) x z (t ) y (t ) 4 4 3 3 3 3 Fig. 4. he error state curves of GFPLS with same dimension. d 2 G( x y(t ) z(t )) B. GFPLS with Different Dimension d<c Choose fractional-order hyperchaotic Lorenz system as drive system D x2 (t ) 2 x (t ) x2 (t ) x (t ) x3 (t ) D x3 (t ) x (t ) x2 (t ) 3 x3 (t ) (7) D x4 (t ) x2 (t ) x3 (t ) 4 x4 (t ) where N [ni j ] Rd d i j ni j n j i i =j ni j R x(t ) x() t [ ].98( 2 3 4 ) ( 288 / 3) x() ( 2.2 68.3 9) drive hyperchaotic attractor shown in Fig. 5. L y2 (t ) y (t ) 3 d 4 where L= c3 ( x ) x3 c4 ( x ) x4 z3 (t ) d2 d3 d4 we can obtain D x (t ) ( x2 (t ) x (t )) x4 (t ) d3 y2 (t ) system i j 23. ence GFPLS with different dimension (d<c) is achieved based on proposition..2 d2 d3 6 d4 3 denotes 3
x2 / 2 x2 x3. x / 2 Choose fractional-order hyperchaotic Lorenz system as reference system D q z (t ) ( z2 (t ) z (t )) z4 (t ) D q z2 (t ) 2 z (t ) z2 (t ) z (t ) z3 (t ) he error state curves of GFPLS with different dimension (d<c) are shown in Fig. 6. D q z3 (t ) z (t ) z2 (t ) 3 z3 (t ) (22) D q z4 (t ) z2 (t ) z3 (t ) 4 z4 (t ) According to (4) we can have error states e (t ) y (t ) c ( x ) x z (t ) e2 (t ) y2 (t ) c2 ( x ) x2 z2 (t ) e3 (t ) y3 (t ) c3 ( x ) x3 z3 (t ) e4 (t ) y4 (t ) c4 ( x ) x3 z4 (t ) where xi xi (t ) i 23. According to (5) and (2) we can have 2 y3 (t ) F ( x y (t ) z (t ))= c2 ( x ) x2 z2 (t ) Fig. 6. he error state curves of GFPLS with different dimension (d<c). C. GFPLS with Different Dimension d>c Choose fractional-order Lüsystem as drive d5 L 3 d6 G ( x y (t ) z (t )) y (t ) D x (t ) ( x2 (t ) x (t )) D x2 (t ) x (t ) x3 (t ) 2 x2 (t ) D x3 (t ) x (t ) x2 (t ) 3 x3 (t ) (2) Where N [ni j ] Rd d i j ni j n j i i =j ni j R i j 23 4. ence GFPLS with different dimension (d>c) is realized based on proposition..5 d5 25 d6 5 d7 4 D y (t ) ( y2 (t ) y (t )) y4 (t ) u ( x y(t ) z(t )) (2) D y4 (t ) y2 (t ) y3 (t ) 5 y4 (t ) u4 ( x y(t ) z(t )) (c2 ( x ) x2 z2 (t )) c ( x ) x z (t ) y3 (t ) (c2 ( x ) x2 z2 (t )) 5 d 7 Choose fractional-order hyperchaotic Chen system as response D y3 (t ) y (t ) y2 (t ) 4 y3 (t ) u3 ( x y (t ) z (t )) (c ( x ) x z (t )) y (t ) 4 y3 (t ) c2 ( x ) x2 z2 (t ) 5 where L =y3 (t ) 2 d5 d6 d7 we can obtain x(t ) x() t [ ] D y2 (t ) 2 y (t ) y (t ) y3 (t ) 3 y2 (t ) u2 ( x y(t ) z(t )) 3.96( 2 3 4 5 ) (35723.5) x2 y() (.22.3..) response system indicates hyperchaotic attractor shown in Fig. 7. x x3 x / 2 2.5 x2 he error state curves of GFPLS with different dimension (d>c) are shown in Fig. 8. Fig. 8. he error state curves of GFPLS with different dimension (d>c). IV. CONCLUSIONS In this paper GFPLS of fractional-order chaotic system is investigated based on fractional-order stability orem. his synchronization scheme is more complex and Fig. 7. he phase trajectory of fractional-order hyperchaotic Chen 32
generalized than existing synchronization method. he numerical simulations with same and different dimension are given to illustrate effectiveness of proposed method. In future GFPLS has more potential to be applied in secure communication. ACKNOWLEDGMEN I appreciate my supervisor Guoan Wu who gives me many opportunities and thank for every classmate in institute of microwave technology and application. I also thank all scholars who contribute to improvement of chaotic synchronization. REFERENCES [] [2] [3] [4] [5] [6] [7] [8] [9] [] [] [2] [3] L. M. Pecora and. L. Carroll Sychronizaiton in chaotic systems Phys. Rev. Lett. vol. 64 pp. 35-38 Feb. 99. J. G. Lu Chaotic dynamics of fractional-order Ikeda delay system and its synchronization Chine. Phys. vol. 5 pp. 3-35 26. C. P. Li W.. Deng and D. Xu Chaos synchronization of Chua system with a fractional order Physica A vol. 36 pp. 7-85 Feb. 26. X. J. Wu J. Li and G. R. Chen Chaos in fractional order unified system and its synchronization J. Franklin Inst. vol. 345 pp. 3924 July 28.. aghvafard and G.. Eqaee Phase and anti-phase synchronization of fractional order chaotic systems via active control Commun. Nonlinear Sci. Numer. Simulat. vol. 6 pp. 479488 Oct. 2. X. Y. Wang and Y. J. e Projective synchronization of fractional order chaotic system based on linear separation Phys. Lett. A vol. 372 pp. 435-44 Jan. 28.. S. Wang and X. Y. Wang Generalized synchronization of fractional order hyperchaotic lorenz system Mod. Phys. Lett. B vol. 23 pp. 267-278 July 29. R Zhou X. R Cheng and N. Y. Zhang Generalized synchronization between different fractional-order chaotic systems Commun. heor. Phys. vol. 5 pp. 93-934 28. K. Suwat Robust synchronization of fractional-order unified chaotic systems via linear control Comput. Math. Appl. vol. 7 pp. 267268 22. P. Zhou and W. Zhu Function projective synchronization for fractional-order chaotic systems Nonlinear Anal. Real World Appl. vol. 2 pp. 8-86 2. W. Sha Y. Y. Guang and W. u Function projective lag synchronization of fractional-order chaotic systems Chin. Phys. B vol. 23 pp. -7 24. R. A. ang Y. L. Liu and J. K. Xue An extended active control for chaos synchronization Phys. Lett. A vol. 373 pp. 449-454 29. L.-X. Yang and J. Jiang Adaptive synchronization of drive-response fractional-order complex dynamical networks with uncertain parameters Commun. Nonlinear Sci. Numer. Simulat. vol. 9 pp. 496-56 24. 33 [4] S. Wen Z. Zeng and. uang Adaptive synchronization of memristor-based Chuaʼs circuits Phys. Lett. A vol. 376 pp. 2775278 22. [5] C. Yin S. Dadras and S. M. Zhong LMI based design of a sliding mode controller for a class of uncertain fractional-order nonlinear systems J. Franklin Inst. vol. 349 pp. 378-3 22. [6] S.. osseinnia R. Ghaderi A. N. Ranjbar M. Mahmoudian and S. Momani Sliding mode synchronization of an uncertain fractional order chaotic system Computers and Mamatics with Application vol. 59 pp. 637-643 March 2. [7] P. Balasubramaniam heoretical and practical applications of fuzzy fractional integral sliding mode control for fractional-order dynamical system Nonlinear Dyn. vol. 8 pp. 249-267 April 25. [8].-C. Lin and.-y. Lee Chaos synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive fuzzy sliding mode control IEEE ran. Fuzzy Sys. vol. 9 pp. 623635 Aug. 2. [9] B. Zhang and. Guo Universal function projective lag synchronization of chaotic systems with uncertainty by using active sliding mode and fuzzy sliding mode control Nonlinear Dyn. vol. 8 pp. 867-879 July 25. [2] B. Zhang and. Li Universal projective synchronization of two different hyperchaotic systems with unknown parameters Journal of Applied Mamatics vol. 5 pp. - March 24. [2] D. Matignon Stability results of fractional differential equations with applications to control processing Computational Engineering in Systems Applications vol. 8 pp. 96-22 996. [22] K. Dielm N. J. Ford and A. D. Freed A predictor-corrector approach for numerical solution of fractional differential equations Nonlinear Dyn. vol. 29 pp. 3-22 July 22. [23] K. Dielm N. J. Ford and A. D. Freed Detailed error analysis for a fractional adams method Numer. Algorithms vol. 36 pp. 3-52 May 24. Yancheng Ma is a doctoral candidate which comes from School of Optical and Electronic Information uazhong University of Science and echnology Wuhan 4374 P. R. China. e was born in Jan. 6 989. Yancheng Ma has published two articles which are indexed by EI and SCI. is research interests are artificial intelligence chaos synchronization optical communication. Guoan Wu is a professor in School of Optical and Electronic Information uazhong University of Science and echnology Wuhan 4374 P. R. China. Guoan Wu s research interests are artificial intelligence microwave technology. Lan Jiang is a teaching assistant in Wuhan Railway Vocational College of echnology. er research interests are intelligent control and chaos synchronization.