Research Article Modified Function Projective Synchronization between Different Dimension Fractional-Order Chaotic Systems

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1 Absrac and Applied Analysis Volume 212, Aricle ID , 12 pages doi:1.1155/212/ Research Aricle Modified Funcion Projecive Synchronizaion beween Differen Dimension Fracional-Order Chaoic Sysems Ping Zhou 1, 2 and Rui Ding 1 1 Research Cener for Sysem Theory and Applicaions, Chongqing Universiy of Poss and Telecommunicaions, Chongqing 465, China 2 Key Laboraory of Indusrial Inerne of Things and Neworked Conrol of he Minisry of Educaion, Chongqing Universiy of Poss and Telecommunicaions, Chongqing 465, China Correspondence should be addressed o Ping Zhou, zhouping@cqup.edu.cn Received 2 Augus 212; Acceped 13 Augus 212 Academic Edior: Jinhu Lü Copyrigh q 212 P. Zhou and R. Ding. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. A modified funcion projecive synchronizaion MFPS scheme for differen dimension fracionalorder chaoic sysems is presened via fracional order derivaive. The synchronizaion scheme, based on sabiliy heory of nonlinear fracional-order sysems, is heoreically rigorous. The numerical simulaions demonsrae he validiy and feasibiliy of he proposed mehod. 1. Inroducion Fracional-order calculus, which can be daed back o he 17h cenury 1, 2. However, only in he las few decades, is applicaion o physics and engineering has been addressed. So, he fracional-order calculus has araced increasing aenion only recenly. On he oher hand, complex bifurcaion and chaoic phenomena have been found in many fracionalorder dynamical sysems. For example, he fracional-order Lorenz chaoic sysem 3, he fracional-order unified chaoic sysem 4, he fracional-order Chua chaoic circui 5, he fracional-order modified Duffing chaoic sysem 6, and he fracional-order Rössler chaoic sysem 7, 8, andsoon. Nowadays, synchronizaion of chaoic sysems and fracional-order chaoic sysems has araced much aenion because of is applicaions in secure communicaion and conrol processing Many approaches have been repored for he synchronizaion of chaoic sysems and fracional-order chaoic sysems In 1999, Mainieri and Rehacek proposed projecive synchronizaion PS 12 for chaoic sysems, which has

2 2 Absrac and Applied Analysis been exensively invesigaed in recen years because of is proporional feaure in secure communicaions. Recenly, a modified projecive synchronizaion, which is called funcion projecive synchronizaion FPS has been repored. In FPS, he maser and slave sysems could be synchronized up o a scaling funcion, bu no a consan. So, he unpredicabiliy of he scaling funcion in FPS can addiionally enhance he securiy of communicaion. To he bes of our knowledge, mos of he exising FPS scheme for he fracional-order chaoic sysems only discuss he same dimension. However, in many real physics sysems, he synchronizaion is carried ou hrough he oscillaors wih differen dimension, especially he sysems in biological science and social science Moreover, in some previous works 16, 17, all he nonlinear erms of response sysem or error sysem was absorbed. Referring o chaoic synchronizaion via fracional-order conroller, here are a few resuls repored unil now. Inspired by he above discussion, in his paper, we presen a modified funcion projecive synchronizaion MFPS scheme beween differen dimension fracionalorder chaoic sysems via fracional-order conroller. The fracional-order conroller is easily designed. The synchronizaion echnique, based on racking conrol and sabiliy heory of nonlinear fracional-order sysems, is heoreically rigorous. Our modified funcion projecive synchronizaion MFPS scheme need no absorb all he nonlinear erms of response sysem. This is differen from some previous works. Two examples are presened o demonsrae he effeciveness of he proposed MFPS scheme. This paper is organized as follows. In Secion 2, a modified funcion projecive synchronizaion MFPS scheme is presened. In Secion 3, wo groups of examples are used o verify he effeciveness of he proposed scheme. The conclusion is finally drawn in Secion The MFPS Scheme for Differen Dimension Fracional-Order Chaoic Sysems The fracional-order chaoic drive and response sysems wih differen dimension are defined as follows, respecively: d q d x d q F d d x, d q r y ( ) ( ) d q F r r y C x, y, where q d <q d < 1 and q r <q r < 1 are fracional order, and q d may be differen wih q r. x R n, y R m n / m are sae vecors of he drive sysem 2.1 and response sysem 2.2, respecively. F d : R n R n, F r : R m R m are wo coninuous nonlinear vecor funcions, and C x, y R m is a conroller which will be designed laer. Definiion 2.1. For he drive sysem 2.1 and response sysem 2.2, i is said o be modified funcion projecive synchronizaion MFPS if here exis a conroller C x, y such ha: lim e lim y M x x, 2.3

3 Absrac and Applied Analysis 3 where is he Euclidean norm, M x is a m n real marix, and marix elemen M ij x i 1, 2,...m, j 1, 2,...,n are coninuous bounded funcions. e i y i n M ijx j i 1, 2,...m are called MFPS error. Remark 2.2. According o he view of racking conrol, M x x can be chosen as a reference signal. The MFPS in our paper is ransformed ino he problem of racking conrol, ha is he oupu signal y in sysem 2.2 follows he reference signal M x x. In order o achieve he oupu signal y follows he reference signal M x x. Now,we define a compensaion conroller C 1 x R m for response sysem 2.2 via fracional-order derivaive d q r M x x /d q r. The compensaion conroller is shown as follows: C 1 x dq r M x x d q r F r M x x, 2.4 and le conroller C x, y as follows: C ) C 1 x C 2 ), 2.5 where C 2 x, y R m is a vecor funcion which will be designed laer. By conroller 2.5 and compensaion conroller 2.4, he response sysem 2.2 can be changed as follows: d q r e d q r D 1 ) e C2 ), 2.6 where D 1 x, y e F r y F r M x x, andd 1 x, y R m m. So, he MFPS beween drive sysem 2.1 and response sysem 2.2 is ransformed ino he following problem: choose a suiable vecor funcion C 2 x, y such ha sysem 2.6 is asympoically converged o zero. In wha follows we presen he sabiliy heorem for nonlinear fracional-order sysems of commensurae order Consider he following nonlinear commensurae fracionalorder auonomous sysem D q x f x, 2.7 he fixed poins of sysem 2.7 is asympoically sable if all eigenvalues λ of he Jacobian marix A f/ x evaluaed a he fixed poins saisfy arg λ >.5πq. Where <q<1, x R n, f : R n R n are coninuous nonlinear funcions, and he fixed poins of his nonlinear commensurae fracional-order sysem are calculaed by solving equaion f x. Now, he following heorem is given based on he above discussion in order o achieve he MFPS beween he drive sysem 2.1 and he response sysem 2.2. Theorem 2.3. Choose he conrol vecor C 2 x, y D 2 x, y e, and if D 1 x, y D 2 x, y saisfy he following condiions: 1 d ij d ji i / j, 2 d ii (all d ii are no equal o zero),

4 4 Absrac and Applied Analysis hen he modified funcion projecive synchronizaion (MFPS) beween 2.1 and 2.2 can be achieved. Where D 2 x, y R m m, and d ij i, j 1, 2,...m, for all d ij R are he marix elemen of marix D 1 x, y D 2 x, y. Proof. Using C 2 x, y D 2 x, y e, so fracional-order sysem 2.6 can be rewrien as follows: d q r e d q r [ D 1 ) D2 )] e. 2.8 Suppose λ is one of he eigenvalues of marix D 1 x, y D 2 x, y and he corresponding non-zero eigenvecor is ψ, hais, [ D1 ) D2 )] ψ λψ. 2.9 Take conjugae ranspose H on boh sides of 2.9, we yield {[ D1 ) D2 )] ψ } T λψ H. 2.1 Equaion 2.9 muliplied lef by ψ H plus 2.1 muliplied righ by ψ, we derive ha ψ H{ [ ( ) ( )] [ ( ) ( )] ( ) H D1 x, y D2 x, y D1 x, y D2 x, y }ψ ψ H ψ λ λ So, ψ H{ [ ( ) ( )] [ ( ) ( )] H D1 x, y D2 x, y D1 x, y D2 x, y }ψ λ λ ψ H ψ Because d ij d ji i / j, for all d ij R in marix D 1 x, y D 2 x, y, so λ λ ( 2d11 ) ψ H 2d 22 ψ 2d mm. ψ H ψ 2.13 Because d ii for all d ii R,andalld ii are no equal o zero. So, λ λ From 2.14, we have arg λ [ D1 ) D2 )].5π >.5qr π. 2.15

5 Absrac and Applied Analysis 5 According o he sabiliy heorem for nonlinear fracional-order sysems of commensurae order 22 25,sysem 2.8 is asympoically sable. Tha is lim e Therefore, lim e lim y M x x This indicaes ha he modified funcion projecive synchronizaion beween drive sysem 2.1 and response sysem 2.2 will be obained. The proof is compleed. Remark 2.4. Theorem 2.3 indicaes ha he condiion of he MFPS beween drive sysem 2.1 and response sysem 2.2 are arg λ D 1 x, y D 2 x, y >.5q r π. So, in pracical applicaions, we can easily choose he marix D 2 x, y according o he marix D 1 x, y. Moreover, in order o reserve all he nonlinear erms in response sysem or error sysem, he conroller in our work may be complex han he conroller repored by 16, 17. Bu,all he nonlinear erms in response sysem or error sysem are absorbed in 16, 17. Remark 2.5. Perhaps our resul can be exended o he modified funcion projecive synchronizaion of complex neworks of fracional order chaoic sysems and he complex fracional-order muli scroll chaoic sysems Bu, he modified funcion projecive synchronizaion for complex neworks and complex fracional-order muli-scroll chaoic sysems would be much more complex. Furher work on his issue is an ongoing research opic in our group. 3. Applicaions In his secion, o illusrae he effeciveness of he proposed MFPS scheme for differen dimension fracional-order chaoic sysems. Two groups of examples are considered and heir numerical simulaions are performed The MFPS beween 3-Dimensional Fracional-Order Lorenz Sysem and 4-Dimensional Fracional-Order Hyperchaoic Sysem The fracional-order Lorenz 3 sysem is described as follows: y 1 1 ( y 2 y 1 ) y 2 28y 1 y 2 y 1 y 3 y 3 y 1 y 2 8y The fracional-order Lorenz sysem exhibis chaoic behavior 3 for q r.993. The chaoic aracor for q r.995 is shown in Figure 1.

6 6 Absrac and Applied Analysis y3 5 y y 1 2 y y 1 a b Figure 1: Chaoic aracors of he fracional-order Lorenz sysem 3.1 for q r x3 x4 5 x x x 3 5 x 2 5 a b Figure 2: Hyperchaoic aracors of he fracional-order sysem 3.2 for q d.95. Recenly, Pan e al. consruced a hyperchaoic sysem 17. Is corresponded fracional-order sysem is described as follows: x 1 1 x 2 x 1 x 4 x 2 28x 1 x 1 x 3 x 3 x 1 x 2 8x 3 3 x 4 x 1 x 3 1.3x The hyperchaoic aracor of sysem 3.2 for q d.95 is shown in Figure 2. Consider he fracional-order hyperchaoic sysem 3.2 wih fracional-order q d.95 as drive sysem, and he fracional-order Loren sysem wih fracional-order q r.995 as response sysem. According o he above menioned, we can obain 1 1 ( ) ( ) 28 y M 1j x j F r y Fr M x x D 1 x, y e e y 2 1j x j M 8 3

7 Absrac and Applied Analysis 7 Now, we can choose ( ) y 2 D 2 x, y 38 y So, 1 1 y 2 ( ) ( ) M 1j x j D 1 x, y D2 x, y 4 y 2 1j x j M According o he above heorem, he MFPS beween he 3-dimensional fracional-order Lorenz sysem 3.1 and he 4-dimensional ( fracional-order ) hyperchaoic sysem 3.2 can 1 x2 2 1 x 2 3 be achieved. For example, choose M x. The corresponding numerical 2 x 1 1 x x 4.5 resul is shown in Figure 3, in which he iniial condiions are x 2, 1, 2, 1 T,andy 18, 13, 13.5 T, respecively The MFPS beween 4-Dimensional Fracional-Order Hyperchaoic Lǔ Sysem and 3-Dimensional Fracional-Order Arneodo Chaoic Sysem In 22, Lü and Chen repored a new chaoic sysem 32, which be called Lü chaoic sysem. The Lü chaoic sysem is differen from he Lorenz and Chen sysem. Based on Lü chaoic sysem, he hyperchaoic Lü chaoic sysem and he fracional-order hyperchaoic Lü sysem have been consruced recenly. The fracional-order hyperchaoic Lüsysem 16 is described by he following y 1 36 ( y 2 y 1 ) y4 y 2 2y 2 y 1 y 3 y 3 y 1 y 2 3y 3 y 4 y 1 y 3 y The hyperchaoic aracor of sysem 3.6 for q r.96 is shown in Figure 4. The fracional order Arneodo chaoic sysem 16 is defined as follows: x 1 x 2 x 2 x 3 x 3 5.5x 1 3.5x 2 x 3 x The chaoic aracor of sysem 3.7 for q d.998 is shown in Figure 5.

8 8 Absrac and Applied Analysis e1 e a b 1 5 e3 5 1 c 1 2 Figure 3: The modified funcion projecive synchronizaion errors beween he fracional-order Lorenz sysem 3.1 and he fracional-order hyperchaoic sysem 3.2. y3 4 2 y4 1 5 y y y 3 5 y 2 5 a b Figure 4: Hyperchaoic aracors of he fracional-order hyperchaoic Lǔ sysem 3.6 for q r.96. Consider he fracional-order Arneodo chaoic sysem 3.7 wih fracional-order q d.998 as drive sysem, and he fracional-order hyperchaoic Lǔ sysem 3.6 wih fracionalorder q r.96 as response sysem. According o he above menioned, we can yield y M 1j x j ( ) ( ) F r y Fr M x x D 1 x, y e 3 y 2 M 1j x j 3 e y 3 M 1j x j 1

9 Absrac and Applied Analysis x3 x3 1 1 x x x 1 a b Figure 5: Chaoic aracors of he fracional-order Arneodo chaoic sysem 3.7 for q d.998. Now, we can choose y 2 ( ) 36 y 3 21 D 2 x, y M 1j x j 1 y 3 So, y M 1j x j ( ) ( ) D 1 x, y D2 x, y 3 y 2 M 1j x j 3 3 M 1j x j M 1j x j 1 According o above heorem, he MFPS beween he 4-dimensional fracional-order hyperchaoic Lǔsysem 3.6 and he 3-dimensional fracional-order Arneodo ) chaoic sysem 3.7 can be achieved. For example, choose M x. The corresponding ( 1 x2 1 x 3.5 x x 1 1 numerical resul is shown in Figure 6, in which he iniial condiions are x 2, 2, 2 T,and y 11, 1, 11, 2 T, respecively. 4. Conclusions In his paper, based on he sabiliy heory of he fracional-order sysem and he racking conrol, a modified funcion projecive synchronizaion scheme for differen dimension fracional-order chaoic sysems is addressed. The derived mehod in he presen paper shows ha he modified funcion projecive synchronizaion beween drive sysem and response sysem wih differen dimensions can be achieved. The modified funcion projecive synchronizaion beween 3-dimensional fracional-order Lorenz sysem and 4-dimensional

10 1 Absrac and Applied Analysis 6 4 e1 4 2 e a b e3 e c d Figure 6: The modified funcion projecive synchronizaion errors beween he fracional-order sysem 3.6 and he following fracional-order sysem: fracional-order hyperchaoic sysem, and he modified funcion projecive synchronizaion beween he 4-dimensional fracional-order hyperchaoic Lǔ sysem, and he 3-dimensional fracional-order Arneodo chaoic sysem, are chosen o illusrae he proposed mehod. Numerical experimens shows ha he presen mehod works very well, which can be used for oher chaoic sysems. Acknowledgmens The auhors are very graeful o he anonymous reviewers for heir valuable commens and suggesions, which have grealy improved he presenaion of his paper. This work is suppored by he Foundaion of Science and Technology projec of Chongqing Educaion Commission under Gran KJ11525, and by he Naional Naural Science Foundaion of China References 1 I. Podlubny, Fracional Differenial Equaions, vol. 198, Academic Press, San Diego, Calif, USA, R. Hilfer, Applicaions of Fracional Calculus in Physics, World Scienific, River Edge, NJ, USA, I. Grigorenko and E. Grigorenko, Chaoic dynamics of he fracional Lorenz sysem, Physical Review Leers, vol. 91, no. 3, pp , X. J. Wu, J. Li, and G. R. Chen, Chaos in he fracional order unified sysem and is synchronizaion, he Franklin Insiue, vol. 345, no. 4, pp , T. T. Harley, C. F. Lorenzo, and H. K. Qammer, Chaos in a fracional order Chua s sysem, IEEE Transacions on Circuis and Sysems I, vol. 42, no. 8, pp , Z. M. Ge and C. Y. Ou, Chaos in a fracional order modified Duffing sysem, Chaos, Solions & Fracals, vol. 34, no. 2, pp , 27.

11 Absrac and Applied Analysis 11 7 Y. G. Yu and H. X. Li, The synchronizaion of fracional-order Rössler hyperchaoic sysems, Physica A, vol. 387, no. 5-6, pp , A. Freiha and S. Momani, Adapaion of differenial ransform mehod for he numeric-analyic soluion of fracional-order Rössler chaoic and hyperchaoic sysems, Absrac and Applied Analysis, Aricle ID , 13 pages, N. X. Quyen, V. V. Yem, and T. M. Hoang, A chaoic pulse-ime modulaion mehod for digial communicaion, Absrac and Applied Analysis, vol. 212, Aricle ID 83534, 15 pages, D. Y. Chen, W. L. Zhao, X. Y. Ma, and R. F. Zhang, Conrol and cynchronizaion of chaos in RCL- Shuned Josephson Juncion wih noise disurbance using only one conroller erm, Absrac and Applied Analysis, vol. 212, Aricle ID , 14 pages, Y. Y. Hou, Conrolling chaos in permanen magne synchronous moor conrol sysem via fuzzy guaraneed cos conroller, Absrac and Applied Analysis, vol. 212, Aricle ID 65863, 1 pages, R. Mainieri and J. Rehacek, Projecive synchronizaion in he hree-dimensional chaoic sysems, Physical Review Leers, vol. 82, no. 15, pp , Y. Yu and H.-X. Li, Adapive generalized funcion projecive synchronizaion of uncerain chaoic sysems, Nonlinear Analysis, vol. 11, no. 4, pp , Y. Chen, H. An, and Z. Li, The funcion cascade synchronizaion approach wih uncerain parameers or no for hyperchaoic sysems, Applied Mahemaics and Compuaion, vol. 197, no. 1, pp , H. An and Y. Chen, The funcion cascade synchronizaion mehod and applicaions, Communicaions in Nonlinear Science and Numerical Simulaion, vol. 13, no. 1, pp , S. Wang, Y. G. Yu, and M. Diao, Hybrid projecive synchronizaion of chaoic fracional order sysems wih differen dimension, Physica A, vol. 389, no. 21, pp , L. Pan, W. Zhou, L. Zhou, and K. Sun, Chaos synchronizaion beween wo differen fracional-order hyperchaoic sysems, vol. 16, no. 6, pp , X. Y. Wang and J. M. Song, Synchronizaion of he fracional order hyperchaos Lorenz sysems wih acivaion feedback conrol, Communicaions in Nonlinear Science and Numerical Simulaion, vol. 14, no. 8, pp , R. Zhang and S. Yang, Adapive synchronizaion of fracional-order chaoic sysems via a single driving variable, Nonlinear Dynamics, vol. 66, no. 4, pp , R. X. Zhang and S. P. Yang, Sabilizaion of fracional-order chaoic sysem via a single sae adapivefeedback conroller, Nonlinear Dynamic, vol. 68, no. 1-2, pp , D. Cafagna and G. Grassi, Observer-based projecive synchronizaion of fracional sysems via a scalar signal: applicaion o hyperchaoic Rössler sysems, Nonlinear Dynamic, vol.68,no.1-2,pp , M. S. Tavazoei and M. Haeri, Chaos conrol via a simple fracional-order conroller, Physics Leers A, vol. 372, no. 6, pp , E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka, Equilibrium poins, sabiliy and numerical soluions of fracional-order predaor-prey and rabies models, Mahemaical Analysis and Applicaions, vol. 325, no. 1, pp , Z. M. Odiba, Adapive feedback conrol and synchronizaion of non-idenical chaoic fracional order sysems, Nonlinear Dynamics, vol. 6, no. 4, pp , Z. Odiba, A noe on phase synchronizaion in coupled chaoic fracional order sysems, Nonlinear Analysis, vol. 13, no. 2, pp , J. Lü, X. Yu, G. Chen, and D. Cheng, Characerizing he synchronizabiliy of small-world dynamical neworks, IEEE Transacions on Circuis and Sysems I, vol. 51, no. 4, pp , J. Zhou, J.-a. Lu, and J. Lü, Adapive synchronizaion of an uncerain complex dynamical nework, IEEE Transacions on Auomaic Conrol, vol. 51, no. 4, pp , J. Lü and G. Chen, A ime-varying complex dynamical nework model and is conrolled synchronizaion crieria, IEEE Transacions on Auomaic Conrol, vol. 5, no. 6, pp , J. Lü and G. Chen, Generaing muliscroll chaoic aracors: heories, mehods and applicaions, Inernaional Bifurcaion and Chaos in Applied Sciences and Engineering, vol. 16, no. 4, pp , J. H. Lü, S. M. Yu, H Leung, and G. R. Cheng, Experimenal verificaion of mulidirecional muliscroll chaoic aracors, IEEE Transacions on Circuis and Sysems I, vol. 53, no. 1, pp , 26.

12 12 Absrac and Applied Analysis 31 J. Lü, F. Han, X. Yu, and G. Chen, Generaing 3-D muli-scroll chaoic aracors: a hyseresis series swiching mehod, Auomaica, vol. 4, no. 1, pp , J. Lü and G. Chen, A new chaoic aracor coined, Inernaional Bifurcaion and Chaos in Applied Sciences and Engineering, vol. 12, no. 3, pp , 22.

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