Research Article Synchronization of the Extended Bonhoffer-Van der Pol Oscillators

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1 Mahemaical Problems in Engineering Volume 212, Aricle ID 96268, 14 pages doi:.11/212/96268 Research Aricle Synchronizaion of he Exended Bonhoffer-Van der Pol Oscillaors Mohamed Zribi and Saleh Alshamali Deparmen of Elecrical Engineering, Kuwai Universiy, P.O. Box 969, Safa 136, Kuwai Correspondence should be addressed o Mohamed Zribi, mohamed.zribi@ku.edu.kw Received 19 February 212; Revised 4 May 212; Acceped 8 May 212 Academic Edior: Jun-Juh Yan Copyrigh q 212 M. Zribi and S. Alshamali. 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. This paper deals wih he synchronizaion of wo exended Bonhoffer-Van der Pol B-VDP oscillaors. A Lyapunov-based conroller and a sliding mode conroller are proposed for he synchronizaion of he oscillaors. Boh design schemes use a single inpu conroller acing on one sae only. Asympoic sabiliy resuls for he closed-loop sysem are derived using Lyapunov heory. I is shown ha he proposed conrollers effecively synchronize he driver and he response sysems for he case when nominal values of he sysem parameers are used, and for he case when parameer perurbaions are inroduced. Simulaion resuls are presened o show he effeciveness of he proposed conrollers. 1. Inroducion Chaoic sysems represen an ineresing class of nonlinear sysems. They are known o be highly sensiive o small changes in he iniial condiions 1. In he pas wo decades, he problem of synchronizaion and anisynchronizaion of chaoic sysems has araced a lo of ineres. The synchronizaion phenomenon has been used in several key applicaions such as nework securiy 2 and biological oscillaors 3. Several conrol sraegies were suggesed for he synchronizaion and ani-synchronizaion of chaoic sysems; he suggesed sraegies include backsepping conrol, adapive conrol, and acive conrol. The resuls of some of hese works are discussed below. In 4, a Lyapunov-based conroller is developed o conrol he chaoic Duffing equaion; he conroller incorporaes an observer ha helps overcome parameric as well as exernal disurbances. An adapive backsepping conrol law which is designed o conrol a class of chaoic sysems is given in ; he chaoic sysem is ransformed ino a nonauonomous sric feedback form, hen he developed backsepping conroller is applied o asympoically rack a given reference signal. In 6, an inegral sliding mode conroller SMC wih a dynamic

2 2 Mahemaical Problems in Engineering sliding surface is proposed for he conrol of an exended Duffing-Van der Pol oscillaor. The work in 7 proposes a combinaion of a sae-feedback conrol and an adapive conrol o synchronize a Van der Pol oscillaor coupled wih a linear oscillaor. A sae-feedback conroller is used o sabilize he equilibrium poin, hen he adapive conrol is uilized o synchronize wo of such oscillaors. In 8, a variable srucure conroller VSC is uilized o synchronize wo chaoic sysems. The chaoic sysems are firs ransformed ino he canonical form using he heory of differenial geomery, hen chaos suppression is achieved by applying he proposed conroller. The performances of an acive conroller and a backsepping conroller in synchronizing a class of chaoic sysems are compared in 9. I is shown ha boh conrol schemes have an accepable ransien performance. However, he acive conroller is more complex han he backsepping conroller. In, a sliding mode conroller is used in he conrol and synchronizaion of a class of uncerain chaoic sysems, where he conroller incorporaes an adapaion law o ensure perfec synchronizaion. A nonlinear feedback conroller, proposed for he synchronizaion of a class of chaoic sysems known as he Lienard chaoic sysem, is given in 11. The conrol design is based on Lyapunov heory. Sufficien condiions ensuring asympoic sabiliy are given in erms of he gains of he conroller. In 12, an acive conrol scheme is used for he synchronizaion and anisynchronizaion of he exended Bonhoffer-Van der Pol oscillaors. The synchronizaion of a class of uncerain chaoic sysems via sliding mode conrol is presened in 13 ; he conroller uilizes a disurbance observer based on a Radial Basis Funcion RBF neural nework. The work in 14 proposes a nonlinear conroller for he synchronizaion of a Van der Pol oscillaor wih a Chen chaoic circui. In his work, he exended Bonhoffer-Van der Pol B-VDP oscillaor model presened in 12 is considered. A Lyapunov-based conroller and a sliding mode conroller SMC are proposed for he synchronizaion of he wo chaoic B-VDP oscillaors. Sliding mode conrol is well known for is fas response and disurbance rejecion properies 1, 16. Unlike he work in 12 where hree conrol acuaors are used, in his work, a single conrol law is fed o he response sysem and designed o achieve complee synchronizaion beween he drive and he response sysems. Furhermore, for he proposed SMC design, a ransformaion is inroduced o ransform he error dynamics ino a canonical form. The ransformaion grealy simplifies he design of he conroller. The paper is organized as follows. The model of he exended B-VDP oscillaor circui is inroduced in Secion 2. Secion 3 presens a Lyapunov-based conroller o synchronize wo B-VDP oscillaors. The design of a sliding mode conroller o synchronize he wo B-VDP Oscillaors is given in Secion 4. Simulaion resuls are given and discussed in Secion. Finally, some concluding remarks are given in Secion The Model of he Exended Bonhoffer-Van der Pol Oscillaor The exended B-VDP oscillaor circui consiss of wo capaciors, an inducor, a linear resisor and a nonlinear resisor see Figure 1. Is dynamic model can be described by he following se of ordinary differenial equaions 12 : C dv 1 d i g v 1,

3 Mahemaical Problems in Engineering 3 L R C + + v 2 v 1 g(v) C Figure 1: The exended B-VDP oscillaor circui. C dv 2 d i v 2 R, L di d v 1 v The saes of sysem 2.1 are v 1, v 2,andi; hese saes are defined as follows: v 1 : he volage across he firs capacior, v 2 : he volage across he second capacior, i: he curren hrough he inducor, R, C, L: he resisance, he capaciances, and he inducance in he circui, g v 1 : he nonlinear resisor defined such ha g v 1 av 1 b anh cv 1, 2.2 where a and b are parameers of he sysem. In order o faciliae he conrol design, he exended B-VDP dynamic model 2.1 is expressed in a normalized form by redefining he parameers and sae variables of he sysem 12. To his end, le he new ime variable be given as τ 1/ LC. Also, define he new sae vecor, x, y, z T, such ha x v 1 /b C/L, y v 2 /b C/L, andz i/b. Finally, define he parameers p 1, p 2,andp 3 such ha p 1 a L/C, p 2 bc L/C, andp 3 1/R L/C. Hence, he normalized model of he exended B-VDP oscillaor is such ha ẋ z p 1 x anh ( p 2 x ), ẏ z p 3 y, 2.3 ż x y, where he denoes he differeniaion wih respec o τ.

4 4 Mahemaical Problems in Engineering Generally, synchronizaion involves wo sysems: he firs sysem is known as he drive maser sysem, and he second one is known as he response slave sysem. The dynamic model of he drive circui of he B-VDP oscillaor be given as ẋ d z d p 1 x d anh ( p 2 x d ), ẏ d z d p 3 y d, 2.4 ż d x d y d, where x d, y d,andz d are he saes of he drive sysem. Moreover, he dynamic model of he response sysem can be wrien as follows: ẋ r z r p 1 x r anh ( p 2 x r ) u, ẏ r z r p 3 y r, 2. ż r x r y r. Noe ha he firs differenial equaion of he response sysem conains he forcing erm u. This erm represens he conroller of he sysem, which will be designed such ha he drive sysem and he response sysem are synchronized afer saring from differen iniial condiions. I is noed ha he synchronizaion of he wo sysems is achieved hrough he use of a single inpu conroller acing on one sae only. Minimizing he number of acuaors is good from an implemenaion poin of view, as he implemenaion becomes cos effecive. For he conrol design, he error vecor e e 1 e 2 e 3 T is defined such ha he errors beween he saes of he response sysem and he drive sysem are given by e 1 x r x d, e 2 y r y d, 2.6 e 3 z r z d. Using , he error dynamics can be wrien as ė 1 e 3 p 1 e 1 anh ( p 2 x r ) anh ( p2 x d ) u, ė 2 e 3 p 3 e 2, 2.7 ė 3 e 1 e 2. Obviously, complee synchronizaion beween he drive sysem and he response sysem is achieved when he errors converge o zero. Therefore, he objecive of his work is o design conrollers which force he errors o asympoically converge o zero as ends o infiniy. In Secions 3 and 4, wo conrol design mehodologies are inroduced o accomplish he synchronizaion ask.

5 Mahemaical Problems in Engineering 3. Design of a Lyapunov-Based Conroller In his secion, a Lyapunov-based conroller is used o force he errors beween he drive and he response circuis o zero as ends o infiniy and hence synchronize he exended B-VDP oscillaors. Le α be a posiive scalar. Proposiion 3.1. The conrol law: u ( p 1 α ) e 1 anh ( p 2 x r ) anh ( p2 x d ) 3.1 renders he error sysem 2.7 globally asympoically sable. Proof. Le he Lyapunov funcion candidae V 1 be such ha V e e e Taking he ime derivaive of V 1, and using expressions 2.7 and 3.1 of he dynamic model of he errors and he conrol law, respecively, gives V 1 e 1 ė 1 e 2 ė 2 e 3 ė 3 e 1 ( e3 p 1 e 1 anh ( p 2 x r ) anh ( p2 x d ) u ) e2 ( e3 p 3 e 2 ) e3 e 1 e αe 2 1 p 3e 2 2. Clearly, V 1 is a posiive definie funcion defined over R 3 such ha V 1 is negaive semidefinie in R 3. In order o show he asympoic sabiliy of he closed-loop sysem, we will use LaSalle s heorem 17. Define he compac se Ω such ha Ω { } e R 3 V Solving he equaion V 1, and using he fac ha α and p 3 are posiive consans, we ge V 1 αe 2 1 p 3e 2 2 e 1, e Noe ha e 1 implies ha ė 1, and ha e 2 implies ha ė 2. Now, using he second differenial equaion given in 2.7, i follows ha ė 2 e 3 p 3 e 2 e Therefore, he only soluion of he equaion V 1 ise 1, e 2, and e 3. This means ha no soluion can say idenically in Ω oher han he rivial soluion,, T.

6 6 Mahemaical Problems in Engineering Hence, from LaSalle s invariance heorem, i can be concluded ha he origin of he error sysem 2.7 is asympoically sable. Finally, he Lyapunov funcion V 1 saisfies lim V 1. e 3.7 Thus V 1 is radially unbounded and hence he origin is globally asympoically sable. Therefore, i can be concluded ha he conrol law 3.1 when applied o he error sysem 2.7 guaranees he convergence of he errors e 1, e 2,ande 3 o zero as ends o infiniy. I should be noed ha since he errors e 1, e 2,ande 3 asympoically converge o zero as ends o infiniy, hen x r, y r,andz r converge o x d, y d,andz d, respecively, as ends o infiniy. Therefore, he saes of he drive and he response chaoic sysems are synchronized. Remark 3.2. The parameer α in he conrol law 3.1 can be aken o be zero. However, he inclusion of he erm αe 1 speeds up he convergence of he error rajecories o zero. 4. Design of a Sliding Mode Conroller This secion deals wih he design of a sliding mode conroller o synchronize wo exended B-VDP oscillaors. To faciliae he design of he conroller, he following ransformaion is inroduced: ζ 1 e 2, ζ 2 e 3 p 3 e 2, ) ζ 3 e 1 (p e 2 p 3 e The inverse of his ransformaion is defined for all values of p 3 and i is such ha: e 1 ζ 1 p 3 ζ 2 ζ 3, e 2 ζ 1, 4.2 e 3 p 3 ζ 1 ζ 2. Using he ransformaion 4.1, he error dynamics given by 2.7 can be wrien using he new coordinaes as ζ 1 ζ 2, ζ 2 ζ 3, 4.3 ζ 3 f u,

7 Mahemaical Problems in Engineering 7 where he funcion f is given by ( ) f e 3 p 1 e 1 p (e3 ) p 3 e 2 p3 e 1 e 2 anh ( ) ( ) p 2 x r anh p2 x d. 4.4 Remark 4.1. The funcion f can be wrien using he ζ 1,ζ 2,ζ 3 coordinaes such ha: f ( p 1 p 3 ) ζ1 ( p 1 p 3 2 ) ζ 2 ( p 1 p 3 ) ζ3 anh ( p 2 x r ) anh ( p2 x d ). 4. The firs sep in he design of a sliding mode conroller is o define a sliding surface owards which he sysem rajecories mus be forced. We will define he linear sliding surface σ such ha: σ ζ 3 β 1 ζ 1 β 2 ζ 2, 4.6 where β 1 and β 2 are posiive scalars such ha he polynomial P x s 2 β 2 s β 1 is Hurwiz. The derivaive of σ wih respec o ime is given by σ ζ 3 β 1 ζ 1 β 2 ζ 2 f β 1 ζ 2 β 2 ζ 3 u. 4.7 The objecive is o design a conrol law, u, such ha he sysem rajecories reach he sliding surface in finie ime and say here for all subsequen ime. Define he sign funcion such ha 1 if σ>, sign σ if σ, 1 if σ<. 4.8 Also, le K and W be posiive scalars. Proposiion 4.2. The sliding mode conroller: u f β 1 ζ 2 β 2 ζ 3 Kσ W sign σ 4.9 renders he ransformed error sysem 4.3 asympoically sable. Proof. Firs, we will show ha he proposed conroller forces he dynamics of he sysem o reach he sliding surface in finie ime. Then, he closed-loop sabiliy of he sysem on he sliding surface is proved. To prove he reachabiliy o σ, i suffices o show ha σ σ <. To his end, consider he Lyapunov funcion candidae V 2 1/2 σ 2. Taking he derivaive of

8 8 Mahemaical Problems in Engineering y() x() Figure 2: Plo of x versus y wih no conrol. V 2 wih respec o ime and using he expressions 4.7 and 4.9 for σ and u, respecively, i follows ha V 2 σ ( f β 1 ζ 2 β 2 ζ 3 u ) σ ( Kσ W sign σ ) 4. Kσ 2 W σ. Since K and W are chosen o be posiive scalars, i follows ha V 2 < forσ / and V 2 for σ. Hence V 2 is a Lyapunov funcion for he sysem. Therefore, i can be concluded ha he finie ime reachabiliy o he surface σ is guaraneed. Noe ha upon reaching he sliding surface, we have σ and σ. On he sliding surface σ,hesaeζ 3 can be wrien as ζ 3 β 1 ζ 1 β 2 ζ 2. Therefore, he closed-loop dynamics of he reduced order sysem i.e., he sysem when σ can be wrien as ζ 1 ζ 2, ζ 2 β 1 ζ 1 β 2 ζ Since β 1 and β 2 are chosen o be posiive scalars such ha he polynomial P x s 2 β 2 s β 1 is Hurwiz, i can be concluded ha he saes ζ 1 and ζ 2 will converge o zero asympoically as ends o infiniy. Moreover, since ζ 3 β 1 ζ 1 β 2 ζ 2, i follows ha ζ 3 will asympoically converge o zero as ends o infiniy. I is clear from he ransformaion 4.1 ha he asympoic convergence of ζ 1, ζ 2 and ζ 3 o zero implies he asympoic convergence of he errors e 1, e 2,ande 3 o zero. Using 2.6, i is noed ha when he errors e 1, e 2 and e 3 converge o zero as ends o infiniy, hen he convergence of x r, y r,andz r o x d, y d,andz d, respecively, is guaraneed as ends o infiniy. Therefore, he saes of he drive and he response chaoic sysems are synchronized.

9 Mahemaical Problems in Engineering 9 xr versus xd x d x r a yr versus yd y d y r b zr versus zd z d z r c Figure 3: Plos of he exended B-VDP oscillaors sae rajecories versus ime when he Lyapunov-based conroller is used. Remark 4.3. To reduce or eliminae he chaering associaed wih he proposed sliding mode conroller, we can use he hyperbolic angen insead of he sign funcion. Moreover, chaering can be reduced or eliminaed by using higher order sliding mode conrollers which can be easily designed since he error sysem given by 4.3 is in he conroller canonical form. Remark 4.4. The erm Kσ K > in 4.9 is added o speed up he convergence of he error rajecories.. Simulaion Resuls Several simulaion sudies are conduced o illusrae he effeciveness of he proposed conrollers in synchronizing wo exended B-VDP oscillaors. Firs, a simulaion of he exended B-VDP sysem under no conrol is carried ou. Figure 2 shows he chaoic aracor of an

10 Mahemaical Problems in Engineering u() Figure 4: Plo of he Lyapunov-based conroller versus ime. xr versus xd x d x r a yr versus yd y d y r b zr versus zd z d z r c Figure : Plos of he exended B-VDP oscillaors sae rajecories versus ime when he Lyapunov-based conroller is used and wih α.

11 Mahemaical Problems in Engineering 11 xr versus xd x d x r a yr versus yd y d y r b zr versus zd z d z r c Figure 6: Plos of he exended B-VDP oscillaors sae rajecories versus ime when he sliding mode conroller is used. exended B-VDP oscillaor for he case when he parameers of he sysem are p 1 1, p 2 1, and p 3 1.2, and for iniial condiions x d,y d,z d T., 1,. T. The performance of he Lyapunov-based conroller used o synchronize he wo exended B-VDP oscillaors, given by 2.4 and 2., is shown in Figures 3 and 4. The parameers of he oscillaors are given by p 1 1,p 2 1, and p The iniial condiions of he drive sysem are aken o be x d 1, y d 1 andz d 1; he iniial condiions of he response sysem are aken o be x r 3, y r 4 andz r 2. The gain of he conroller α is chosen such ha α 2. Furhermore, he conroller is applied o he response sysem a sec.figure 3 shows he rajecories of he drive and response sysems. I is clear ha complee synchronizaion akes place a abou 9 sec. The verical doed lines in he figure indicae he ime a which he conroller is applied. Figure 4 shows he conrol acion.

12 12 Mahemaical Problems in Engineering 2 1 u() Figure 7: Plo of he sliding mode conroller versus ime. A small spike appears when he conroller is swiched on. Finally, Figure depics he rajecories of he wo oscillaors for he case when he parameer α is equal o zero. As indicaed in Remark 3.2, his parameer can be aken o zero a he expense of he speed of he convergence of he errors o zero. Figure indicaes ha he synchronizaion is aained a around 2 seconds or 1 seconds afer he conroller is swiched on. The performance of he sliding mode conroller used o synchronize he exended B- VDP oscillaors is depiced in Figures 6 and 7. The iniial condiions are he same as he ones used previously. The parameers of he conroller are chosen such ha K 2, W 1, β 1 2, and β 2 3. Figure 6 depics he evoluion of he sae rajecories of he drive and response sysems. Synchronizaion is achieved 4 seconds afer he conroller is applied. Figure 7 depics he conrol acion rajecory, where a spike appears a he ime when he conroller is applied a sec. Finally, in order o check he robusness of he proposed sliding mode conrol scheme, a % change on he parameers p 1, p 2,andp 3 is inroduced. Figure 8 depics he resuls of his case. The wo oscillaors synchronize few seconds afer he conroller is applied. Therefore, i can be concluded from he simulaion resuls ha he proposed conrollers work well as hey are able o synchronize he drive and he response sysems. Also, he robusness of he sliding mode conroller under parameer perurbaions is eviden from he simulaion resuls. 6. Conclusion A Lyapunov-based conroller and a sliding mode conroller are proposed for he synchronizaion of wo exended Bonhoffer-Van der Pol oscillaors which sar from differen iniial condiions. The single inpu conroller acs on he firs sae of he response sysem only, which makes i easier o implemen. I is shown ha he proposed conrollers guaranee he asympoic convergence o zero of he errors beween he response and he drive sysems, hence,

13 Mahemaical Problems in Engineering 13 xr versus xd x d x r a yr versus yd y d y r b zr versus zd z d z r c Figure 8: Plos of he perurbed B-VDP oscillaors sae rajecories versus ime when he sliding mode conrol is used. ensuring complee synchronizaion beween he wo sysems. Furhermore, a nonlinear ransformaion is used o ransform he error dynamics ino he conroller canonical form; his ransformaion simplifies he design of he sliding mode conroller. Simulaion sudies show he effeciveness of he proposed conrol schemes. References 1 P. Glendinning, Sabiliy, Insabiliy and Chaos: An Inroducion o he Theory of Nonlinear Differenial Equaions, Cambridge Universiy, Cambridge, UK, Q. A. Memon, Synchronized choas for nework securiy, Compuer Communicaions, vol. 26, no. 6, pp. 498, 23.

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