Chaos synchronization of identical Sprott systems by active control
CChaos synchronization of identical Sprott systems byactive control
Marcin Daszkiewicz
Institute of Theoretical PhysicsUniversity of Wroclaw pl . Maxa Borna 9 , −
206 Wroclaw , Polande − mail : marcin@ift . uni . wroc . pl Abstract
In this article we synchronize by active control method all 19 identical Sprottsystems provided in paper [10]. Particularly, we find the corresponding active con-trollers as well as we perform (as an example) the numerical synchronization of twoSprott-A models. a r X i v : . [ n li n . C D ] S e p Introduction
In the last four decades there appeared a lot of papers dealing with so-called chaoticmodels, i.e., with such models whose dynamics is described by strongly sensitive withrespect initial conditions, nonlinear differential equations. The most popular of themare: Lorenz system [1], Roessler system [2], Rayleigh-Benard system [3], Henon-Heilessystem [4], jerk equation [5], Duffing equation [6], Lotka-Volter system [7], Liu system[8], Chen system [9] and Sprott system [10]. A lot of them have been applied in variousfields of industrial and scientific divisions, such as, for example: Physics, Chemistry,Biology, Microbiology, Economics, Electronics, Engineering, Computer Science, SecureCommunications, Image Processing and Robotics.One of the most important problem of the chaos theory concerns so-called chaos syn-chronization phenomena. Since Pecora and Caroll [11] introduced a method to synchronizetwo identical chaotic systems, the chaos synchronization has received increasing attentiondue to great potential applications in many scientific discipline. Generally, there areknown several methods of chaos synchronization, such as: OGY method [12], active con-trol method [13]-[16], adaptive control method [17]-[21], backstepping method [22], [23],sampled-data feedback synchronization method [24], time-delay feedback method [25] andsliding mode control method [26]-[29].In this article we synchronize by active control scheme all 19 identical Sprott systemsprovided in publication [10]. Particularly, we establish the proper so-called active con-trollers with use of the Lyapunov stabilization theory [30]. It should be noted, however,that some types of Sprott model have been already synchronized with use of the activecontrol method in papers [31] and [32]. Apart of that there has been synchronized theSprott systems in the framework of adaptive control scheme in articles [33] and [34].The paper is organized as follows. In second Section we recall the main result of Sprottarticle [10], i.e., we provide Table 1 including dynamics of all 19 Sprott chaotic systems.In Section 3 we remaind the basic concepts of active synchronization method, while inSection 4 we consider as an example the synchronization of two identical Sprott-A models.The fifth Section is devoted to the main result of the paper, and it provides in Table 2the active controllers which synchronize all identical Sprott systems. The conclusions andfinal remarks are discussed in the last Section.
In this Section we recall the main result of paper [10], in which there has been performed asystematic examination of general three-dimensional ordinary differential equations withquadratic nonlinearities. Particularly, it has been uncovered 19 distinct simple examplesof chaotic flows (so-called Sprott systems) listed in
Table 1 .2ype 1st equation 2nd equation 3rd equationA ˙ x = x ˙ x = − x + x x ˙ x = 1 − x B ˙ x = x x ˙ x = x − x ˙ x = 1 − x x C ˙ x = x x ˙ x = x − x ˙ x = 1 − x D ˙ x = − x ˙ x = x + x ˙ x = x x + 3 x E ˙ x = x x ˙ x = x − x ˙ x = 1 − x F ˙ x = x + x ˙ x = − x + 0 . x ˙ x = x − x G ˙ x = 0 . x + x ˙ x = x x − x ˙ x = − x + x H ˙ x = − x + x ˙ x = x + 0 . x ˙ x = x − x I ˙ x = − . x ˙ x = x + x ˙ x = x + x − x J ˙ x = 2 x ˙ x = − x + x ˙ x = − x + x + x K ˙ x = x x − x ˙ x = x − x ˙ x = x + 0 . x L ˙ x = x + 3 . x ˙ x = 0 . x − x ˙ x = 1 − x M ˙ x = − x ˙ x = − x − x ˙ x = 1 . . x + x N ˙ x = − x ˙ x = x + x ˙ x = 1 + x − x O ˙ x = x ˙ x = x − x ˙ x = x + x x + 2 . x P ˙ x = 2 . x + x ˙ x = − x + x ˙ x = x + x Q ˙ x = − x ˙ x = x − x ˙ x = 3 . x + x + 0 . x R ˙ x = 0 . − x ˙ x = 0 . x ˙ x = x x − x S ˙ x = − x − x ˙ x = x + x ˙ x = 1 + x Table 1.
The Sprott systems.
In this Section we remaind the general scheme of chaos synchronization of two systems byso-called active control procedure [13]-[16]. Let us start with the following master model ˙ x = Ax + F ( x ) , (1)where x = [ x , x , . . . , x n ] is the state of the system, A denotes the n × n matrix ofthe system parameters and F ( x ) plays the role of the nonlinear part of the differentialequation (1). The slave model dynamics is described by˙ y = By + G ( y ) + u , (2)with y = [ y , y , . . . , y n ] being the state of the system, B denoting the n -dimensionalquadratic matrix of the system, G ( y ) playing the role of nonlinearity of the equation (2)and u = [ u , u , . . . , u n ] being the active controller of the slave model. Besides, it shouldbe mentioned that for matrices A = B and functions F = G the states x and y describe dodt = ˙ o . A (cid:54) = B or F (cid:54) = G they correspond to the twodifferent chaotic models.Let us now provide the following synchronization error vector e = y − x , (3)which in accordance with (1) and (2) obeys˙ e = By − Ax + G ( y ) − F ( x ) + u . (4)In active control method we try to find such a controller u , which synchronizes the stateof the master system (1) with the state of the slave system (2) for any initial condition x = x (0) and y = y (0). In other words we design a controller u in such a way that forsystem (4) we have lim t →∞ || e ( t ) || = 0 , (5)for all initial conditions e = e (0). In order to establish the synchronization (4) we usethe Lyapunov stabilization theory [30]. It means, that if we take as a candidate Lyapunovfunction of the form V ( e ) = e T P V ( e ) e , (6)with P being a positive n × n matrix, then we wish to find the active controller u so that˙ V ( e ) = − e T QV ( e ) e , (7)where Q is a positive definite n × n matrix as well. Then the systems (1) and (2) remainsynchronized. In accordance with two pervious Sections the master Sprott-A system is described by thefollowing dynamics (see
Table 1 ) ˙ x = x ˙ x = − x + x x ˙ x = 1 − x , (8)4here functions x , x and x denote the states of the system; its slave Sprott-A partneris given by ˙ y = y + u ˙ y = − y + y y + u ˙ y = 1 − y + u , (9)with active controllers u , u and u respectively. Using (8) and (9) one can check thatthe dynamics of synchronization errors e i = y i − x i is obtained as ˙ e = e + u ˙ e = − e + y y − x x + u ˙ e = − e ( y + x ) + u . (10)Besides, if we define the positive Lyapunov function by V ( e ) = 12 (cid:0) e + e + e (cid:1) , (11)then for the following choice of control functions u = − ( e + e ) u = e − e − y y + x x u = e ( y + x ) − e , (12)we have ˙ V ( e ) = − (cid:0) e + e + e (cid:1) . (13)Such a result means (see general prescription) that the identical Sprott-A systems (8) and(9) are synchronized for all initial conditions with active controllers (12).Let us now illustrate the above considerations by the proper numerical calculations.First of all, we solve the Sprott-A system with two different sets of initial conditions( x , x , x ) = (1 , . , . , (14) See also formula (4). The matrix P = 1 in the formula (6). The matrix Q = 1 in the formula (7). y , y , y ) = (1 . , . , , (15)respectively. The results are presented on Figure 1 - one can see that there exist (infact) the divergences between both trajectories. Next, we find the solutions for the mastersystem (8) (the x -trajectory) and for its slave partner (9) with active controllers (12) (the y -trajectory) for initial data (14) and (15) respectively. Now, we see that the correspond-ing trajectories become synchronized - the vanishing in time error functions e i = y i − x i are presented on Figure 2 . Additionally, we repeat the above numerical procedure fortwo another sets of initial data: x = (0 , . , .
05) and y = (0 . , . , Figures 3 and respectively. The used in pervious Section algorithm can be applied to the case of all remaining Sprottsystems as well. The obtained results are summarized in
Table 2 , i.e., there are listed con-trollers u , u and u for which the proper identical Sprott systems become synchronizedfor arbitrary initial conditions x , x and x as well as y , y and y . However, as itwas already mentioned in Introduction, the control functions for Sprott-L and Sprott-Mmodels have been provided in paper [31].type 1st controller 2nd controller 3rd controllerA u = − ( e + e ) u = e − e − y y + u = e ( y + x ) − e + x x B u = x x − y y − e u = − e u = y y − x x − e C u = x x − y y − e u = − e u = ( x + y ) e − e D u = e − e u = − ( e + e + e ) u = x x − y + x ) ·· e − e − y y E u = x x − y y − e u = − ( y + x ) e u = 4 e − e F u = − ( e + e + e ) u = e − . e u = − ( y + x ) e G u = − (1 . e + e ) u = x x − y y u = e − ( e + e )H u = e − ( y + x ) · u = − ( e + 1 . e ) u = − e · e − e I u = 0 . e − e u = − ( e + e + e ) u = − ( e + ( y + x ) ·· e )J u = − ( e + 2 e ) u = e − e u = e − (1 + y + x ) ·· e − e )K u = e − e − y y + u = − e u = − ( e + 1 . e )+ x x L u = − ( e + e + 3 . e ) u = − . y + x ) e u = e − e u = e − e u = − e ( y + x ) u = − (1 . e + e + e )N u = 2 e − e u = − ( e + ( y + x ) · u = 2 e − e − e · e + e )O u = − ( e + e ) u = − ( e + e ) + e u = − ( e + 2 . e ++ e ) + x x − y y P u = − ( e + 2 . e + u = e − ( y + x ) · u = − ( e + e + e )+ e ) · e − e )Q u = e − e u = e u = − (3 . e + ( y + x ) ·· e ) − . e R u = − e + e u = − ( e + e ) u = x x − y y S u = 4 e u = − ( e + ( y + x ) · u = − ( e + e ) · e ) − e Table 2.
The active controllers for Sprott systems.
In this article we synchronize all identical Sprott systems defined in paper [10] with useof the active control method. Particularly, we find the corresponding so-called activecontrollers listed in
Table 2 . As an example we also study numerically synchronizationof Sprott-A model defined by the formulas (8) and (9).It should be noted that the presented investigations can be extended in various ways.For example, one may consider synchronization of Sprott models with use of others men-tioned in Introduction methods. The works in this direction already started and are inprogress.
References [1] E.N. Lorenz, J. Atmos. Sci. 20, 130 (1963)[2] O.E. Roessler, Phys. Lett. A 57, 397 (1976)[3] A.V. Getling, ”Rayleigh-Benard Convection: Structures and Dynamics”, World Sci-entific, 1998[4] M. Henon, C. Heiles, AJ. 69, 73 (1964)[5] J.C. Sprott, Am J Phys. 65, 537 (1997)[6] G. Duffinng, ”Erzwungene Schwingungen bei Vernderlicher Eigenfrequenz”, F.Vieweg u. Sohn, Braunschweig, 1918. 77] V. Volterra ”Variations and fluctuations of the number of indviduals in animal speciesliving together. In Animal Ecology” McGraw-Hill, 1931. Translated from 1928 editionby R. N. Chapman.[8] C. Liu, T. Liu, L. Liu, K. Liu, Chaos, Solitons and Fractals 22, 1031 (2004)[9] G. Chen, T. Ueta, Journal of Bifurcation and Chaos 9, 1465 (1999)[10] J.C. Sprott, Phys. Rev. E 50, 647 (1994)[11] L.M. Pecora, T.L. Carroll, Phys. Rev. Lett 64, 821 (1990)[12] E. Ott, C. Grebogi, J.A. Yorke, Phys. Rev. Lett 64, 1196 (1990)[13] M.C. Ho, Y.C. Hung, Phys. Lett. A 301, 424 (2002)[14] H.K. Chen, Chaos, Solitons and Fractals 23, 1245 (2005)[15] V. Sundarapandian, Int. Journ. of Comp. Sc. and Eng. 3, 2145 (2011)[16] V. Sundarapandian, Int. Journ. of Appl. Sc. and Tech. 3, 1 (2011)[17] T.L. Liao, S.H. Tsai, Chaos, Solitons and Fractals 11, 1387 (2000)[18] V. Sundarapandian, Int. Journ. of Cont. Theory and Comp. Model. 1, 1 (2011)[19] V. Sundarapandian, Int. Journ. of Cont. Theory and Comp. Model. 1, 1 (2011)[20] V. Sundarapandian, Int. Journ. of Comp. Eng. and Applications 1, 127 (2011)[21] V. Sundarapandian, R. Karthikeyan, Int. Journ. of Inf. Tech. and Comp. Sc. 1, 49(2011)[22] Y.G. Yu, S.C. Zhang, Chaos, Solitons and Fractals, 27, 1369 (2006)[23] X. Wu, J. Lu, Chaos, Solitons and Fractals, 18, 721 (2003)[24] T. Yang, L.O. Chua, Int. J Bifurcat. Chaos 9, 215 (1999)[25] J.H. Park, O.M. Kwon, Chaos, Solitons and Fractals 17, 709 (2003)[26] V. Sundarapandian, Int. Journ. of Cont. Theory and Comp. Model. 1, 15 (2011)[27] V. Sundarapandian, Int. Journ. of Comp. Sc. and Eng. 3, 2163 (2011)[28] V. Sundarapandian, Int. Journ. on Inf. Sys. and Tech. 1, 20 (2011)[29] V. Sundarapandian, S. Sivaperumal, Int. Journ. of Arch. Comp. 9, 274 (2012)830] A.M. Lyapunov, ”The General Problem of the Stability of Motion” (In Russian),Doctoral dissertation, Univ. Kharkov 1892. English translation: ”Stability of Mo-tion”, Academic Press, New-York and London, 1966[31] S. Vaidyanathan, Int. Journ. of Cont. Theory and Comp. Model. 2, 21 (2012)[32] D. Xu, Adv. Theor. Appl. Mech. 3, 195 (2010)[33] S. Vaidyanathan, Int. Journ. of Soft Comp. Math. and Cont. 1, 1 (2012)[34] S. Vaidyanathan, Int. Journ. of Inf. and Comp. Sec. 1, 13 (2011)9igure 1: The error functions e i = y i − x i for Sprott-A model defined by the system (8)with the initial conditions (14) (the x -trajectory) and with the initial conditions (15) (the y -trajectory). The blue line corresponds to the e -error function, the orange one - to e and the green one - to e respectively.igure 2: The error functions e i = y i − x i for synchronized Sprott-A model defined bythe master system (8) with the initial conditions (14) (the x -trajectory) and by the slavesystem (9) with the initial conditions (15) (the y -trajectory). The blue line correspondsto the e -error function, the orange one - to e and the green one - to e respectively.igure 3: The error functions e i = y i − x i for Sprott-A model defined by the system(8) with the initial conditions x = (0 , . , .
05) (the x -trajectory) and with the initialconditions y = (0 . , . ,
0) (the y -trajectory). The blue line corresponds to the e -errorfunction, the orange one - to e and the green one - to e respectively.igure 4: The error functions e i = y i − x i for synchronized Sprott-A model defined by themaster system (8) with the initial conditions x = (0 , . , .
05) (the x -trajectory) andby the slave system (9) with the initial conditions y = (0 . , . ,
0) (the y -trajectory).The blue line corresponds to the e -error function, the orange one - to e and the greenone - to e3