 Applied Mathematics, 2013, 4, 1-6 Published Online November 2013 (http://www.scirp.org/journal/am) http://dx.doi.org/10.4236/am.2013.411A2001 Open Access AM Controlling Unstable Discrete Chaos and Hyperchaos Systems* Xin Li, Suping Qian School of Mathematics and Statistics, Changshu Institute of Technology, Changshu, China Email: lovelixin0412@163.com Received July 23, 2013; revised August 23, 2013; accepted August 30, 2013 Copyright © 2013 Xin Li, Suping Qian. 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. ABSTRACT A method is introduced to stabilize unstable discrete systems, which does not require any adjustable control parameters of the system. 2-dimension discrete Fold system and 3-dimension discrete hyperchaotic system are stabilized to fixed points respectively. Numerical simulations are then provided to show the effectiveness and feasibility of the proposed chaos and hyperchaos controlling scheme. Keywords: 2-Dimension Discrete Fold System; 3-Dimension Discrete Hyperchaotic System; Lyapunov Stability Theory; Controlling Chaos 1. Introduction In many engineering and other practical problems, chaos is undesirable and therefore needs to be controlled. Thus, a large number of control methods have been developed and are being applied to real systems [1-10]. The method given by Ott, Grebogi and Yorke (OGY)  is to stabi- lize an unstable orbit in the neighborhood of a hyperbolic fixed point by forcing the orbit onto the stable manifold. The method proposed by Romeiras, Grebogi, Ott and Dayawansa (RGOD)  is not yet suitable for control- ling hyperchaos since the method changes the stability property of the fixed point completely. However, the method proposed by Yang, Liu and Jian-min Mao  gives a new idea to stabilize unstable orbits even if there is no preexisting stable manifold nearby. For a finite- dimensional dynamical system, whose governing equa- tions may or may not be analytically available, Yang, Liu and Mao show how to stabilize an unstable orbit in a neighborhood of a “fully” unstable fixed point. The ad- vantage of this method is: only one of the unstable direc- tions is to be stabilized via time-dependent adjustments of control parameters. The parameter adjustments can be optimized. Recently, Bu  and Li  stabilized un- stable discrete systems by a method which does not re- quire any adjustable control parameters of the system. Consider an n-dimensional dynamical system defined by 1,kkxFx (1) where nxRis an n-dimensional vector, F is a nonlin-ear vector valued function. Let xf be the fixed point of the map (1). To stabilize a chaotic orbit to this fixed point, we take a variable feedback control described by 1kk kkxFx MFx x , (2) Define an infinitesimal deviation of xk from xf as kkfxxx. Then from Equation (2), one has 1,kk kxJxMJ Ix  (3) where kfkxxJFx  is the Jacobian matrix of the original system F evaluated at the fixed point xf and I is the nn identity matrix. The goal of controlling here is to make likm 0kx . For this aim, one requires 1,kkxQx (4) where Q is an nn matrix and takes the form 120,0qQq (5) where are constants. Substituting Equation (4) and Equa- tion (5) into Equation (3) and eliminating kx, choosing one special form of the matrix one have 1,1QqI ,q ,*The work was supported by the Special Funds of the National Natural Science Foundation of China (Grant No. 11141003). X. LI, S. P. QIAN 2 1.MqIJJI (6) This needs to use numeric computation to do. There- fore the above scheme based on the symbolic numeric computation is summarized as follows. Input: 1) The unstable system (1); 2) The system (2) with a variable feedback controller; 3) Choose the initial values of systems (2). Output: 1) ;kfkxxJFx  2) M in (6); 3) Deduce the system (2) according to the results (6); 4) Numerical simulations of the states xk when . kIn this paper, we use the method to stabilize 2-dimen- sion discrete Fold system  and 3-dimension discrete hyperchaotic system due to Wang  to fixed points respectively. 2. Stabilizing 2-Dimensional Discrete Fold System Using the above method, we stabilize 2-dimension dis- crete Fold system presented as:  1212211,1xkxkxkxk xk  (7) where ,  are the parameters, and we choose  = −0.1,  = −1.7. In the following based on the method mentioned above, we will make the Fold system stabilize at the fixed point. It is easy to get the two fixed points (1.965097170, 2.161606887), and (−0.8650971698, −0.9516068868) of Equation (7). The Jacobian matrix corresponding the fixed point 12,ffxx is 11.20fJx (8) From (6) one can have 1111112112 12,21212 12ffffffqx qxxM1fxqxqxx  q (9) here we choose 12,1.965097170, 2.161606887ffxx  as our re- search object. Choosing the parameter 0.5,q and 0.3q respectively, one gets 121.1766663130.1766663133 ,0.6943329445 1.1943329441.2473328380.2473328387 .0.9720661223 1.272066122MM (10) From (2), respectively substitute (10) into (7), we can obtain    121 211212221 2112121( )1.766663130.1766663133 ,1 0.69433294451.194332944 ,xk xkxkxkxkxkxk xkx kxkx kxkxkxk xk   (11) and   121 211212221 2112121 1.2473328380.2473328387 ,1 0.97206612231.272066122 .xkxkxk xkxkxkxk xkx kxkx kxkxkxk xk   (12) In the following, we give the orbit of 2-dimension dis- crete Fold system before being stabilized in Figure 1(a). And in Figure 1(b), three orbits starting from different initial points are stabilized to the fixed point (1.965097170, 2.161606887). It is shown that the unsta- ble orbit is stabilized to the desired fixed point mono- tonically. Then the orbits stabilized of xk and yk versus tk are depicted contrasting with the ones before being stabi- lized in Figures 2 and 3, respectively. 3. Stabilizing 3-Dimension Discrete Hyperchaotic System In this section, we consider 3-dimension discrete hyper- haotic system c Open Access AM X. LI, S. P. QIAN 3 (a) (b) Figure 1. (a) 2-dimension discrete Fold system; (b) Three orbits starting from different initial points are stabilized to the fixed point (1.965097170, 2.161606887), for q = 0.5. (a) (b) Figure 2. (a) x1(k) versus k before being stabilized; (b) x1(k) versus k after being stabilized for q = 0.5 and q = 0.3. (a) (b) Figure 3. (a) x2(k) versus k before being stabilized; (b) x2(k) versus k after being stabilized for q = 0.5 and q = 0.3.    13241223112356237311,111ykaykaykykaykaykykykaa ykykayk    ,,(13) which was derived from the generalized Rössler system via a first-order difference algorithm . We take the fixed point (0.09610764055, 0.4420951466, 0.9130225853) as our research object, that is 123,,fffyyy = (0.09610764055, 0.4420951466, 0.9130225853). Following the procedure above, the Jacobian matrix of Open Access AM X. LI, S. P. QIAN 4 map (13) is 431263 62 71010ffaaJa aayay a. (14) 1Here we let a1 = −1.9, a2 = 0.2, a3 = 0.5, a4 = −2.3, a5 = 2, a6 = −0.6, a7 = −1.9 and 1. From (6), choosing 0.5q and respec- tively, the matrix M at the fixed point (0.09610764055, 0.4420951466, 0.9130225853) is correspondingly ob- tained as following 0.2q 0.9762747382 0.2344378413 0.021654503970.89086379700.07841407030.09961071835 ,0.22538998420.2728405070 0.7942822117M (15) and 0.94305937220.5626508191 0.051970809562.1380731131.5881937680.2390657240 .0.54093596150.6548172166 0.562773086M (16) From (2), respectively substitute (15) and (16) into (13), one can obtain    132411123115623732231123241 0.02372526180.97627473820.23443784120.02165450397 ,1 1.078414070.8908637970.099610yka ykaykykykayk aykaaykyk aykykaykayk ykaykayk          562 37335623734123 11371835 ,1 0.205717788310.22538998420.2728405070.7942822117 ,aaykykayk132ykaaykykaykaaykayk aykyk      yk4132 (17) and     13241112311562373223112321 0.05694062780.94305937220.56265081910.05197080956 ,1 2.5881937682.1380731130.23906yka ykaykykykayk aykaaykyk aykykayk aykykayk ayk        56237335623734123 1135724 ,1 0.493722691410.54093596150.65481721660.5062773086 ,aaykykaykykaa ykykaykaykaykayk aykyk       (18) The numerical results are shown in the followed fig- ures. The orbit of 3-dimension discrete time hyperchaotic system is given by Figure 4(a). In Figure 4(b), three orbits starting from different initial points are stabilized to the fixed point (0.09610764055, 0.4420951466, 0.9130225853). We can also get the result that 3-dimension discrete time hyperchaotic system is stabilized. In Figures 5-7, the stabilized orbits of  123,,ykykyk versus tk are plotted contrasting with the ones before being stabi- lized, respectively. 4. Conclusion In summary, we have introduced a method to stabilize unstable discrete systems, which does not require any adjustable control parameters of the system. 2-dimension discrete Fold system and 3-dimension discrete hypercha- otic system are stabilized to fixed points respectively. From the process we finish, it is shown that stabilizing the unstable discrete systems neither requires a prior analytical knowledge of the underlying system nor any adjustable control parameters in advance. Numerical imulations are then provided to show the effectiveness s Open Access AM X. LI, S. P. QIAN 5 (a) (b) Figure 4. (a) 3-dimension discrete time hyperchaotic system; (b) Three orbits starting from different initial points are stabi- lized to the fixed point (0.09610764055, 0.4420951466, 0.9130225853), for q = 0.5. (a) (b) Figure 5. (a) yk1 versus k before being stabilized; (b) yk1 versus k after being stabilized. (a) (b) Figure 6. (a) 2yk versus k before being stabilized; (b) 2yk versus k after being stabilized. (a) (b) Figure 7. (a) 3yk versus k before being stabilized; (b) 3yk versus k after being stabilized. Open Access AM X. LI, S. P. QIAN 6 and feasibility of the proposed chaos and hyperchaos controlling Scheme. 5. Acknowledgements The authors are grateful to the reviewers for their valu- able comments and suggestions. REFERENCES  E. Ott, C. Grebogi and J. A. York, “Controlling Chaos,” Physical Review Letters, Vol. 64, No. 11, 1990, pp. 1196- 1199.  D. Liu, L. 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