Controlling Unstable Discrete Chaos and Hyperchaos Systems *

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.


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][2][3][4][5][6][7][8][9][10].The method given by Ott, Grebogi and Yorke (OGY) [1] is to stabilize 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) [2] is not yet suitable for controlling hyperchaos since the method changes the stability property of the fixed point completely.However, the method proposed by Yang, Liu and Jian-min Mao [11] gives a new idea to stabilize unstable orbits even if there is no preexisting stable manifold nearby.For a finitedimensional dynamical system, whose governing equations 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 advantage of this method is: only one of the unstable directions is to be stabilized via time-dependent adjustments of control parameters.The parameter adjustments can be optimized.Recently, Bu [12] and Li [13] stabilized unstable discrete systems by a method which does not require any adjustable control parameters of the system.Consider an n-dimensional dynamical system defined by where n x R  is an n-dimensional vector, F is a nonlinear vector valued function.Let x f 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 Define an infinitesimal deviation of x k from x f as where is the Jacobian matrix of the original system F evaluated at the fixed point x f and I is the n n  identity matrix.The goal of controlling here is to make li


For this aim, one requires where Q is an n n  matrix and takes the form where are constants.Substituting Equation (4) and Equation (5) into Equation (3) and eliminating k x  , choosing one special form of the matrix one have This needs to use numeric computation to do.Therefore 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).
3) Deduce the system (2) according to the results (6); 4) Numerical simulations of the states x k when .k   In this paper, we use the method to stabilize 2-dimension discrete Fold system [14] and 3-dimension discrete hyperchaotic system due to Wang [15] to fixed points respectively.

Stabilizing 2-Dimensional Discrete Fold System
Using the above method, we stabilize 2-dimension discrete Fold system presented as: where ,  are the parameters, and we choose  = −0.1, 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 From ( 6) one can have here we choose From ( 2), respectively substitute (10) into (7), we can obtain In the following, we give the orbit of 2-dimension discrete 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 unstable orbit is stabilized to the desired fixed point monotonically.Then the orbits stabilized of x k and y k versus t k are depicted contrasting with the ones before being stabilized in Figures 2 and 3, respectively.

Stabilizing 3-Dimension Discrete Hyperchaotic System
In this section, we consider 3-dimension discrete hyperhaotic system c Open Access AM which was derived from the generalized Rössler system via a first-order difference algorithm [15].

 
From ( 6), choosing 0.5 q  and respectively, the matrix M at the fixed point (0.09610764055, 0.4420951466, 0.9130225853) is correspondingly obtained as following 0.2 q   0.9762747382 0.2344378413 0.02165450397 0.8908637970 0.0784140703 0.09961071835 , 0.2253899842 0.2728405070 0.7942822117 From ( 2), respectively substitute ( 15) and ( 16) into ( 13), one can obtain and The numerical results are shown in the followed figures.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 , , y k y k y k versus t k are plotted contrasting with the ones before being stabilized, respectively.

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 hyperchaotic 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 6 and feasibility of the proposed chaos and hyperchaos controlling Scheme.