The control of Kamel chaotic system’s

Djeddi kamel1

University of Constantine, Algeria

djeddi.kamel@gmail.com

ABSTRACT

We present a control Kamel system’s for stabilizing chaotic orbits on an un-stable focus point embedded within a chaotic attractor. This control system doesnot require the location of the unstable focus point and the local linearized dy-namics at the point. Even if a parameter of the chaotic system changes slowly,the control system can automatically track the unstable focus point both withinand outside the chaotic regimes. We test the control system using a nonlinearKamel map in numerical experiments.

Subject headings:

Chaos, Control, Kamel system’s, Point fixe.

1. Introduction

In the past few years there has been considerable interest in controlling chaotic systems.The pioneers of controlling chaos are Ott, Grebogi, and Yorke (OGY). They have proposeda method which utilizes the existence of unstable fixed points within a chaotic attractor[1]. The OGY method can stabilize chaotic orbits on a desired unstable fixed point bymaking small parameter perturbations. The OGY method has been developed for variouspurposes [2-5], and these methods have been applied to many mathematical systems. Onthe other hand, several methods have been proposed for tracking unstable fixed points overlarge ranges of a parameter and through bifurcations. The tracking methods are very useful

Mathematics Department,University of Constantine, Algeria 25000.

January 2015

—2 —

to maintain the control of practical chaotic systems which are influenced by a change in theirenvironment.

The idea of chaos control was enunciated at the beginning of this decades at the Univer-sity of Maryland [6], the ideas for controlling chaos were outlined and a method for stabilizingan unstable periodic orbit was suggested, as a proof of principle. The main idea consistedin waiting for a natural passage of the chaotic orbit close to the desired periodic behavior,and then applying a small judiciously chosen perturbation, in order to stabilize such peri-odic dynamics (which would be, in fact, unstable for the unperturbed system). Throughthis mechanism, one can use a given laboratory system for producing an infinite number ofdifferent periodic behavior (the infinite number of its unstable periodic orbits), with a greatflexibility in switching from one to another behavior. Much more, by constructing appro-priate goal dynamics, compatible with the chaotic attractor, an operator may apply smallperturbations to produce any kind of desired dynamics, even not periodic, with practicalapplication in the coding process of signals.

In this Letter, we present a control Kamel system’s which automatically stabilizes anunstable focus point embedded within a chaotic attractor. This control system does notrequire the location of the unstable focus point and the local dynamics at the point. Oncethe chaotic orbit is stabilized on the unstable focus point in a chaotic regime, the trackingof the point is achieved automatically even if the parameter of the chaotic system changesslowly compared to the state dynamics. We test the control system by using a nonlinearKamel map on numerical experiments.

2. The Kamel map

Consider the two-dimensional iterated map function given by

{xn+1 = a cos (xn) + byn

yn+1 = cxn + d(1)

where a, b, c and d reals constantes.

The Kamel map displays periodicity, mixing, and sensitivity to initial conditions. Thesystem can also display hysteresis, and bistability can be observed in the bifurcation dia-grams. Each of these phenomena will now be discussed briefly in turn.

Suppose that the discrete nonlinear system

—3 —

xn+1 = P (xn, yn) , yn+1 = Q (xn, yn) ,

has a fixed point at (x1, y1), where P and Q are at least quadratic in xn and yn. The fixedpoint can be transformed to the origin and the nonlinear terms can be discarded after takinga Taylor series expansion. The Jacobian matrix is given by

J (x1, y1) =

(∂P∂x

∂P∂y

∂Q∂x

∂Q∂y

)∣∣∣∣∣(x1,y1)

Definition 1 Suppose that the Jacobian has eigenvalues λ1 and λ2. A fixed points calledhyperbolic if both |λ1| 6= 1 and |λ2| 6= 1. If either |λ1| = 1 and |λ2| = 1, then the fixed pointis called nonhyperbolic.

In the discrete case, the fixed point is stable as long as |λ1| < 1 and |λ2| < 1, otherwisethe fixed point is unstable. For example, the fixed points of period one for the Kamel mapcan be found by solving the equations given by xn+1 = xn and yn+1 = yn simultaneously.Therefore, period-one points satisfy the equation

x = a cos (x) + by, y = cx+ d

consider system (1) with a = 4, b = 0.8, c = 1 and d = 2. There are four fixed points given byA(4.538434732, 6.538434732), B(1.881695898, 3.881695898), C(−4.064829821,−2.064829821)and D(−2.100189256,−0.1001892556).

Consider the fixed point at A; the fixed point is a saddle point. B, C and D it is criticalspoints is nonhyperbolic Jacobian is given by

J =

(−a sinx b

c 0

)The Lyapunov exponents of the Kamel map is λ1 ≈ 0.78 and λ2 ≈ −1.

Example 2 The reader can verify these results using the Maple program. Some iterativeplots are given in Figure

—4 —

Fig. 1.– Points fixes of Kamel system’s whith a=4, d=0.8, c=1and d=2

Fig. 2.– Kamel’s map a=4, b=0.8, c=1 and d=2

—5 —

Fig. 3.– Kamel map a=-4, b=0.9, c=-0.8 and d=1

Fig. 4.– Kamel map a=4, b=0.7, c=1 and d=1

Fig. 5.– Kamel map a=4, b=1, c=1 and d=1

—6 —

Fig. 6.– Kamel map a=4, b=1.1, c=1 and d=1

Fig. 7.– Kamel map a=3.8, b=0.7, c=1 and d=1

Fig. 8.– Kamel map a=4, b=0.9, c=1 and d=2

—7 —

Fig. 9.– Kamel map a=4, b=0.5, c=1 and d=1

2.1. Controlling Chaos in the Kamel Map

Ott et al. [1] used the Kamel map to illustrate the control method. A simple example.Consider the system (1) or values of b close to b0 in a small neighborhood of A, the map canbe approximated by a linear map

Xn+1 −Xs (b0) = J (Xn −Xs (b0)) + C (b− b0) (2)

where Xn = (xn, yn)T , A = Xs (b0) , J is the Jacobian, and

C =

(∂P∂b∂Q∂b

)=

(y

0

)Let xf noted fixed point A.

C =

(6.538434732

0

)∣∣∣∣xf

The following system is considered:

.x = F (x, s) , x ∈ Rn

where s represent the parameter of control.

Let xf noted fixed point A. then

—8 —

.xf = F (xf , s0)

where s0 represent the face value of the parameter and all partial derivatives are evaluated

at b0 and Xs (b0). Assume that in a small neighborhood of A,

b− b0 = −K (Xn −Xs (b0))

where

K =

(k1k2

)Substitute (2) in to (3) to obtain

Xn+1 −Xs (b0) = (J − CK) (Xn −Xs (b0)) . (3)

Therefore, the fixed point at A = Xs (b0) is stable if the matrix J −CK has eigenvalues(or regulator poles) with modulus less than unity. In this particular case,

J − CK =

(3.939632297− 6.538434732k1 0.8− 6.538434732k2

1 0

)and the characteristic polynomial is given by

λ2 + (6.538434732k1 − 3.939632297)λ+ (6.538434732k2 − 0.8) = 0

Suppose that the eigenvalues (regulator poles) are given by λ1 and λ2; then

λ1λ2 = (6.538434732k2 − 0.8)− (λ1 + λ2) = (6.538434732k1 − 3.939632297) .

The lines of marginal stability are determined by solving the equations λ1 = ±1 and λ1λ2 = 1.These conditions guarantee that the eigenvalues λ1 and λ2 have modulus less than unity.Suppose that λ1λ2 = 1. Then

k2 = 0.2752952463

Suppose that λ1 = +1. Then

λ2 = (6.538434732k2 − 0.8) and λ2 = −6.538434732k1 + 2.939632297.

—9 —

Fig. 10.– intersection of two plots

Therefore,

k2 = −k1 + 0.4495926651.

If λ1 = −1. Then

λ2 = −6.538434732k2 + 0.8 and λ2 = −6.538434732k1 + 4.939632297.

Therefore,

k2 = k1 − 0.6331228293.

The stable eigenvalues (regulator poles) lie within a triangular region, as depicted in Figure(10)

Select k1 = −36.16145038 and k2 = 0.9095501950. This point lies well inside the trian-gular region, as depicted in Figure 10. The perturbed Kamel map becomes

{xn+1 = a cos (xn) + (−k1 (xn − x1.1)− k2 (yn − y1.1) + b0) yn

yn+1 = cxn + d(4)

Applying (1) and (4) without and with control, respectively, it is possible to plot time seriesdata for these maps. Figure (11) shows a time series plot when the control is switched onafter the 200th iterate; the control is left switched on until the 500th iterate. Remember tocheck that the point is in the control region before switching on the control.

—10 —

Fig. 11.– Determine the square of the distance of each point from the origin. Iterate thesystem and switch on the control after 500 iterations.

3. Conclusion

In this paper, we have proposed the OGY idea of controlling chaos gives flexibility. Byswitching the small control, one can switch the time asymptotic behavior from one periodicorbit to another. In some situations, where the flexibility offered by the ability to do suchswitching is desirable, it may be advantageous to design the system so that it is chaotic. Inother situations, where one is presented with a chaotic system, the method may allow one toeliminate chaos and achieve greatly improved behavior at relatively low cost. The OGY ideacan also be used to stabilize a desired chaotic trajectory, which has potential applicationsto problems such as synchronization of chaotic systems, conversion of transient chaos intosustained chaos communication with chaos, and selection of a desired chaotic phase.

REFERENCES

[1] E. Ott, C. Grebogi and J.A. Yorke, Phys. Rev. Lett. 64 (1990) 1196.

[2] F.J. Romeiras, C. Grebogi, E. Ott and W.P. Dayawansa, Physica D 58 (1992) 165.

[3] Y,L Pyragas, Phys. Lett. A 170 (1992) 412.

[4] V. Petrov, B. Peng and K. Showalter, J. Chem. Phys. 96(1992) 7506.

—11 —

[5] Y.C. Lai, M. Ding and C. Grebogi, Phys. Rev. E 47 (1993)86.

[6] S. Boccaletti et al, The theory of chaos, Physics Reports 329 (2000) 103.197.

[7] T.Ushio, S Yamamoto, Prediction-based control of chaos, Physics Letters A 264 1999.30—35.

[8] K.Konishi, H.Kokame, Stabilizing and tracking unstable focus points in chaotic systemsusing a neural network, Physics Letters A 206 (1995) 203-210.

[9] K.Konishi, M.Hirai, Sliding mode control for a class of chaotic systems, Physics LettersA 245 ( 1998) 51l-517.

[10] K.Konishi, H.Kokame, Observer-based delayed-feedback control for discrete-time chaoticsystems, Physics Letters A 248 (1998) 359-368.

[11]M.P. Fernández de Córdoba, E. Liz ,Prediction-based control of chaos and a dynamicParrondo’s paradox , Physics Letters A 377 (2013) 778—782.

[12] David J. Christini and James J. Collins, Real-Time, Adaptive, Model-Independent Con-trol, IEEE Trans. Circuits Syst., VOL. 44, NO. 10, 1997.

[13] H Broer. F Takens, Dynamical Systems and Chaos, Springer Science+Business Media,LLC 2011.

[14] C.Grebogi, Y.C.Lai, Controlling chaotic dynamical systems, Systems & Control Letters31 (1997) 307 312.

[15] Stephen Wiggins, Global Bifurcations and Chaos, 1988 by Springer-Verlag New YorkInc.

[16] Ying-C Lai · Tamas Tél, Transient Chaos, Springer Science+Business Media, LLC 2011.

This preprint was prepared with the AAS LATEX macros v5.2.