Electronic Journal of Mathematical Analysis and Applications

Vol. 1(2) July 2013, pp. 318-325.

ISSN: 2090-792X (online)

http://ejmaa.6te.net/

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THE UNIFIED SYSTEM BETWEEN LORENZ AND CHEN SYSTEMS:

A DISCRETIZATION PROCESS

A. M. A. EL-SAYED, S. M. SALMAN

Abstract. The unified chaotic system that contains the Lorenz and Chen systems was

introduced. In this paper we are interested in discretizing that system. Fixed points

and their asymptotic stability of the discrete system obtained are investigated. Somedynamical behaviors such as chaotic attractor, bifurcation and chaos are discussed.

1. Introduction

It was notable that both Lorenz and Chen system share some common properties such asthey have the same symmetry, stability and dissipativity. In [8], a unified chaotic systemwas introduced that contains the Lorenz and Chen system as two dual systems as thetwo extremes to its spectrum. On the other hand, Some examples of dynamical systemsgenerated by piecewise constant arguments have been studied in [2]-[3]. Here we proposea discretization process to obtain the discrete version of that unified system.

Consider the unified chaotic system

x. = (25β + 10)(y − x), t ∈ (0, T ],y. = (28− 35β)x− xz + (29β − 1)y,

z. = xy − β + 83

z,

(1.1)

where β ∈ [0, 1].

2000 Mathematics Subject Classification. 2010 MSC: 39B05, 37N30, 37N20.Key words and phrases. Unified chaotic system, piecewise constant arguments, fixed points, chaotic

attractor, bifurcation, chaos.Submitted April 1, 2013.

318

EJMAA-2013/1(2) ON THE FRACTIONAL-ORDER GAMES 319

Meanwhile, consider the corresponding system with piecewise constant arguments givenby

x(t) = (25β + 10)(y(r[t

r])− x(r[ t

r])),

y(t) = (28− 35β)x(r[ tr

])− x(r[ tr

])z(r[t

r]) + (29β − 1)y(r[ t

r]),

z(t) = x(r[t

r])y(r[

t

r])− β + 8

3z(r[

t

r]).

(1.2)

Let t ∈ [0, r), then [ tr ] = 0, and the solution of (1.2) is given byx1(t) = xo + t((25β + 10)(yo − xo)), t ∈ [0, r)y1(t) = yo + t((28− 35β)xo − xozo + (29β − 1)yo), t ∈ [0, r)

z1(t) = zo + t(xoyo −β + 8

3zo), t ∈ [0, r).

Let t ∈ [r, 2r), then [ tr ] = 1, and the solution of (1.2) is given byx2(t) = x1(r) + (t− r)((25β + 10)(y1 − x1)), t ∈ [0, r)y2(t) = y1(r) + (t− r)((28− 35β)x1 − x1z1 + (29β − 1)y1), t ∈ [0, r)

z2(t) = z1(r) + (t− r)(x1y1 −β + 8

3z1), t ∈ [0, r).

Repeating the process we get

xn+1(t) = xn(nr) + (t− nr)((25β + 10)(yn(nr)− xn(nr))), t ∈ [r, 2r)yn+1(t) = yn(nr) + (t− nr)((28− 35β)xn(nr)− xn(nr)zn(nr) + (29β − 1)yn(nr)),

zn+1(t) = zn(nr) + (t− nr)(xn(nr)yn(nr)−β + 8

3zn(nr)),

as t→ (n+ 1)r,xn+1((n+ 1)r) = xn(nr) + r((25β + 10)(yn(nr)− xn(nr))), t ∈ [nr, (n+ 1)r)yn+1((n+ 1)r) = yn(nr) + r((28− 35β)xn(nr)− xn(nr)zn(nr) + (29β − 1)yn(nr)),

zn+1((n+ 1)r) = zn(nr) + r(xn(nr)yn(nr)−β + 8

3zn(nr)).

That is

xn+1((n+ 1)r) = xn + r((25β + 10)(yn − xn)), t ∈ [nr, (n+ 1)r)yn+1((n+ 1)r) = yn + r((28− 35β)xn − xnzn + (29β − 1)yn),

zn+1((n+ 1)r) = zn + r(xnyn −β + 8

3zn).

(1.3)

Moreover, if we consider the equations

x(t) = (25β + 10)(y(r[t− rr

])− x(r[ t− rr

])),

y(t) = (28− 35β)x(r[ t− rr

])− x(r[ t− rr

])z(r[t− rr

]) + (29β − 1)y(r[ t− rr

]),

z(t) = x(r[t− rr

])y(r[t− rr

])− β + 83

z(r[t− rr

]).

320 A. M. A. EL-SAYED, S. M. SALMAN EJMAA-2013/1(2)

We can apply the same procedure to obtain the discretization of the second order differenceequation

xn+1 = xn + r((25β + 10)(yn−1 − xn−1)), t ∈ [0, r)yn+1 = yn + r((28− 35β)xn−1 − xn−1zn−1 + (29β − 1)yn−1),

zn+1 = zn + r(xn−1yn−1 −β + 8

3zn−1).

(1.4)

Remark 1. It should be noticed that the system (1.3) can be obtained also by applyingEuler’s discretization method [12]. However, Euler’s method fails in obtaining system(1.4).

2. Fixed points and their asymptotic stability

Now we study the asymptotic stability of the fixed points of the system (1.3) which hasthree fixed points if (β + 8)(9− 2β) > 0:• fix1 = (0, 0, 0)• fix2 = (

√(β + 8)(9− 2β),

√(β + 8)(9− 2β), 27− 6β)

• fix3 = (−√

(β + 8)(9− 2β),−√

(β + 8)(9− 2β), 27− 6β)

The last two fixed points are symmetrically placed with respect to z−axis.

By considering a Jacobian matrix for one of these fixed points and calculating their eigen-values, we can investigate the stability of each fixed point based on the roots of the systemcharacteristic equation [7].

Linearizing the system (1.3) about fix1 yields the following characteristic equation

P (λ) ≡ λ3 + (B + C − 1)λ2 + (BC −B +A)λ+ (AC −A−D) = 0,

where:A = 1 + r(4β − 11)− r2(25β + 10)(29β − 1)),B = −2− r(−54β − 11) + r2(25β + 10)(28− 35β),C = r β+83 ,

D = r3(25β + 10)(28− 35β)(β+83 )− r2(25β + 10)(28− 35β).

Now let a1 = (B+C−1), a2 = (BC−B+A), a3 = (AC−A−D). From the Jury test, ifP (1) > 0, P (−1) < 0, and a3 < 1, |b3| > b1, c3 > |c2|, where b3 = 1−a23, b2 = a1−a3a2,b1 = a2 − a3a1, c3 = b23 − b21, and c2 = b3b2 − b1b2, then the roots of P (λ) satisfy λ < 1and thus fix1 is a asymptotically stable.

While linearizing the system (1.3) about fix2 or fix3 yields the following characteristicequation

F (λ) ≡ λ3 + a11λ2 + a22λ+ a33 = 0,where:a11 = 3− 11r + 4βr,

EJMAA-2013/1(2) ON THE FRACTIONAL-ORDER GAMES 321

a22 = 25.3βr3 + 31β2r2 + 315βr2 + 717.3βr + 29.6r − 59.3r2 − 7.3r3 − 3,

a33 = −r2(25β + 10)(1− 29β)(1− r(β+83 )− r3(25β + 10)(β + 8)(9− 2β),

From the Jury test, if F (1) > 0, F (−1) < 0, and a33 < 1, |b33| > b11, c33 > |c22|,where b33 = 1 − a233, b22 = a11 − a33a22, b11 = a22 − a33a11, c33 = b233 − b211, andc22 = b33b22 − b11b22, then the roots of F (λ) satisfy λ < 1 and thus fix2 or fix3 is aasymptotically stable.

3. Attractors, bifurcation and chaos

In this section we show by numerical experiments that the dynamical behavior of the dy-namica systems (1.3) and (1.4) is strongly affected by the change in β. In all simulationswe take r = 0.01Take β = −4.2 in (1.3) (Figure (1)).Take β = −2.5 in (1.3) (Figure (2)).Take β = 0 in (1.3) (Figure (3)).Take β = 0.3 in (1.3) (Figure (4)).Take β = −1.5 in (1.4) (Figure (5)).Take β = −1 in (1.4) (Figure (6)).Take β = −0.9 in (1.4) (Figure (7)).Take β = −0.7 in (1.4) (Figure (8)).

Figure 1. Regular attractorof (1.3) with β = −4.2

Figure 2. Regular attractorof (1.3) with β = −2.5

322 A. M. A. EL-SAYED, S. M. SALMAN EJMAA-2013/1(2)

Figure 3. Chaotic attractorof (1.3) with β = 0

Figure 4. Chaotic attractorof (1.3) with β = 0.3

Figure 5. Regular attractorof (1.4) with β = −1.5

Figure 6. Chaotic attractorof (1.4) with β = −1

EJMAA-2013/1(2) ON THE FRACTIONAL-ORDER GAMES 323

Figure 7. Chaotic attractorof (1.4) with β = −0.9

Figure 8. Chaotic attractorof (1.4) with β = −0.7

Figure 9. Bifurcation dia-gram of (1.3) as a function ofx

Figure 10. Bifurcation dia-gram of (1.3) as a function ofz

324 A. M. A. EL-SAYED, S. M. SALMAN EJMAA-2013/1(2)

Figure 11. Bifurcation dia-gram of (1.4) as a function ofx

Figure 12. Bifurcation dia-gram of (1.4) as a function ofz

4. Conclusion

A discretization process is applied to descretize the unified dynamical system betweenthe Lorenz and Chen systems with piecewise constant arguments. We obtained first andsecond difference equations of the unified system. Euler’s discretzation method was ableto obtain first order difference equation while we were able here to obtain a second orderdifference equation. The range of the parameter β of the original system has been changedfrom [0, 1] to [−0.9, 0.3].

References

[1] A. El-Sayed, A. El-Mesiry and H. EL-Saka, On the fractional-order logistic equation, AppliedMathematics Letters, 20, 817-823, (2007).

[2] A. M. A. El-Sayed and S. M. Salman, Chaos and bifurcation of discontinuous dynamicalsystems with piecewise constant arguments, Malaya Journal of Matematik, Volume 1, Issue1, 2012, 14-18.

[3] A. M. A. El-Sayed and S. M. Salman, Chaos and bifurcation of the Logistic discontinu-ous dynamical systems with piecewise constant arguments, Malaya Journal of Matematik,’accepted manuscript’.

[4] A. M. A. El-Sayed and S. M. Salman, On a discretization process of fractional order Riccatidifferential equation, Journal of Fractional Calculus and Applications, Vol. 4(2), 2013, pp.251-259.

[5] A. M. A. El-Sayed and S. M. Salman, On a discretization process of fractional order differ-ential equations: Dynamic behavior, ’submitted’.

[6] D. Altintan, Extension of the Logistic equation with piecewise constant arguments and pop-ulation dynamics, Master dissertation, Turkey 2006 .

[7] Holmgren, R. A First Course in Discrete Dynamical Systems. Springer-Verlag, New York(1994).

[8] J. H. L, G. Chen, D. Cheng, S. Celikovsky, Bridge the gap between the Lorenz system and theChen system, International Journal of Bifurcation and Chaos. vol. 12, No.12(2002) 2917-2926

EJMAA-2013/1(2) ON THE FRACTIONAL-ORDER GAMES 325

[9] M. U. Akhmet, Stability of differential equations with picewise constant arguments of gen-eralized type, Nonlinear Anal. 68(2008), No. 4, 794-803.

[10] M. U. Akhmet, D. Altntana, T. Ergenc, Chaos of the logistic equation with piecewise con-stant arguments, arXiv:1006.4753,2010.

[11] N. A. Bohai, Continuous solutions of systems of nonlinear difference equations with contin-uous arguments and their properties, Journal of Nonlinear oscillations,Vol. 10, No. 2, 2007.

[12] S.N. Elaidy, An introduction to difference equations, Third Edition, Undergradute Texts inMathematics, Springer, New York, 2005.

Ahmed M. A. El-SayedFaculty of Science, Alexandria University, Alexandria, Egypt

E-mail address: amasayed@gmail.com

S. M. Salman

Faculty of Education, Alexandria University, Alexandria, Egypt

E-mail address: samastars9@gmail.com

1. Introduction2. Fixed points and their asymptotic stability3. Attractors, bifurcation and chaos4. ConclusionReferences