A New Modified Hyperchaotic Finance System and its Control

ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.8(2009) No.1,pp.59-66 A New Modified Hyperchaotic Finance System and its Control Juan Ding, Weiguo Yang, Hongxing Yao Faculty of science, Jiangsu University, Zhenjiang, Jiangsu, 212013, China (Received 30 July 2008, accepted 9 April 2009) Astract: In this letter, a new four-dimensional continuous autonomous hyperchaotic system, which is constructed ased on a modified finance system y introducing a nonlinear state feedack controller. The detailed dynamical ehaviors of this hyperchaotic system are further investigated, including equliria and staility, various attractors, ifurcation analysis, and Lyapunov exponents spectrum. Furthermore, effective speed feedack controllers are designed for stailizing hyperchaos to unstale equilirium points. Numerical simulations are given to illustrate and verify the results. Keywords: hyperchaos; speed feedack control; Lyapunov exponent; ifurcation MSC: O415.5;TP.183 1 Introduction In recent years, chaos control and synchronization, including chaotification of dynamical system, have een received more attention due to its potential applications to physics, chemical reactor, control theory, iological networks,artificial neural networks, telecommunications and secure communication [1 6]. Many methods have een used to control dynamical system [1-6,22]. For instance, OGY method, differential geometric method, linear state space feedack, inverse optimal control and output feedack control, among many others. For feedack control, we often multiply the independent variale of system functions with coefficient and take the result as a feedack gain, so the method is called of displacement feedack control. Similarly, if we multiply the derivative of independent variale with coefficient, we call it speed feedack control[7]. If the feedack gain satisfies certain conditions, the chaotic system can e controlled to unstale equilirium points. Hyperchaotic system is usually defined as a chaotic system with more than one positive Lyapunov exponent[5,8-9]. Historically, hyperchaos was firstly reported y Rössler. That is, the noted four-dimensional hyperchaotic Rössler system [5]. Over the last two decades, some interesting hyperchaos generators were demonstrated and their dynamics have een investigated extensively in [10-14] over the past decades ecause of their useful application in engineering. As we know now, there are many hyperchaotic systems discovered in the high-dimensional social and economical systems [15-17]. A new hyperchaotic finance system is constructed in this Letter, and stailization of the hyperchaotic finance system is achieved. This letter is presented as follows: in the next section, the controlled finance system showing hyperchaotic ehavior is constructed via introducing a state feedack. In the third section, properties and dynamics of the controlled system are investigated numerically via ifurcation diagram, Lyapunov exponents. And in the last section, simple ut effective speed feedack controllers are designed for stailizing the hyperchaotic system to unstale equilirium. Furthermore, all aove dynamical ehaviors are verified y numerical simulation. Corresponding author. E-mail address: doudou@ujs.edu.cn Copyright c World Academic Press, World Academic Union IJNS.2009.xx.15/xxx

60 International Journal of Nonlinear Science,Vol.8(2009),No.1,pp. 59-66 2 Construction of the hyperchaotic finance system Recent works [18-20] have reported a dynamic model of finance, composed of three first-order differential equations.the model descries the time variations of three state variales: the interest rate x, the investment demand y, and the price index z, By choosing an appropriate coordinate system and setting appropriate dimensions for each state variale, references [18-20] offer the simplified finance system as ẋ = a(x + y) ẏ = y axz z = + axy where a, are system parameters.when a = 3, = 15, it shows chaotic ehavior.the strange attractor of the system is illustated in Fig.1. In light of the thought of G. Chen [17], we construct a hyperchaotic finance (1) Figure 1: Strange attractor of finance system (1). system y introducing a state feedack controller w to the system (1). The new controlled system has the form of ẋ = a(x + y) + w ẏ = y axz (2) ż = + axy ẇ = cxz dw where a, are the parameters of the system (1), and c is constant(where c=0.2), and d is the control parameter.when parameters a = 3, = 15, c = 0.2 and d= 0.12, the four Lyapunov exponents of the controlled system (2) calculated with Wolf algorithm [21] are 0.708300, 0.032247, 0 and 4.713900. Therefore, the controlled system (2) with parameter d= 0.12 shows hyperchaotic ehavior. Our numerical experiments show that system (2) has hyperchaotic attractors for a = 3, = 15, c = 0.2,d= 0.12 as depicted in Fig. 2(a) (j). 3 Dynamics analysis the hyperchaotic finance system This section further investigates the asic dynamical ehaviors of system (2). Oviously, from system (2), one has V = ẋ x + ẏ y + ż z + ẇ w Therefore, to make system (2) e dissipative, it is required that a + d + 1 > 0. = (a + d + 1) IJNS email for contriution: editor@nonlinearscience.org.uk

J. Ding, W. Yang, H. Yao: A New Modified Hyperchaotic Finance System and its Control (a) 3D view in the x y w space () 3D view in the x z w space. (c) 3D view in the y z w space. (d) 3D view in the x y z space. (e) Projection on the x w plane. (f) Projection on the x y plane. (g) Projection on the z w plane. (h) Projection on the y z plane. (i) Projection on the x z plane. (j) Projection on the y w plane. Figure 2: Phase portraits of hyperchaotic finance system (2). IJNS homepage:http://www.nonlinearscience.org.uk/ 61

62 International Journal of Nonlinear Science,Vol.8(2009),No.1,pp. 59-66 The equiliria of system (2) satisfies the following equations: a(x + y) + w = 0 y axz = 0 + axy = 0 cxz dw = 0 When the parameters a,, c, d satisfy ad(a 2 d c) > 0, the system (2) has two equilirium points: P 1 ( a a 2 d c ad, ad a 2 d c, ad a 2 d c, c ad ad a 2 d c ) = ( 2,,, c ad ), P 2 ( a 2 d c ad a ad, a 2 d c, ad a 2 d c, c ad ad a 2 d c ) = ( 2,,, c ad ), ad where = a 2 d c.oviously, P 1 and P 2 are symmetric aout x, y, w-axis for any parameters a,, c, d. At the equilirium pointsp 1, the Jacoian matrix is a a 0 1 2 J P1 = 1 0 0 0 (3) c 2 c 0 d which results in the characteristic polynomial: λ 4 + (1 + a + d)λ 3 + (a + d + ad + ac2 d+ad a 3 d (a 2 d c) )λ 2 +(2a 2 c + ad ac+dc ad )λ + 2(a 2 d c) = 0. (4) Using Routh Hurwitz criterion, it is easy to show that when a = 3, = 15, c = 0.2, d = 0.12. Some eigenvalues of the characteristic polynomial of the Jacoian matrix (3) has positive real parts. Thus the equilirium point P 1 is unstale. Similarly, the equilirium point P 2 is unstale. To investigate the impact of parameter d on the dynamics of the controlled system, we extend the range of d to an interval [0,0.35] and give the initial condition (0.1,-0.1,0.1,0.1), ifurcation diagram with respect to parameter d generated from the Poincaré section method is shown in Fig. 3, and the corresponding Lyapunov exponent spectrum calculated with the Wolf algorithm [21] is given in Fig. 4. Figs. 3 and 4 show how the dynamics of the controlled system(2) changes with the increasing value of the parameter d. Figure 3: Bifurcation diagram of the controlled system (2) versus parameter d, generated y the Poincaré section method. IJNS email for contriution: editor@nonlinearscience.org.uk

J. Ding, W. Yang, H. Yao: A New Modified Hyperchaotic Finance System and its Control 63 Assume that the Lyapunov exponents of system (2) are L i for i = 1, 2, 3, 4satisfying L 1 > L 2 > L 3 > L 4. The dynamical ehaviors of system (2) can e classified as follows ased on the Lyapunov exponents: (1) For L 1 > L 2 > 0, L 3 = 0 or L 3 > 0, L 4 < 0 and L 1 + L 2 + L 4 < 0,system (2) is hyperchaos. (2) For L 1 > 0, L 2 = 0, L 4 < L 3 < 0 and L 1 + L 3 + L 4 < 0,system (2) is chaos. (3) For L 1 < 0, L 2 < 0, L 3 < 0, L 4 < 0, system (2) is an equilirium point. (a) () Figure 4: Corresponding Lyapunov exponents of the controlled system (2) versus parameter d. 4 Hyperchaos control for the hyperchaotic system For the feedack control, the independent variale of a system function is often multiplied y a coefficient as the feedack gain, so the method is called displacement feedack control. Similarly, if the derivative of an independent variale is multiplied y a coefficient as the feedack gain, it is called speed feedack control. Suppose the following autonomic chaos system: Ẋ = AX + f(x), where X = (x 1, x 2,..., x n ) T, A = (a ij ) n n, f(x) is a nonlinear function, whenx 0 0, the system is chaotic or hyperchaotic. Then speed feedack control is presented as kẋ i feedack in the right side of the equation of x j (where k > 0, i j). As a whole system, changes of a certain variale can have relative influence upon the other variales. when x i is increasing, ẋ i > 0 and the feedack gain kẋ i < 0; when x i is decreasing, ẋ i < 0and the feedack gain kẋ i > 0. As a result, the system could achieve an anti-stailization alanced degree y control when coefficient k satisfies some conditions. Let the controlled hyperchaotic system is ẋ = a(x + y) + w ẏ = y axz ż = + axy kẋ ẇ = cxz dw where k is the feedack coefficient. When k < 3.0045 the system (5) will gradually converge to unsteadily equilirium point P 1 ( 2,,, c ad ) andp 2( 2,,, c ad ). Proof. At the equilirium pointsp 1,the Jacoian matrix of the congtrolled system (5) is a a 0 1 2 J = 1 0 ak + ak 0 k, c 2 c 0 d then the characteristic equation is (5) λ 4 + (a + d + 1)λ 3 + (a + d + ad + ck a2 k ( c + ck +ad2 +a 2 +a 3 2 a 2 dk + 2 2 c+a2 2 )λ 2 + 2 )λ + 2(a 2 d c) = 0 (6) IJNS homepage:http://www.nonlinearscience.org.uk/

64 International Journal of Nonlinear Science,Vol.8(2009),No.1,pp. 59-66 Because it is difficult to solve Esq. (6), we will give out the value data area of simulation in the following text. After complicated calculations, we get the range of the control gain, k < 3.0045which can ensure that the controlled system (5) is asymptocally stale at the equilirium points P 1. To verify the validity of the staility condition otained aove, we choose the control gaink = 3.0045. The four characteristic values of the Jacoian matrix are -23..9344, -1.1767 and -9.9523 ± 19.9366i. Consequently, the controlled system is asymptotically stale at the origin. The time responses of states of the controlled system withk = 3.0045are shown in Fig. 5(a). The controllers are activated at t = 10. It can e seen from the simulations that the hyperchaotic state is quickly settled down to the unstale equilirium points P 1. But the controlled system will ecome unstale if feedack coefficientk=3.00451. They are shown in Fig. 5().Therefore the original proposition k < 3.0045 is resulted. Selecting the control gain k = 3.0045, the Lyapunov exponents of the controlled system (5) are 0,0,- 0.03 and -4.05. The time evolution of the Lyapunov exponents of the controlled system (5) with k=3.0045 are shown in Fig.6. (a) Time responses of the states of the controlled system(5) with control gain.k=3.0045. () Time responses of the states of the controlled system(5) with control gain k=3.00451. Figure 5: The difference of the states of the controlled system (5) with the control gain change. Figure 6: Time evolution of the Lyapunov exponents of the controlled system(5) with k=3.0045. Figure 7: Time evolution of the Lyapunov exponents of the controlled system (7) with k=-13. Choose the controller kẋ, and adds it to the second equation of the hyperchaotic system, we otain the following controlled hyperchaotic system: ẋ = a(x + y) + w ẏ = y axz kẋ ż = + axy ẇ = cxz dw (7) IJNS email for contriution: editor@nonlinearscience.org.uk

J. Ding, W. Yang, H. Yao: A New Modified Hyperchaotic Finance System and its Control 65 where k is the control gain. When k < 1 the system (7) will gradually converge to unstale equilirium point P 1 and P 2 Selecting the control gain k= -13, and using the Wolf algorithm [21], we otain the following Lyapunov exponents 0, -0.5718, -0.6415 and 34.3091. From these Lyapunov exponents,we can know that the controller kẋ can stailize the hyperchaotic system (7) to the unstale equilirium pointsp 1. The time evolutions of the Lyapunov exponents are shown in Fig. 7. The time responses of states of the controlled system with k = 13 are shown in Fig. 8. The equilirium point of controlled system (7) with k = 13 are shown in Fig. 9. Figure 8: Time responses of the states of the controlled (7). Figure 9: The equilirium point of controlled system (7) with k=-13 k=-13. 5 Conclusion In this Letter, a new hyperchaotic finance system is uilt. Some asic dynamical ehaviors are further explored y calculating its Lyapunov exponent spectrum and ifurcation diagrams. The new hyperchaotic system has more complex dynamical ehaviors than the normal chaotic systems. Effective speed feedack controllers are designed for stailizing hyperchaos to unstale equilirium. Numerical simulations are proposed to verify and illustrate the effectiveness of these controllers. It is elieved that the system will have road applications in various chaos-ased information systems. Acknowledgements This research was supported y the National Nature Science Foundation of China ( No.10571076) and (No.70401013). References [1] G. Chen, X. Dong: From Chaos to Order: Perspectives, Methodologies and Applications,World Scientific,Singapore.157-( 1998) [2] E.Ott, C.Greogi, J.A.Yorke: Controlling chaos. Phys. Rev. Lett. 64:1196-1123(1990) [3] Dianchen Lu,Aicheng Wang, Xiandong Tian: Control and Synchronization of a New Hyperchaotic System With Unknown Parameters. International Journal of Nonlinear Science. 3(6):224-229(2008) [4] Xuein Zhang, Honglan Zhu: Anti- synchronization of Two Different Hyperchaotic Systems via Active and Adaptive Control. International Journal of Nonlinear Science. 3(6):216-223(2008) [5] O.E. Rössler: An equation for hyperchaos. Phys. Lett. A.71(2-3):155 157 (1979) IJNS homepage:http://www.nonlinearscience.org.uk/

66 International Journal of Nonlinear Science,Vol.8(2009),No.1,pp. 59-66 [6] Ju H. Park.: Adaptive controller design for modified projective synchronization of Genesio Tesi chaotic system with uncertain parameters. Choas, Solitons and Fractals. 34:1154-1159(2007) [7] H. N. Agiza: Controlling chaos for the dynamical system of coupled dynamos. Chaos, Solitons & Fractals.13:341 352(2002) [8] A. Ĉenys, A.Tamaŝeviĉius, A. Baziliauskas: Hyperchaos in coupled Colpitts oscillators. Chaos, Solitons Fractals. 17 (2-3):349 353(2003) [9] D.Cafagna, G.Grassi: New 3D scroll attractors on hyperchaotic Chua s circuits forming a ring. Int. J. Bifurcat. Chaos. 13 (10):2889-2903(2003) [10] T. Matsumoto, L.O. Chua, K. Koayashi: Hyperchaos: laoratory experiment and numerical confirmation. IEEE. Trans. Circuits. Syst. CAS. 33:1143-1147 (1986) [11] Hua Chen, Mei Sun: Generalized projective synchronization of the energy resource system. International Journal of Nonlinear Science. 2(3):166-170(2006) [12] Y.Takahashi, H.Nakano, T.Saito: A simple hyperchaos generator ased on impulsive switching. IEEE. Trans. Circuits. Syst. II. 51:468-472 (2004) [13] A.Tamasevicuius, A.Cenys: Hyperchaotic oscillator with gyrators. Electron. Lett.33:542-544(1997) [14] T. Tsuone, T. Saito: Hyperchaos from a 4-D manifold piecewiselinear system, IEEE. Trans. Circuits. Syst. I. 45:889-894 (1998) [15] C.Z.Ning, H.Haken: Detuned lasers and the complex Lorenz equations: sucritical and super-critical Hopf ifurcations.phys. Rev. A. 41:3826-3837(1990) [16] T.Kapitaniak, L.O.Chua: Hyperchaotic attractor of unidirectionally coupled Chua s circuit, Int. J. Bifurcat. Chaos. 4(2):477-482(1994) [17] Y.X. Li, W.K.S.Tang, G.Chen: Generating Hyperchaos via State Feedack Control.Int. J. Bifurcat. Chaos.15(10):3367-3375(2005) [18] J.H. Ma, Y.S.Chen: Study for the ifurcation topological structure and the gloal complicated character of a kind of nonlinear finance system (I).Appl. Math. Mech. (Englished.) 22:1240-1251(2001) [19] J.H. Ma, Y.S.Chen: Study for the ifurcation topological structure and the gloal complicated character of a kind of nonlinear finance system (II). Appl. Math. Mech. (Englished.) 22:1375-1382(2001) [20] J.H.Ma, R.B, Y.S Chen: Impulsive control of chaotic Attractors in Nonlinear Chaotic systems. Appl. Math. Mech. 25:889-894(2004) [21] A.Wolf, J.B.Swift, H.L.Swinney, A.W.John: Determining Lyapunov exponents from a time series. Physica D. 16:285-317(1985) [22] Guoliang Cai, Wentao Tu: Adaptive Backstepping Control of the Uncertain Unified Chaotic System. International Journal of Nonlinear Science. 4(1):17-24(2007) IJNS email for contriution: editor@nonlinearscience.org.uk