Stability and hybrid synchronization of a time-delay financial hyperchaotic system

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1 ISSN England UK Journal of Information and Computing Science Vol. No. 5 pp Stability and hybrid synchronization of a time-delay financial hyperchaotic system Lingling Zhang Guoliang Cai + and Xiulei Fang Nonlinear Scientific Research Center Jiangsu University Zhenjiang Jiangsu PR China (Received January 8 5 accepted May 8 5 Abstract. his paper studies stability and hybrid synchronization of a time-delay financial hyperchaotic system. Based on Lyapunov stability theorem and differential inequalities stability is obtained by intermittent linear state feedback control. Furthermore hybrid synchronization method is firstly proposed to synchronize a financial hyperchaotic system and globally synchronization is obtained by proper hybrid controllers and Lyapunov stability theorem. he corresponding numerical simulations are performed to verify and illustrate the effectiveness and correctness of proposing methods. Keywords: time-delay financial hyperchaotic system intermittent linear state feedback control hybrid synchronization hybrid control.. Introduction In nonlinear science chaos and hyperchaos study has attracted much attention from scientists and engineers for its applications in diverse areas such as physical systems biological networks secure communications and so on []. Hyperchaotic systems have more complex behaviors and abundant dynamics than chaotic system because of possessing at least two positive Lyapunov exponents. herefore it is extremely important to research on hyperchaotic systems nowadays. Some classical hyperchaotic systems have been proposed such as the hyperchaotic Chen system the hyperchaotic Lü system etc. Due to its applications in many areas the studies of hyperchaos have not only emphasized on proposing and analyzing new interesting hyperchaotic systems but also studying hyperchaos control and synchronization. Chaos control is an important subject and it could control system to a predictive target many methods which include adaptive control linear feedback control nonlinear feedback control fuzzy control time delay control have been used to achieve it. Up to now various kinds of synchronization have been proposed to synchronized different systems or identity system with different values like phase synchronization lag synchronization projective synchronization hybrid synchronization complete synchronization antisynchronization and so on[-9]. Recently economy is the hot topic. Basing on the global economic crisis in 7 we firstly construct a financial hyperchaotic system in []. Its dynamical behaviors are more complex and more effective controls are proposed to control it. In fact hyperchaos system has a strong sensitivity to initial value so discrete control method is more in line with the actual situation. he intermittent control is active in work time and rest in other time(rest time [-].It reduce control input and save cost. So we main investigate intermittent control in this paper. Nowadays most works focus on studying the same kind synchronization between drive system and response system that is the states of response system synchronized to the states of drive system by the same kind synchronization. Whether it has the same phenomenon by two or more kinds of synchronization which defined as hybrid synchronization or not? here is no doubt that it is an interesting problem. Some scholars has investigate hybrid synchronization [] study the alternating between complete synchronization and hybrid synchronization of hyperchaotic Lorenz system with time delay [5] investigate hybrid synchronization of time-delay hyperchaotic D systems via partial variables However up to now fewer scholars investigate hybrid synchronization by hybrid control which combine continuous control with discrete control. As a result in this paper periodically intermittent linear control is firstly proposed to stabilize the financial hyperchaotic system which we introduce. Basing on Lyapunov stability thermo and differential inequalities stability of financial hyperchaotic system achieves. We also proposed hybrid synchronization scheme which is firstly applied to financial hyperchaotic system. his paper is organized as follows. In section describing the time-delay financial hyperchaotic system model. In section presenting stability scheme of a time-delay financial hyperchaotic system. In section hybrid Published by World Academic Press World Academic Union

2 z w 9 Lingling Zhang et.al. : Stability and hybrid synchronization of a time-delay financial hyperchaotic system synchronization method of a time-delay financial hyperchaotic system is shown. In section 5 corresponding numerical simulation are given. Finally making a conclusion in section 6.. he time-delay financial hyperchaotic system model Yu Cai etc proposed a financial hyperchaotic system without time-delay in [] however time-delay phenomenon often happed in actual situation. herefore in this paper we add time-delay to the system that has been proposed in [] we got a novel time-delay system as follows: x z y a x w( t y by x ( z x cz w dxy kw( t where the interest rate x the investment demand y the price exponent z the average profit margin w they are the state variables is time delay a b c d k are the positive parameters of the system(. When parameters a=.9 b=. c=.5 d=. and k=.7 the four Lyapunov exponents of the system ( calculated with Wolf algorithm are L =. L =.8 L = and L =-.99. here are three unstable equilibrium points k ack P P d ac kb abck where b ck d c cd k. According to the c( d k hyperchaotic parameter values given above the equilibrium points are calculated as: P ( 5 P ( and P ( Figure shows the Lyapunov exponents of system (. Figure (a-(d shows the -dimensional phase portraits of financial hyperchaotic system (. Fig.: Lyapunov exponents spectrum of system ( with = y x -.5 y x (a D view in the x-y-z space (b D view in the x-y-w space JIC for contribution: editor@jic.org.uk

3 w w Journal of Information and Computing Science Vol. (5 No. pp z x -.5 z y (c D view in the x-z-w space (d D view in the y-z-w space Fig. : Phase portraits of system ( with =.. Stability schemes of a time-delay financial hyperchaotic system via intermittent linear state feedback control In this section we will give the control scheme of a time-delay financial system in detail. At first we will give a stability scheme of a class of time-delay hyperchaotic system by intermittent linear state feedback control. Secondly this method will be applied to a time-delay financial hyperchaotic system. Finally stability is obtained by Lyapunov stability and proper controllers... Stability scheme of a time-delay financial hyperchaotic system via intermittent control Consider a class of certain hyperchaotic system with time delay described by: x& ( t Ax( t f ( x( t g( x( t ( he controlled system is designed as x& ( t Ax( t f ( x( t g( x( t u( t ( n nn where x R is the state vector of the systems ( and ( A R is the constant matrix n n n n f R n n nn : R and g : R R are continuous nonlinear vector functions satisfying f(= g(= τ is the time delay and u(t is the intermittent linear state feedback controller that is defined as the following: Kx( t n t n ut ( n t n nn where K R is a constant control gain > is the control period and δ> is called the control width. Our goal is to design suitable δ K such that the system ( stable in this section. Assumption. We assume f (x g(x(t-τ are bounded functions. hat is exist constants matrices L and P for any x such that f ( x L x x Lx gx( t P x( t x ( t Px( t where diag... with L l l l n P p p p n diag L. Assumption. We assume x is bounded variables. hat is exist constants D for any x such that x D. n Lemma []. For any vectors x y R and a positive definite matrix Q n n R the following matrix inequality holds: x y x Qx y Q y. ( v v Lemma []. V ( t M exp( t M V ( e e t>. If there exist positive constants u u v v such that the following condition holds: (a r( ( v v( V ( t uv ( t uv ( t n t n (b V ( t vv ( t vv ( t n t ( n where r is the unique positive solution to r=-u +u e rτ τ is the time delay δ is the control width ω is the control period. JIC for subscription: publishing@wau.org.uk

4 9 Lingling Zhang et.al. : Stability and hybrid synchronization of a time-delay financial hyperchaotic system heorem. (stabilization criterion he system ( is stabilized if there exist positive constants α >α c c c and positive matrix L which defined in Assumption such that the following conditions hold: (a A A KI I I L c I (b A A KI I L c I (c r c c ( r where r is the unique positive solution to r c c e I is the identity matrix. Proof. Let us choose the following Lyapunov function: t x ( t x( t V ( t x (5 and calculate the derivation Vt ( with respect to time to along the trajectories of the controlled system (. For n t n in addition to Assumption Lemma and condition (a in the heorem are used to get the following estimate: V & t x & t x t x t x & t ( ( ( ( = ( ( Ax t f x t g x t Kx t xt x t( Ax t f ( x( t g xt + Kx (t = x t A A K xt f x( t xt g xt xt A A K Kxt g xt Qg xt xt x t Q xt f x f x t x t x t ( ( ( ( ( x t A A K K Q x t x ( t Lx( t PQx ( t x( t x t A A K K L c I x t c V ( t cv ( t cv t c V t. Notice that c c PQ I Q I. Namely we have V & t cv t c V t for n t n. For n t ( n in addition to Assumption and condition (b in the heorem are used to get the following estimate: V & t x & t x t x t x & t ( ( ( ( = Axt f ( x( t gxt xt x t Axt f ( x( t gxt = x t A Axt f ( xt xt g xt xt x t A Axt f ( x( t f ( x( t x ( t xt + g xt Qgxt x t Q xt x ( t A A L c I x t c V t c x t x t c V t c V t. Notice that PQ=c I Q I. Namely we have V & t c V t c V t for n t ( n. By Lemma we have ( c c M V e e where V t t r( ( c c(. By Eq. (5 and Eq.(6 we have M e for t> (6 JIC for contribution: editor@jic.org.uk

5 Journal of Information and Computing Science Vol. (5 No. pp x t t M e. herefore we obtained the following: x t t M e for t>. his means that the control system is get stabilization. he proof is complete... Periodically intermittent linear state feedback control with time delay stabilized P (/b At first we transform system ( in P let x =x x =y-/b x =z x =w. So the Eq. ( changes into Eq. (7 and unstable equilibrium point P changes into P (. x x xx a x x( t b x bx x x x cx d x dxx x kx ( t b (7 he controlled system is designed as x x a x x( t xx u b x bx x u (8 x x cx u d x x kx( t dxx u b he controllers are designed as kixi( t n t n ui i. n t ( n In system (8 the parameters a=.9 b=. c=.5 d=. and k=.7 therefore linear matrix is calculated as:.. A.5 nonlinear function vector f ( x x x x.x x and the time delay function g( x( t x ( t.7 x ( t We choose positive matrices L=P=.5I α =α = c = c =.5 c = then system (8 satisfied all conditions in heorem. It has proved that system (8 stabilized in equilibrium point P (. herefore the system ( stabilized in P ( /b.. Hybrid synchronization scheme of a time-delay financial hyperchaotic system via hybrid control JIC for subscription: publishing@wau.org.uk

6 9 Lingling Zhang et.al. : Stability and hybrid synchronization of a time-delay financial hyperchaotic system In this section we will discuss hybrid synchronization of a time-delay financial system by hybrid control. he definitions of hybrid synchronization and hybrid control will be given. Base on proper hybrid controllers hybrid synchronization will be achieve between drive system and response system. Definition. It is defined as hybrid synchronization if e i (t=y i (t-x i (t e j (t=y j (t+x j (t ij i j= n where y i (t x i (t are the state variables of response system and drive system respectively. Definition. U(t is called hybrid controller if there exist different kinds of controllers such that U(t=U i (t+u j (t i j= n ij. Rewritten system ( as: x x ( x a x x( t x bx x (9 x x cx x dxx kx ( t Choosing system (9 as the drive system and response system is designed as: y y ( y a y y( t u y by y u ( y y cy u y dyy ky( t u where u u u u are controllers to be constructed. he goal of this section is to choose proper controllers so that the state variables y y y in response system are complete synchronized to x x x in drive system respectively while the state variable y in response system is anti-synchronized to x in drive system. For this target let error vector e (t=y (t-x (t e (t=y (t-x (t e (t=y (t+x (t e (t=y (t-x (t e (t-=y (t--x (t- from systems (9 and ( we get error system: e ae e( t y y y x xx u e be y x u ( e ce y x u e ke ( t dy y dxx u he purpose is to propose hybrid input controllers so that the state errors in ( satisfy: lim e ( t. i. t i With this in mind the proper hybrid controllers will be designed as follows: ui ui ui ( i ( where bie i n t n ui t i ( n t n and u y x y y xx u y x ( u y x u dy y dxx where b i (i= are positive intermittent linear state feedback control gain. heorem. Hybrid synchronization can be achieved between systems (9 and ( with above hybrid control law( and proper positive constants m m m m d d d satisfied the following conditions: k k min{ a b b b c b b } (a JIC for contribution: editor@jic.org.uk

7 Journal of Information and Computing Science Vol. (5 No. pp k min{ a m b m c m m } r( ( d d ( (b (c r where r is the unique positive solution to r d d e is the time delay is the control width is the control period. Proof. Constructing a positive definite Lyapunov function as following: V e e e e (5 Calculating the time derivative of the Lyapunov function (5 along the trajectory of system ( arrives at: V& e e& e e& e e& e e& e ( ae e y y y x x x u e ( be y x u e ( ce y x u e ( ke dy y dxx u For n t n Adding Eq.( to Eq.(6 by Lemma and condition (a in heorem we obtain V& ( a b e ( b b e ( c b e b e ke e ( t e e ( t k k ( a b e ( b b e ( c b e ( b e ( e ( t dv ( t dv ( t k k where d max{ a b b b c b b } d For n t ( n by Assumption and condition (b in heorem we get V& ae be ce ke e ( t e e ( t k k ( a e be ce e ( e ( t k k ( a m e ( b m e ( c m e ( m e ( e ( t m e m e m e m e dv ( t dv ( t k where d max{ m m m m} d herefore dv ( t dv ( t n t n V dv ( t dv ( t n t ( n By Lemma we get t V M exp{ } M V ( exp{( d d } herefore by Eq.(6 we get t ei( t Mei{ } i. that is if t then ei i. herefore if appropriate intermittent linear state feedback control gains and d d d are selected appropriately it can be obtained that the error system ( will be convergence to zero and globally asymptotically stable while the coexistence of anti-synchronization and complete synchronization of systems (9 and ( with the hybrid control ( can be achieved. his completes the proof. (6 JIC for subscription: publishing@wau.org.uk

8 x (t x (t x (t x (t Numerical simulation Lingling Zhang et.al. : Stability and hybrid synchronization of a time-delay financial hyperchaotic system o verify stability scheme of financial hyperchaotic system via intermittent linear state feedback control we choose system (8 as the controlled system. he parameters are chosen as a=.9 b=. c=.5 d=. k =.7. he initial values of the controlled system (x ( x ( x ( x (=( the time delay τ=.5 and control gain K is designed as 6. Assuming the controllers are switched on the =s and δ=.8s. Using MALAB we get the time response of states for the controlled system (8 in Fig.. he state variables of controlled system (8 converge to zero when t>s therefore the unstable equilibrium point P get stabilization. 5 5 x x x 5 x Fig.: ime response of x ( t ( x ( t x( t x( t x( t of the controlled system (8 with intermittent linear state feedback control. o verify hybrid synchronization scheme of financial hyperchaotic system we choose system (9 as the drive system system ( as the response system. he parameters are chosen as a=.9 b=. c=.5 d=. k =.7. he initial values of the drive system and the response system are (x ( x ( x ( x (=( and (y ( y ( y ( y (=(. Control gain b i (i= are designed as the time delay Assuming the controllers are switched on the =s and δ=.8s. Using MALAB we get the time evolution of states for the error system ( in Fig. and the time evolution of state variables in drive system (9 and response system ( in Fig.5. In Fig. the states for the error system ( converge to zero quickly and the fluctuation is small it also need less energy by intermittent linear state feedback control. In Fig.5 it is easy to find that the states x (t x (t x (t complete synchronized to y (t y (t y (t and x (t antisynchronized to y (t. All of those have been illustrated the effectiveness and correctness of hybrid synchronization method. JIC for contribution: editor@jic.org.uk

9 e (t e (t e (t e (t Journal of Information and Computing Science Vol. (5 No. pp e e e e Fig.: he time evolution of errors system (. y y x x y x y x y y x x y x y x Conclusion Fig.5: he time evolution of state variables in drive system (9 and response system (. In this paper a new time-delay financial hyperchaotic system is introduced and analysis its stability by intermittent linear state feedback control with time-delay τ. Hybrid synchronized method is firstly proposed to a financial hyperchaotic system. By proper hybrid control and Lyapunov stability theory hybrid synchronization is achieve between drive system and response system the state variables y y y in response system are complete synchronized to x x x in drive system respectively while the state variable y in response system is anti-synchronized to x in drive JIC for subscription: publishing@wau.org.uk

10 98 Lingling Zhang et.al. : Stability and hybrid synchronization of a time-delay financial hyperchaotic system system. Corresponding numerical simulation are offered to show the effectiveness and correctness of proposing methods. Acknowledgements his work was supported by the National Nature Science foundation of China (Grants the Society Science Foundation from Ministry of Education of China (Grants YJAZH 8JA7957 he Priority Academic Program Development of Jiangsu Higher Education Institutions the Advanced alents Foundation of Jiangsu University (Grants 7JDG5JDG and the Students Research Foundation of Jiangsu University (No.YA6No.A9. Especially thanks for the support of Jiangsu University. 7. References [] L. Pan D. Y. Xu and W. N. Zhou Controlling a Novel Chaotic Attractor using Linear Feedback Journal of Information and Computing Science 5 ( pp. 7-. [] H. X. Wang and G. L. Cai Controlling hyperchaos in a novel hyperchaotic System Journal of Information and Computing Science (9 pp [] X. F. Li A-C. S. Leung and X. P. Han Complete (anti-synchronization of chaotic systems with fully uncertain parameters by adaptive control Nonlinear Dynamics 6 ( pp [] G. L. Cai P. Hu and Y. X. Li Modified function lag projective synchronization of a finance hyperchaotic system Nonlinear Dynamics 69 ( pp [5] X. L. Fang G. L. Cai L. Yao and L. L. Zhang A single-input linear controller for complete synchronization of a delay financial hyperchaotic system Journal of Information and Computing Science 9( pp. -. [6] X. R. Shi and Z. L. Wang Complete synchronization of delay hyperchaotic Lüsystem via a single linear input Nonlinear Dynamics 69( pp.5-5. [7] G. L. Cai H. X. Wang and S. Zheng Adaptive function projective synchronization of two different hyperchaotic systems with unknown parameters Chinese Journal of Physics 7( [8] J. H. Park Adaptive control for modified projective synchronization of a four-dimensional chaotic system with uncertain parameters Journal of Computational and Applied Mathematics (8 pp [9] G. L. Cai L. Yao P. Hu and X. L. Fang Adaptive full state hybrid function projective synchronization of financial hyperchaotic systems with uncertain parameters Discrete and Continuous Dynamical Systems Series B 8( pp.9-8. [] H. J. Yu G. L. Cai and Y. X. Li Dynamic analysis and control of a new hyperchaotic finance system Nonlinear Dynamics 67( pp.7 8. []. W. Huang C. D. Li X. Z. Liu Synchronization of chaotic systems with time delay using intermittent linear state feedback Chaos 8(8. [] S. M. Cai J. J. Hao Q. B. He Z. R. Liu Exponential stabilization of complex delayed dynamical networks via pinning periodically intermittent control Physics. Letter. A 75( pp []. W. Huang C. D. Li Chaotic synchronization by the intermittent feedback method Journal of Computation and Applied Mathematics ( pp.97-. [] X. R. Shi and Z. L. Wang he alternating between complete synchronization and hybrid synchronization of hyperchaotic Lorenz system with time delay Nonlinear Dynamics 69( pp [5] Z. L. Wang W. Chao X. R. Shi J. M K. M. ang H. S. Cheng Realizing hybrid synchronization of time-delay hyperchaotic D systems via partial variables Applied Mathematics and Computation 5( pp.7-7. JIC for contribution: editor@jic.org.uk