# Modified Function Projective Synchronization of Financial Hyperchaotic Systems via Adaptive Impulsive Controller with Unknown Parameters

Authors: Guoliang Cai; Lingling Zhang; Lan Yao; Xiulei Fang
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

Modified function projective synchronization via adaptive impulsive controller between two different financial hyperchaotic systems is investigated, where the external uncertainties are considered. The updated laws of the unknown parameters are given and the sufficient conditions are deduced based on Lyapunov stability theorem and the stability analysis of impulsive system. Finally, the two financial hyperchaotic systems are taken for example and the numerical examples are worked through for illustrating the main results.


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## Body

## 1. Introduction

Chaos synchronization, an important topic in nonlinear science, has been developed and studied extensively in various fields including secure communication, biological systems, information science, chemical reactions, and plasma technologies [1–3]. Since Pecora and Carroll [4] introduced a method to synchronize two identical systems with different initial conditions, a variety of approaches have been proposed for the synchronization of chaotic systems including complete synchronization [5], phase synchronization [6], generalized synchronization [7], lag synchronization [8], exponential synchronization [9], projective synchronization [10, 11], modified function projective synchronization [12], and adaptive synchronization [13].Among those synchronization methods, in recent years, function projective synchronization (FPS) of special form is introduced by some researches [14], where the drive system and response system could be synchronized up to a scaling function. It was modified in a more general form by Li and Zhao [15], function projective synchronization, which is called modified function projective synchronization (MFPS), where the drive system and response system are synchronized up to a desire scaling function matrix in the Lorenz system by an active control method. Up to now, MFPS which consider external uncertainties has been proposed by some researchers in [16].As we all know, financial systems are concerned with our daily life [17]. Chaos will appear when economic crisis happens. Actually, the 2007 global economic crisis shows the existence of the chaos. That is the background of our previous system in this paper [18]. The dynamical behaviors of the system are more complex because they have more than one positive Lyapunov exponent and they are expanded in more than one direction. So an effective and rapid control method is very necessary for government to take when chaotic phenomenon appears. And, in this paper, we study its MFPS via an adaptive impulsive control method in this financial hyperchaotic system.Impulsive phenomena exist in many biological science and mechanics fields actually. Consequently, the impulsive control becomes an interesting and useful synchronization approach [19, 20]. Its main idea is to change the states of a system by the “sudden jumps” instantaneously. Impulsive control may provide an efficient method for some cases in which the systems can not endure continuous disturbance. Therefore, by using the impulsive control method, the response system needs to receive the information from the drive system and achieve synchronization faster only at some discrete instants. So it has very strong advantage in practice due to reduced control cost and has strong robustness and anti-interference ability. Impulsive control is chosen aiming at applying some simple and easily verified sufficient conditions to stabilize the financial system. It is simpler to achieve practically, and the stabilization time is shorter than adaptive control. To the best of our knowledge, MFPS via impulsive control in this new financial hyperchaotic system has not been studied.The rest of this paper is organized as follows. In Section2, the new financial hyperchaotic system is presented. In Section 3, some sufficient conditions for MFPS are derived by impulsive control. In Section 4, MFPS of the financial hyperchaotic systems with uncertain parameters is investigated. Numerical examples are given in Section 5. Finally, the conclusions are drawn in Section 6.


## 2. System Description

In this section, we will give some description of our system studied. The novel nonlinear hyperchaotic finance system [18] can be described by the following differential equations:(1)x˙=z+y-ax+w,y˙=1-by-x2,z˙=-x-cz,w˙=-dxy-kw,where the model describes the time variations of four state variables: the interest rate x, the investment demand y, the price exponent z, and the average profit margins w, a, b, c, d, and k are the parameters of system (1), and they are all positive constants. As parameters are chosen as a=0.9, b=0.2, c=1.5, d=0.2, and k=0.17, the four Lyapunov exponents of system (1) calculated with Wolf algorithm are L1=0.034432, L2=0.018041, L3=0, and L4=-1.1499, which are presented in Figure 1. Figures 2(a)–2(d) show the 3-dimensional phase portraits of financial hyperchaotic system (1) and the time responses of (x,y,z,w)T in system (1) are presented in Figure 3. System (1) is constructed in the background of the global economic crisis which occurred in 2007 and still exists nowadays. As this global economic crisis did not cause the great depression like in the 1920s–1930s, system (1) is a weak hyperchaotic finance system. The largest Lyapunov exponent of system (1) is a little small, but this hyperchaotic finance system exactly reflects this finance phenomenon. By calculation, hyperchaotic system (1) has three equilibrium points: P0 (0,1/b,0,0) and P1,2(±θ,(k+ack)/c(k-d),∓θ/c,dθ(1+ac)/(cd-ck)) when the parameters a, b, c, d, and k satisfy (kb+dc+abck-ck)/c(d-k)>0 and θ=(kb+abck)/c(d-k)+1. As the eigenvalues of the Jacobian matrix at the equilibrium point P0 are 3.6433, 0.1278, −0.200, and −1.3411, respectively, hence, equilibrium P0 is an unstable saddle point in this nonlinear four-dimensional autonomous system. Similarly, system (1) undergoes a Hopf bifurcation at the equilibrium points P1,2. Therefore, the equilibrium points P1,2 are unstable saddle points. According to the hyperchaotic parameter values given above, the equilibrium points are calculated as: P0 (0,5,0,0), P1 (1.66,-8.87,-1.11,17.4), and P2 (-1.66,-8.87,1.11,-17.4). More dynamic behaviors have been analyzed about system (1) in [18].Figure 1
Lyapunov exponents spectrum of system (1).Figure 2
Phase portraits of hyperchaotic finance system (1).
(a)
3D view in thex-y-z space
(b)
3D view in thex-y-w space
(c)
3D view in thex-z-w space
(d)
3D view in they-z-w spaceFigure 3
Time series of(x,y,z,w)T in system (1).Remark 1.
Sometimes we need to propose some assumptions to simplify mathematical model for complexity of question when we apply this model to discuss some practical questions.


## 3. MFPS via Impulsive Control with Parameter Identification

In this section, we discuss the modified function projective synchronization of the uncertain dynamical system.Consider a class of chaotic systems with unknown parameters:(2)x˙=fx+ΦxΘ,where x∈Rn denotes the state vector of the system, Φ(x):Rn→Rn×p, and Θ∈Rp represent the vector of unknown parameters. Take system (2) as the drive system, and introduce the response system with a controller U∈Rn as the following equation:(3)y˙=gy+ΨyΩ+D+U,t≠tk,Δy=ytk+-ytk-=ytk+-ytk=Bke,hhhhhhhhhhhht=tkk=1,2,…,n,yt0+=y0,where y∈Rn is the state vector, y(tk+)=limt→tk+y(t), y(tk-)=limt→tk-y(t), Ψ(y):Rn→Rn×q, Ω∈Rq composes the unknown parameter vector of response system, U=[u1,u2,…,un]Τ∈Rn is the control input vector, and Bk denotes the impulsive control gains. The impulsive instant sequence tkk=1+∞ satisfies 0≤t1<t2<⋯<tk<tk+1<⋯ and limk→∞tk=∞, τk=tk-tk-1<∞.Remark 2.
ConsiderD=[d1,d2,…,dn]Τ∈Rn in (3) represents the external disturbance; that is, there exists positive bounded function ρi making the following inequalities hold: (4)di(t)≤ρii=1,2,…,n.Then the synchronization error dynamic system can be obtained as follows: (5)et=Λhtyt-xt,where Λ=diag{Λ1,Λ2,…,Λn} is a constant diagonal matrix, ht=diagh1t,h2t,…,hnt, and hi(t):Rn→Rn (i=1,2,…,n) are called scaling function factors.Definition 3.
The two systems described by system (2) and system (3) achieved modified function projective synchronization (MFPS), if there exists a continuously differentiable scaling function with hi(t)≠0 and a vector controller U such that (6)limt→∞e(t)=limt→∞Λh(t)y(t)-x(t)=0.Remark 4.
Ifh1t=h2t=⋯=hn(t), the scheme of synchronization is developed to synchronize two systems to a scaling function matrix with different initial values; that is, the modified function projective synchronization problem is simplified to function projective synchronization. If h=diag{h1,h2,…,hn}, where hi are real constants for all i∈R, the modified function projective synchronization converts to the modified projective synchronization. If Λh=diag{1,1,…,1}, modified function projective synchronization converts to the complete synchronization.From (3), we know that the error system between (2) and (3) is (7)e˙t=Λh(t)y˙(t)+Λh˙(t)y(t)-x˙(t)=Λhtgy+ΨyΩ+D+U+Λh˙(t)y(x)-f(x)-ΦxΘt≠tk,k=1,2,…,n,etk+=y(tk+)-x(tk+)=y(tk)+Bke(tk)-x(tk)=(I+Bk)e(t)t=tk,where I is the identical matrix with the same dimension as Bk.For given synchronization scaling functionhi(t) and scaling matrix Λ, the adaptive impulsive controller is as the following:(8)U=Λ-1h(t)-Λh˙tyt+fx+ΦxΘ^-me-g(y)-Ψ(y)Ω^-sgndiage1Λ1h1(t),e2Λ2h2(t),…,enΛnhn(t)ρ,where Θ^, Ω^ are the estimations of the parameters of the two systems, sgn(·) denotes the sign function, and m is a negative constant, which would influence the rate of the convergence. The larger m, the faster rate. So we can adjust the value of m according to the desirable convergent rate.The parameter update laws are as follows:(9)Θ^˙=-ΦΤxet,(10)Ω^˙=ΨΤyΛhtet,where Θ^ and Ω^ are the estimated values of unknown parameters Θ and Ω, respectively. Θ~=Θ-Θ^ and Ω~=Ω-Ω^ are estimate errors.Remark 5.
In the control law,sgn(diag{e1Λ1ht,e2Λ2ht,…,enΛnht})ρ is the compensator that is introduced to eliminate the influence of the external uncertainties.Lemma 6 (see [21]).
Suppose theq≥0, u(t) satisfies the scalar impulsive different inequality:(11)D+u(t)≤-pu(t)+qsup⁡t-τ≤s≤tus,t≠tk,t≥t0,u(tk)≤αku(tk-),0<αk-1≤1,k∈Z,u(t)=ϕ(t),t∈[t0-τ,t0],where u(t) is continuous at t≠tk, t≥t0 and u(tk)=u(tk+) and u(tk-) exist. Consider ϕ∈PC([t0-τ,t0],R). And the following condition is satisfied:(12)-p+qαk-1tk-tk-1<-ln⁡αk-1,k∈Z+.Then error system (7) is asymptotically stable.Now, the goal is to find some conditions on the control gains. Because the impulsive intervalτk=tk-tk-1=τ is a positive constant, the drive system is synchronized with the response system for arbitrary initial conditions. Thus we can present the following theorem.Theorem 7.
System (2) and system (3) can realize MFPS under adaptive impulse controller (8) and updated laws of the parameters (9) and (10) and satisfy the following conditions:(1)  Δk=sup⁡k{τk-τk-1}<∞,(2)  αk=λmax⁡I+BkΤI+Bk<1,(3)  -p+q/αk-1tk-tk-1<-ln⁡αk-1, k∈Z+,where λmax⁡(·) is the maximum eigenvalue.Proof.
Let us choose the following Lyapunov function:(13)V=12eΤe+Θ~ΤΘ~+Ω~ΤΩ~.With the controllers and the parameter update laws, here noting ρ=[ρ1,ρ2,…,ρn]Τ, the time derivation of the Lyapunov function along the trajectory of error system is given in two cases. In the case when t≠tk, we have(14)V˙=eΤe˙+Θ~ΤΘ~+Ω~ΤΩ~=eΤ+Λh˙(t)y(x)-f(x)-Φ(x)ΘΛh(t)[g(y)+Ψ(y)Ω+D+U]+Λh˙(t)y(x)-f(x)-Φ(x)Θ+Θ-Θ^Τ-Θ^˙+Ω-Ω^Τ-Ω^˙=-meΤe+eΤΛht·D-sgndiage1Λ1ht,e2Λ2ht,…,enΛnhtdiage1Λ1ht,e2Λ2htρ=-meΤe+h(t)eΤΛD-hteΤΛsgndiage1Λ1ht,e2Λ2ht,…,enΛnhtρ.
In the case whent=tk, we have(15)Vtk+=12eΤ(tk+)e(tk)+12α~Τα~+β~Τβ~=121+BiekΤ1+Biek+12α~Τα~+β~Τβ~≤12λmax⁡I+BiΤI+BieΤ(tk)e(tk)+12α~Τα~+β~Τβ~≤αkVtk,where αk=λmax⁡[(I+Bk)Τ(I+Bk)]<1.
Letξ1=h(t)eΤΛD, ξ2=h(t)eΤΛsgn(diag{e1Λ1h(t),e2Λ2h(t),…,enΛnh(t)})ρ. We see that ξ1,ξ2∈R and ξ2≥0. According to the definition and assumption of ρ and D, it is guaranteed that ξ1-ξ2≤0; therefore it can be obtained that(16)V˙t=-meΤe+ξ1-ξ2≤-meΤe≤-2mVt,where 2m=p; then V˙(t)≤-pV(t).
Then, we can obtain the following comparison system:(17)D+u(t)≤-pu(t)+qsup⁡t-τ≤s≤tus,t≠tk,t≥t0,u(tk)≤αku(tk-),0<αk-1<1,k∈Z,ut=ϕt,t∈t0-τ,t0.
Whenq=0, αk=αk-1<1, and (-p+q/αk-1)(tk-tk-1)<-ln⁡αk-1, k∈Z+, the stability conditions in comparison theorem are satisfied. We know that the error system is asymptotically stable, that is, the synchronization of the chaotic system under the impulsive controllers.
The proof is completed.Based on Lyapunov stability theory, we discussed MFPS between the drive and response system. By using the impulsive control, this drive-response hyperchaotic system can easily be synchronized. Some sufficient conditions for impulsive synchronization are derived. So we present this method to control financial system.


## 4. MFPS of the Financial Hyperchaotic Systems with Uncertain Parameters

In this section, new financial hyperchaotic system (1) will be adopted to verify the effectiveness of the MFPS schemes with the parameter identification obtained in Section 3.To further illustrate the effectiveness of the controller, we select the new financial hyperchaotic system as the drive system:(18)x˙1=x3+x2-a1x1+x4,x˙2=1-b1x2-x12,x˙3=-x1-c1x3,x˙4=-d1x1x2-k1x4,where u1, u2, u3, and u4 are controllers to be designed and a, b, c, d, and k are unknown parameters.System (18) can be presented in the form of (19)x˙1x˙2x˙3x˙4=x3+x1x2+x41-x12-x10+-x100000-x200000-x300000-x1x2-x4a1b1c1d1k1.We choose the novel hyperchaotic system [22] as the response system:(20)y˙1=-a2y1+y2+y4+d1+u1,y˙2=-y2-a2y1y3+d2+u2,y˙3=b2+a2y1y2+d3+u3,y˙4=-c2y1y3-d2y4+d4+u4,supposing all the parameters in the two systems are unknown.Then the above system can be described in the form of(21)y˙1y˙2y˙3y˙4=y4-y200+-y1-y2000-y1y3000y1y210000-y1y3-y4a2b2c2d2+d1d2d3d4+u1u2u3u4.According to Theorem7, the controllers can be obtained as(22)u1=1Λ1h1tx3+(x2-a^1)x1+x4-Λ1h˙1(t)y1-me1+a^2(y1+y2)-y4-ρ1sgn(Λ1h(t)e1),u2=1Λ2h2t1-b^1x2-x12-Λ2h˙2ty2-me2+y2+a^2y1y3-ρ2sgn(Λ2h(t)e2),u3=1Λ3h3t-x1-c^1x3-Λ3h˙3ty3-me3-b^2-a^2y1y2-ρ3sgnΛ3hte3,u4=1Λ4h4t-d^1x1x2-k^1x4-Λ4h˙4ty4-me4+c^2y1y3+d^2y4-ρ4sgn(Λ4h(t)e4).And the parameters estimation update laws are given as(23)a^˙1=x1e1,b^˙1=x2e2,c^˙1=x3e3,d^˙1=x1x2e4,k^˙1=x4e4,a^˙2=-(y1+y2)Λ1h(t)e1-y1y3Λ2h(t)e2+y1y2Λ3h(t)e3,b^˙2=Λ3h(t)e3,c^˙2=-y1y3Λ4h(t)e4,d^˙2=-y4Λ4h(t)e4.Through the above results, error system can be asymptotically stable.


## 5. Numerical Simulations

To verify the results obtained in Section4, some numerical simulations will be performed in this section. We choose system (18) as the drive system and system (20) as the response system. The parameters are chosen as a1=0.9, b1=0.2, c1=1.5, d1=0.2, a2=3, b2=15, c2=0.2, d2=0.12, and k2=0.17. The initial values of the drive system and the response system are (x1(0),x2(0),x3(0),x4(0))=(-0.5,-5,-5,-4) and (y1(0),y2(0),y3(0),y4(0))=(-4,-2,-2,-1). And the external disturbances are supposed as d1=0.5cos⁡(2πt), d2=0.2sin(5πt), d3=0.5cos⁡(6πt), and d4=0.2sin(5πt). Thus the boundaries of the uncertainties are ρ1=ρ2=ρ3=ρ4=1. The scaling matrix is taken as Λ=diag{1,-1,-1,-1}, while the scaling function is taken as h1(t)=sin(t), h2(t)=cos⁡(t), h3(t)=sin(t), and h4(t)=cos⁡(t). And choose m=-1, p=-2, τ=0.01, Bk=diag{-0.9,-0.9,-0.9,-0.9}, and αk=λmax⁡[(I+Bk)Τ(I+Bk)]=0.01<1; the value of the parameters has obtained the conditions in Theorem 7. The initial values of the unknown parameters are as a^˙1=b^˙1=c^˙1=d^˙1=k^˙1=1, a^˙2=b^˙2=c^˙2=d^˙2=1. Assume the controllers are switched on; then t=0 s. Using MATLAB, we get the time responses of MFPS errors with impulses in Figure 4 and the estimations of the unknown parameters of hyperchaotic systems in Figure 5.Figure 4
Time responses of MFPS errors with impulses.Figure 5
The estimations of the unknown parameters of hyperchaotic systems.In order to illustrate the superiority of MFPS method, we let the drive system, response system with impulses, parameters, and initial values be the same as before. The time responses of FPS errors with impulses are shown in Figure6. Comparing Figure 4 with Figure 6, we can observe two facts about the errors of MFPS method with impulse compared to the errors of FPS method with impulse: first, they converge to zero faster; second, their fluctuation is smaller.Figure 6
Time responses of FPS errors with impulses.Considering the traditional FPS method without impulse control, we let the drive system, response system without impulse, parameters, and initial values be the same as before. The time responses of FPS errors without impulses are given in Figure7. The time responses of MFPS errors without impulses are shown in Figure 8.Figure 7
Time responses of FPS errors without impulses.Figure 8
Time responses of MFPS errors without impulses.From Figures7 and 8, it is easy to find that the errors of MFPS without impulse have smaller fluctuation than those of FPS without impulse. This proved that the MFPS method is better than the FPS method. To further explain the advantages of MFPS method, we compare Figure 4 with Figure 7. It is obvious that the response and drive system synchronize in a shorter time under the MFPS method with impulse control than the FPS method without impulse. Those above results show the correctness and effectiveness of our methods via impulsive controller. As we all know, hyperchaotic financial system (18) is just one of the hyperchaotic systems. Furthermore, the MFPS method is a generalized form of FPS method from Definition 3 and Remark 2. So the MFPS method has wide application and can be applied to many other hyperchaotic systems such as hyperchaotic Chen system and hyperchaotic Lorenz system.In a summary, the MFPS method has three advantages: quick convergence, smaller fluctuation, and wide application. The MFPS method consuming more energy under impulse is a disadvantage.Remark 8 (the practical meaning of MFPS method).
There exist different levels of financial development in different regions. However, after using MFPS method, that is, taking some measures to promote the controlled financial system, it can make two different financial systems of different levels achieve synchronization level finally.


## 6. Conclusion

In this paper, we have introduced a new financial hyperchaotic system. Then we studied its MFPS method via an impulsive control. Through this paper, some less conservative conditions for its MFPS method are obtained. In the case of the equal impulsive distances, we use some simple and easily verified sufficient conditions to guarantee MFPS method. At last, simulated examples have been presented to show the effectiveness of the proposed impulsive control scheme.



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