# Chaos Control of a Fractional-Order Financial System

Authors: Mohammed Salah Abd-Elouahab; Nasr-Eddine Hamri; Junwei Wang
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

Fractional-order financial system introduced by W.-C. Chen (2008) displays chaotic motions at order less than
3. In this paper we have extended the nonlinear feedback control in ODE systems to fractional-order systems, in order to eliminate the chaotic behavior. The results are proved analytically
by applying the Lyapunov linearization method and stability condition for fractional system.
Moreover numerical simulations are shown to verify the effectiveness of the proposed control
scheme.


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## Body

## 1. Introduction

Nonlinear chaotic systems have attracted more attention of researchers in various fields of natural sciences. This is because these systems are rich in dynamics, and possess great sensitivity to initial conditions. Since the chaotic phenomenon in economics was first found in 1985, great impact has been imposed on the prominent western economics at present, because the chaotic phenomenon's occurring in the economic system means that the macroeconomic operation has in itself the inherent indefiniteness. Although the government can adopt such macrocontrol measures as the financial policies or the monetary policies to interfere, the effectiveness of the interference is very limited. The instability and complexity make the precise economic prediction greatly limited, and the reasonable prediction behavior has become complicated as well. In the fields of finance, stocks, and social economics, because of the interaction between nonlinear factors, with all kinds of economic problems being more and more complicated and with the evolution process from low dimensions to high dimensions, the diversity and complexity have manifested themselves in the internal structure of the system and there exists extremely complicated phenomenon and external characteristics in such a kind of system. So it has become more and more important to study the control of the complicated continuous economic system, and stabilize the instable periodic or stationary solutions, in order to make the precise economic prediction possible [1, 2].Great interest has been paid to the application of fractional calculus in physics, engineering systems, and even financial analysis [3, 4]. The fact that financial variables possess long memories makes fractional modelling appropriate for dynamic behaviors in financial systems. Moreover, the control and synchronization of fractional-order dynamic systems is also performed by various researchers [5–10]. Fractional-order financial system proposed by Chen in [11] displays many interesting dynamic behaviors, such as fixed points, periodic motions, and chaotic motions. It has been found that chaos exists in this system with orders less than 3, period doubling, and intermittency routes to chaos were found. In this paper, we propose to eliminate the chaotic behaviors from this system, by extending the non-linear feedback control in ODE systems to fractional-order systems. This paper is organized as follows. In Section 2, we present the financial system and its fractional version. In Section 3 general approach to feedback control scheme is given, and then we have extended this control scheme to fractional-order financial system, numerical results are shown. Finally, in Section 4 concluding comments are given.


## 2. Financial System

### 2.1. Integer-Order Financial System

Recently, the studies in  [1, 2] have reported a dynamic model of finance, composed of three first-order differential equations. The model describes the time-variation of three state variables: the interest rate x, the investment demand y, and the price index z. The factors that influence the changes of x mainly come from two aspects: firstly, it is the contradiction from the investment market, (the surplus between investment and savings); secondly, it is the structure adjustment from goods prices. The changing rate of y is in proportion with the rate of investment, and in proportion by inversion with the cost of investment and the interest rate. The changes of z, on one hand, are controlled by the contradiction between supply and demand of the commercial market, and on the other hand, are influenced by the inflation rate. Here we suppose that the amount of supplies and demands of commercials is constant in a certain period of time, and that the amount of supplies and demands of commercials is in proportion by inversion with the prices. However, the changes of the inflation rate can in fact be represented by the changes of the real interest rate and the inflation rate equals the nominal interest rate subtracts the real interest rate. The original model has nine independent parameters to be adjusted, so it needs to be further simplified. Therefore, by choosing the appropriate coordinate system and setting an appropriate dimension to every state variable, we can get the following more simplified model with only three most important parameters:(2.1)ẋ=z+(y-a)x,ẏ=1-by-x2,ż=-x-cz,
where a≥0 is the saving amount, b≥0 is the cost per investment, and c≥0 is the elasticity of demand of commercial markets. It is obvious that all three constants, a, b, and c, are nonnegative, For more detail about the study of the local topological structure and bifurcation of this system; see [1, 2]. We assume that a is control parameter and b=0.1,c=1.


#### 2.1.1. Analysing the System

(i)
Ifa≥9, system (2.1) has one fixed point:
(2.2)p1=(0,10,0).(ii)
Ifa<9, system (2.1) has three fixed points:
(2.3)p1=(0,10,0),p2,3=(∓9-a10,a+1,±9-a10).To study the stability of equilibrium points we apply the Lyapunov's first (indirect) method [12] so we have the following theorem.Theorem 2.1.
Letx=x* be an equilibrium point of a nonlinear system:
(2.4)ẋ=f(x),
where f:D→ℝn is continuously differentiable and D⊂ℝn is the neighborhood of the equilibrium point x*. Let λi denote the eigenvalues of the Jacobian matrix A=∂f/∂x|x* then the following are considered. (i)
IfReλi<0 for all i, then x=x* is asymptotically stable.(ii)
IfReλi>0 for one or more i, then x=x* is unstable.(iii)
IfReλi≤0 for all i and at least one Reλj=0, then x=x* may be either stable, asymptotically stable, or unstable.SinceA is only defined at x*, stability determined by the indirect method is restricted to infinitesimal neighborhoods of x*.To study the signs of the real parts of eigenvalues, we have the following famous criterion [13].Criterion (Routh-Hurwitz).
Given the polynomialP(λ)=λn+a1λn-1+⋯+an-1λ+an, where the coefficients ai,i=1,2,…,n, are real constants, define the n Hurwitz matrices
(2.5)H1=(a1),H2=(a11a3a2)⋮Hn=(a1100⋯0a3a2a11⋯0a5a4a3a2⋯0⋮⋮⋮⋮⋯⋮0000⋯an),
where ai=0 if i>n.
All of roots of the polynomial have negative real part if and only if the determinants of all Hurwitz matrices are positive:detHi>0,i=1,2,…,n.Routh-Hurwitz criteria forn=3 are a1>0,a3>0 and a1a2-a3>0.Stability ofp1
The Jacobian matrix of system (2.1) at the equilibrium point p1 is
(2.6)Jp1=(10-a010-1100-10-1),
its characteristic polynomial is
(2.7)P(λ)=λ3+(a-8910)λ2+(11a-9910)λ+(a-910).
By applying the Routh-Hurwitz criterion we find that the real parts of these eigenvalues are all negative if and only if(2.8)a-8910>0,a-9>0,(a-8910)(11a-9910)-(a-910)>0.
Then it follows that a>9, and thus p1 is locally asymptotically stable if and only if a>9.Stability ofp2,3
The Jacobian matrix of system (2.1) at the equilibrium points p2,3 is
(2.9)Jp2,3=(1±9-a101∓29-a10-0.10-10-1),
and its characteristic polynomial is
(2.10)p̃(λ)=λ3+110λ2+(-15a+1810)λ+(-15a+1810).
The real parts of these eigenvalues are all negative if and only if
(2.11)-15a+1810>0,110(-15a+1810)-(-15a+1810)>0.
Then it follows that
(2.12)a<9,a>9.
So p2,3 are unstable for every value of a.
In order to detect the chaos we calculate the largest Lyapunov exponentλmax using the scheme proposed by Wolf et al. [14]. The initial states are taken as x(0)=2, y(0)=3,z(0)=2, Figure 1(a) displays the evolution of λmax according to a and Figure 1(b) displays chaotic attractor for a=3. System (2.1) displays chaotic behavior in the windows 0<a<7 (λmax>0), periodic behavior in 7≤a≤9 (λmax≈0) and stationary behavior for a>9 (λmax<0).(a) Largest Lyapunov exponent according toa. (b) Chaotic attractor for a=3.
(a)(b)


### 2.2. Fractional-Order Financial System

Chen has introduced in [11] the generalization of system (2.1) for fractional incommensurate-order model which takes the form(2.13)Dq1x=z+(y-a)x,Dq2y=1-by-x2,Dq3z=-x-cz.
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary order but there are several definitions of fractional derivatives.In this paper, we use the Caputo-type fractional derivative defined in [15] by:(2.14)Dqf(t)=1Γ(n-q)∫0t(t-τ)n-q-1f(n)(τ)dτ=jn-q(dndtnf(t)),
where n=[q] is the value of q rounded up to the nearest integer, Γ is the gamma function and jα is the Riemann-Liouville integral operator defined by(2.15)jαf(t)=1Γ(α)∫0t(t-τ)α-1f(τ)dτ.
For the numerical solutions of system (2.13) we use the Adams-Bashforth-Moulton predictor-corrector scheme [16].We assume thatq (q1=q2=q3=q) is the control parameter, and c=1,b=0.1,a=3. Fractional system (2.13) has the same fixed points p1,2,3 as integer system (2.1), but for the stability analysis we have this theorem introduced in [17, 18].Theorem 2.2.
The fractional linear autonomous system(2.16)DαX=AXX(0)=X0X∈Rn,0<α<2,A∈Rn×Rn,
is locally asymptotically stable if and only if
(2.17)mini|arg(λi)|>απ2,i=1,2,…,n.Proposition 2.3.
Letx=x* be an equilibrium point of a fractional nonlinear system
(2.18)Dαx=f(x),0<α<2.
If the eigenvalues of the Jacobian matrix A=∂f/∂x|x* satisfy
(2.19)mini|arg(λi)|>απ2,i=1,2,…,n,
then the system is locally asymptotically stable at the equilibrium point x*.Proof.
Letx=x*+δx. Substituting in (2.18), we find
(2.20)Dα(x*+δx)=f(x*+δx).
so
(2.21)Dα(δx)=f(x*)+Aδx+◯(∥δx∥2).
Sincef(x*)=0 (x* is the equilibrium point of system (2.18)) and lim∥δx∥→0(◯(∥δx∥2)/∥δx∥)=0, then(2.22)Dαδx≈Aδx.
Taking into account Theorem 2.2, we deduce that If the eigenvalues of the matrix A satisfy
(2.23)mini|arg(λi)|>απ2,i=1,2,…,n,
then x* is locally asymptotically stable.
This completes the proof.Stability ofp1
The Jacobian matrix of system (2.13) at the equilibrium point p1 is
(2.24)Jp1=(7010-1100-10-1),
and its characteristic polynomial is
(2.25)P(λ)=λ3-5910λ2-6610λ-610.
its eigenvalues are λ1≈-0.87298,λ2=-1/10,λ3≈6.8730, we note that λ3 is real positive then |arg(λ3)|=0<q(π/2), for all q∈]0,2[, so p1 is unstable for all q∈]0,2[.Stability ofp2,3
The Jacobian matrix of system (2.13) at the equilibrium point p2,3 is
(2.26)Jp2,3=(1±351∓235-1100-10-1),
its characteristic polynomial is
(2.27)p̃(λ)=λ3+110λ2+65λ+65,
and its eigenvalues are λ1≈0.31278+1.2474i,λ2≈0.31278-1.2474i,andλ3≈-0.72556, we have
(2.28)|arg(λ1,2)|≈1.3251,|arg(λ3)|=π,
so mini|arg(λi)|≈1.3251, then the critical value of q is
(2.29)qc=2mini|arg(λi)|π≈0.8436,(i)
Ifq<0.8436, then p2,3 are locally asymptotically stable.(ii)
Ifq>0.8436, then p2,3 are unstable.  In order to detect the chaos, we calculate the largest Lyapunov exponent λmax.
The initial states are taken asx(0)=2, y(0)=3, z(0)=2, Figure 2(a) shows the evolution of λmax according to q. System (2.13) exhibits chaotic behaviors for q≥0.86.(a) Largest Lyapunov exponent according toq. (b) Phase diagram for some values of q.
(a)(b)


## 2.1. Integer-Order Financial System

Recently, the studies in  [1, 2] have reported a dynamic model of finance, composed of three first-order differential equations. The model describes the time-variation of three state variables: the interest rate x, the investment demand y, and the price index z. The factors that influence the changes of x mainly come from two aspects: firstly, it is the contradiction from the investment market, (the surplus between investment and savings); secondly, it is the structure adjustment from goods prices. The changing rate of y is in proportion with the rate of investment, and in proportion by inversion with the cost of investment and the interest rate. The changes of z, on one hand, are controlled by the contradiction between supply and demand of the commercial market, and on the other hand, are influenced by the inflation rate. Here we suppose that the amount of supplies and demands of commercials is constant in a certain period of time, and that the amount of supplies and demands of commercials is in proportion by inversion with the prices. However, the changes of the inflation rate can in fact be represented by the changes of the real interest rate and the inflation rate equals the nominal interest rate subtracts the real interest rate. The original model has nine independent parameters to be adjusted, so it needs to be further simplified. Therefore, by choosing the appropriate coordinate system and setting an appropriate dimension to every state variable, we can get the following more simplified model with only three most important parameters:(2.1)ẋ=z+(y-a)x,ẏ=1-by-x2,ż=-x-cz,
where a≥0 is the saving amount, b≥0 is the cost per investment, and c≥0 is the elasticity of demand of commercial markets. It is obvious that all three constants, a, b, and c, are nonnegative, For more detail about the study of the local topological structure and bifurcation of this system; see [1, 2]. We assume that a is control parameter and b=0.1,c=1.


### 2.1.1. Analysing the System

(i)
Ifa≥9, system (2.1) has one fixed point:
(2.2)p1=(0,10,0).(ii)
Ifa<9, system (2.1) has three fixed points:
(2.3)p1=(0,10,0),p2,3=(∓9-a10,a+1,±9-a10).To study the stability of equilibrium points we apply the Lyapunov's first (indirect) method [12] so we have the following theorem.Theorem 2.1.
Letx=x* be an equilibrium point of a nonlinear system:
(2.4)ẋ=f(x),
where f:D→ℝn is continuously differentiable and D⊂ℝn is the neighborhood of the equilibrium point x*. Let λi denote the eigenvalues of the Jacobian matrix A=∂f/∂x|x* then the following are considered. (i)
IfReλi<0 for all i, then x=x* is asymptotically stable.(ii)
IfReλi>0 for one or more i, then x=x* is unstable.(iii)
IfReλi≤0 for all i and at least one Reλj=0, then x=x* may be either stable, asymptotically stable, or unstable.SinceA is only defined at x*, stability determined by the indirect method is restricted to infinitesimal neighborhoods of x*.To study the signs of the real parts of eigenvalues, we have the following famous criterion [13].Criterion (Routh-Hurwitz).
Given the polynomialP(λ)=λn+a1λn-1+⋯+an-1λ+an, where the coefficients ai,i=1,2,…,n, are real constants, define the n Hurwitz matrices
(2.5)H1=(a1),H2=(a11a3a2)⋮Hn=(a1100⋯0a3a2a11⋯0a5a4a3a2⋯0⋮⋮⋮⋮⋯⋮0000⋯an),
where ai=0 if i>n.
All of roots of the polynomial have negative real part if and only if the determinants of all Hurwitz matrices are positive:detHi>0,i=1,2,…,n.Routh-Hurwitz criteria forn=3 are a1>0,a3>0 and a1a2-a3>0.Stability ofp1
The Jacobian matrix of system (2.1) at the equilibrium point p1 is
(2.6)Jp1=(10-a010-1100-10-1),
its characteristic polynomial is
(2.7)P(λ)=λ3+(a-8910)λ2+(11a-9910)λ+(a-910).
By applying the Routh-Hurwitz criterion we find that the real parts of these eigenvalues are all negative if and only if(2.8)a-8910>0,a-9>0,(a-8910)(11a-9910)-(a-910)>0.
Then it follows that a>9, and thus p1 is locally asymptotically stable if and only if a>9.Stability ofp2,3
The Jacobian matrix of system (2.1) at the equilibrium points p2,3 is
(2.9)Jp2,3=(1±9-a101∓29-a10-0.10-10-1),
and its characteristic polynomial is
(2.10)p̃(λ)=λ3+110λ2+(-15a+1810)λ+(-15a+1810).
The real parts of these eigenvalues are all negative if and only if
(2.11)-15a+1810>0,110(-15a+1810)-(-15a+1810)>0.
Then it follows that
(2.12)a<9,a>9.
So p2,3 are unstable for every value of a.
In order to detect the chaos we calculate the largest Lyapunov exponentλmax using the scheme proposed by Wolf et al. [14]. The initial states are taken as x(0)=2, y(0)=3,z(0)=2, Figure 1(a) displays the evolution of λmax according to a and Figure 1(b) displays chaotic attractor for a=3. System (2.1) displays chaotic behavior in the windows 0<a<7 (λmax>0), periodic behavior in 7≤a≤9 (λmax≈0) and stationary behavior for a>9 (λmax<0).(a) Largest Lyapunov exponent according toa. (b) Chaotic attractor for a=3.
(a)(b)


## 2.1.1. Analysing the System

(i)
Ifa≥9, system (2.1) has one fixed point:
(2.2)p1=(0,10,0).(ii)
Ifa<9, system (2.1) has three fixed points:
(2.3)p1=(0,10,0),p2,3=(∓9-a10,a+1,±9-a10).To study the stability of equilibrium points we apply the Lyapunov's first (indirect) method [12] so we have the following theorem.Theorem 2.1.
Letx=x* be an equilibrium point of a nonlinear system:
(2.4)ẋ=f(x),
where f:D→ℝn is continuously differentiable and D⊂ℝn is the neighborhood of the equilibrium point x*. Let λi denote the eigenvalues of the Jacobian matrix A=∂f/∂x|x* then the following are considered. (i)
IfReλi<0 for all i, then x=x* is asymptotically stable.(ii)
IfReλi>0 for one or more i, then x=x* is unstable.(iii)
IfReλi≤0 for all i and at least one Reλj=0, then x=x* may be either stable, asymptotically stable, or unstable.SinceA is only defined at x*, stability determined by the indirect method is restricted to infinitesimal neighborhoods of x*.To study the signs of the real parts of eigenvalues, we have the following famous criterion [13].Criterion (Routh-Hurwitz).
Given the polynomialP(λ)=λn+a1λn-1+⋯+an-1λ+an, where the coefficients ai,i=1,2,…,n, are real constants, define the n Hurwitz matrices
(2.5)H1=(a1),H2=(a11a3a2)⋮Hn=(a1100⋯0a3a2a11⋯0a5a4a3a2⋯0⋮⋮⋮⋮⋯⋮0000⋯an),
where ai=0 if i>n.
All of roots of the polynomial have negative real part if and only if the determinants of all Hurwitz matrices are positive:detHi>0,i=1,2,…,n.Routh-Hurwitz criteria forn=3 are a1>0,a3>0 and a1a2-a3>0.Stability ofp1
The Jacobian matrix of system (2.1) at the equilibrium point p1 is
(2.6)Jp1=(10-a010-1100-10-1),
its characteristic polynomial is
(2.7)P(λ)=λ3+(a-8910)λ2+(11a-9910)λ+(a-910).
By applying the Routh-Hurwitz criterion we find that the real parts of these eigenvalues are all negative if and only if(2.8)a-8910>0,a-9>0,(a-8910)(11a-9910)-(a-910)>0.
Then it follows that a>9, and thus p1 is locally asymptotically stable if and only if a>9.Stability ofp2,3
The Jacobian matrix of system (2.1) at the equilibrium points p2,3 is
(2.9)Jp2,3=(1±9-a101∓29-a10-0.10-10-1),
and its characteristic polynomial is
(2.10)p̃(λ)=λ3+110λ2+(-15a+1810)λ+(-15a+1810).
The real parts of these eigenvalues are all negative if and only if
(2.11)-15a+1810>0,110(-15a+1810)-(-15a+1810)>0.
Then it follows that
(2.12)a<9,a>9.
So p2,3 are unstable for every value of a.
In order to detect the chaos we calculate the largest Lyapunov exponentλmax using the scheme proposed by Wolf et al. [14]. The initial states are taken as x(0)=2, y(0)=3,z(0)=2, Figure 1(a) displays the evolution of λmax according to a and Figure 1(b) displays chaotic attractor for a=3. System (2.1) displays chaotic behavior in the windows 0<a<7 (λmax>0), periodic behavior in 7≤a≤9 (λmax≈0) and stationary behavior for a>9 (λmax<0).(a) Largest Lyapunov exponent according toa. (b) Chaotic attractor for a=3.
(a)(b)


## 2.2. Fractional-Order Financial System

Chen has introduced in [11] the generalization of system (2.1) for fractional incommensurate-order model which takes the form(2.13)Dq1x=z+(y-a)x,Dq2y=1-by-x2,Dq3z=-x-cz.
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary order but there are several definitions of fractional derivatives.In this paper, we use the Caputo-type fractional derivative defined in [15] by:(2.14)Dqf(t)=1Γ(n-q)∫0t(t-τ)n-q-1f(n)(τ)dτ=jn-q(dndtnf(t)),
where n=[q] is the value of q rounded up to the nearest integer, Γ is the gamma function and jα is the Riemann-Liouville integral operator defined by(2.15)jαf(t)=1Γ(α)∫0t(t-τ)α-1f(τ)dτ.
For the numerical solutions of system (2.13) we use the Adams-Bashforth-Moulton predictor-corrector scheme [16].We assume thatq (q1=q2=q3=q) is the control parameter, and c=1,b=0.1,a=3. Fractional system (2.13) has the same fixed points p1,2,3 as integer system (2.1), but for the stability analysis we have this theorem introduced in [17, 18].Theorem 2.2.
The fractional linear autonomous system(2.16)DαX=AXX(0)=X0X∈Rn,0<α<2,A∈Rn×Rn,
is locally asymptotically stable if and only if
(2.17)mini|arg(λi)|>απ2,i=1,2,…,n.Proposition 2.3.
Letx=x* be an equilibrium point of a fractional nonlinear system
(2.18)Dαx=f(x),0<α<2.
If the eigenvalues of the Jacobian matrix A=∂f/∂x|x* satisfy
(2.19)mini|arg(λi)|>απ2,i=1,2,…,n,
then the system is locally asymptotically stable at the equilibrium point x*.Proof.
Letx=x*+δx. Substituting in (2.18), we find
(2.20)Dα(x*+δx)=f(x*+δx).
so
(2.21)Dα(δx)=f(x*)+Aδx+◯(∥δx∥2).
Sincef(x*)=0 (x* is the equilibrium point of system (2.18)) and lim∥δx∥→0(◯(∥δx∥2)/∥δx∥)=0, then(2.22)Dαδx≈Aδx.
Taking into account Theorem 2.2, we deduce that If the eigenvalues of the matrix A satisfy
(2.23)mini|arg(λi)|>απ2,i=1,2,…,n,
then x* is locally asymptotically stable.
This completes the proof.Stability ofp1
The Jacobian matrix of system (2.13) at the equilibrium point p1 is
(2.24)Jp1=(7010-1100-10-1),
and its characteristic polynomial is
(2.25)P(λ)=λ3-5910λ2-6610λ-610.
its eigenvalues are λ1≈-0.87298,λ2=-1/10,λ3≈6.8730, we note that λ3 is real positive then |arg(λ3)|=0<q(π/2), for all q∈]0,2[, so p1 is unstable for all q∈]0,2[.Stability ofp2,3
The Jacobian matrix of system (2.13) at the equilibrium point p2,3 is
(2.26)Jp2,3=(1±351∓235-1100-10-1),
its characteristic polynomial is
(2.27)p̃(λ)=λ3+110λ2+65λ+65,
and its eigenvalues are λ1≈0.31278+1.2474i,λ2≈0.31278-1.2474i,andλ3≈-0.72556, we have
(2.28)|arg(λ1,2)|≈1.3251,|arg(λ3)|=π,
so mini|arg(λi)|≈1.3251, then the critical value of q is
(2.29)qc=2mini|arg(λi)|π≈0.8436,(i)
Ifq<0.8436, then p2,3 are locally asymptotically stable.(ii)
Ifq>0.8436, then p2,3 are unstable.  In order to detect the chaos, we calculate the largest Lyapunov exponent λmax.
The initial states are taken asx(0)=2, y(0)=3, z(0)=2, Figure 2(a) shows the evolution of λmax according to q. System (2.13) exhibits chaotic behaviors for q≥0.86.(a) Largest Lyapunov exponent according toq. (b) Phase diagram for some values of q.
(a)(b)


## 3. Feedback Control

### 3.1. Integer Case

A general approach to control a nonlinear dynamical system via feedback control can be formulated as follows:(3.1)ẋ(t)=f(x,u,t),
where x(t) is the system state vector, and u(t) the control input vector. Given a reference signal x̃(t), the problem is to design a controller in the state feedback form: (3.2)u(t)=g(x,t),
where g is vector-valued function, so that the controlled system(3.3)ẋ(t)=f(x,g(x,t),t)
can be driven by the feedback control g(x,t) to achieve the goal of target tracking so we must have(3.4)limt→tf∥x(t)-x̃(t)∥=0.Proposition 3.1.
Let us consider the nonlinear system(3.5)ė=F(e,t).
where e=x-x̃,x̃(t) is a periodic orbit (or fixed point) of the given system (3.1) with u=0, and F(e,t)=f(x,g(x,t),t)-f(x̃,0,t).
If0 is a fixed point of system (3.5) and all eigenvalues of the jacobian matrix A=∂F/∂x|0 have negative real parts then the trajectory x(t) of system (3.3) converge to x̃(t)Proof.
Sincex̃(t) is a periodic orbit (or fixed point) of the given system (3.1) with u=0, so it satisfies
(3.6)x̃̇(t)=f(x̃,0,t),
a subtraction of (3.6) from (3.1) gives
(3.7)ẋ(t)-x̃̇(t)=f(x,g(x,t),t)-f(x̃,0,t),
so
(3.8)ė=F(e,t).
Since all eigenvalues of the jacobian matrix A have negative real parts, it follows from Theorem 2.1 that 0 is asymptotically stable, so we have limt→+∞∥e(t)∥=0 then limt→+∞∥x(t)-x̃(t)∥=0, finally x(t)→t→tfx̃(t).


### 3.2. Fractional Case

Let us consider the fractional system(3.9)Dαx(t)=f(x,u,t).
We proceed as in the integer case. the controlled system can be written as(3.10)Dαx(t)=f(x,g(x,t),t).
Let x̃(t) be a periodic orbit (or fixed point) of the given system (3.9) with u=0, then we obtain the system error(3.11)Dαe(t)=F(e,t)Proposition 3.2.
If0 is a fixed point of system (3.11) and the eigenvalues of the jacobian matrix A=∂F/∂x|0 satisfies the condition
(3.12)mini|arg(λi)|>απ2,i=1,2,…,n
then the trajectory x(t) of system (3.10) converge to x̃(t).Proof.
It follows directly from Proposition2.3.


### 3.3. Application to the Fractional Financial System

Let us consider the fractional financial system (2.13), we propose to stabilize unstable periodic orbit (or fixed point) (x̃,ỹ,z̃), the controlled system is as follows:(3.13)Dq1x=z+(y-a)x+u1(t),Dq2y=1-by-x2+u2(t),Dq3z=-x-cz+u3(t).
Since (x̃,ỹ,z̃) is solution of (2.13), then we have:(3.14)Dq1x̃=z̃+(ỹ-a)x̃,Dq2ỹ=1-bỹ-x̃2,Dq3z̃=-x̃-cz̃.
Subtracting (3.14) from (3.13) with notation, e1=x-x̃,e2=y-ỹ,e3=z-z̃, we obtain the system error:
(3.15)Dq1e1=e3-ae1+xy-x̃ỹ+u1(t),Dq2e2=-be2-e1(x+x̃)+u2(t),Dq3e3=-e1-ce3+u3(t).
We define the control functions as follow: (3.16)u1(t)=-(xy-x̃ỹ),u2(t)=e1(x+x̃),u3(t)=e1.
So the system error (3.15) becomes(3.17)Dq1e1=e3-ae1,Dq2e2=-be2,Dq3e3=-ce3.
The Jacobian matrix is  [-a010-b000-c] and its characteristic polynomial is: (3.18)p(x)=x3+(a+b+c)x2+(ab+c(a+b))x+abc
so we have the eigenvalues λ1=-a,λ2=-b,λ3=-c. Since all eigenvalues are real negatives one has arg(λi)=π, therefore |arg(λi)|>q(π/2), for all q satisfies 0<q<2, it follows from Proposition 3.2 that the trajectory x(t) of system (3.13) converges to x̃(t) and the control is completed.


### 3.4. Simulation Results

In this section we give numerical results which prove the performance of the proposed scheme. As mentioned in Section2.3 we have implemented the improved Adams-Bashforth-Moulton algorithm for numerical simulation.The initial states are taken asx(0)=2,y(0)=3,z(0)=2.


#### 3.4.1. Stabilizing the Unstable Fixed Points

The control can be started at any time according to our needs, so we choose to activate the control whent≥20, in order to make a comparison between the behavior before activation of control and after it.Forq=0.9 unstable point p1 has been stabilized, as shown in Figure 3(a), note that u1(t)=-(x(t)y(t)-0×10)=-x(t)y(t), so the control is activated when t≥20 and |x(t)y(t)|≤0.2 (more precisely t=22.5) in order to make the perturbation u1(t) smaller. firstly the evolution of x(t),y(t),z(t) is chaotic, then when the control is started at t=22.5 we see that p1 is rapidly stabilized.(a) Stabilizing the equilibrium pointp1 for q=0.9. (b) Evolution of the perturbation u(t).
(a)(b)In Figure3(b) we observe the evolution of the perturbation u(t), when the control is started we see that u2(t) and u3(t) are very small but u1(t) is a bit larger, after that the perturbation u(t) becomes close to zero rapidly.Forq=0.95, the unstable point p2 has been stabilized, as shown in Figure 4(a).(a) Stabilizing the equilibrium pointp2 for q=0.95. (b) Stabilizing the equilibrium point p3 for q=1.4.
(a)(b)Forq=1.4 the fixed point p3 was stabilized, Figure 4(b) shows the results of control.Whent is less than 20, there is a chaotic behavior, but when the control is activated at t=20, the two points p2 and p3 are rapidly stabilized.In the real world of finance if we want to have a good investment demand we can choose to stabilizep1, and in this case the interest rate and price index will be near zero. During the recent financial crisis in 2009 many banks decided to reduce interest rates to nearly zero in order to control this situation.


#### 3.4.2. Stabilizing Unstable Periodic Orbit

Although the unstable periodic orbits are dense in the chaotic attractor, we can choose one of them (which represent the performance of the system), by analyzing data experimental, after that we stabilize it. In this paper the close-return (CR) method [19] is used for the detection of UPO embedded in the attractor.Forq=0.97 we choose an unstable periodic orbit with period T=9, localized in the interval [78.2,87.2] as shown in Figure 5, then the control is started at t=87.2, when the trajectory x(t) begins to emerge from the unstable orbit, Figure 6 displays the results of control, if t is less then 78.2 there is chaotic behavior (the error e(t) is large), after the activation of control, this chaotic behavior is replaced by a periodic behavior and we note that the error e(t) becomes very close to zero.Figure 5
Selecting an unstable periodic orbit in the chaotic attractor of periodT=9 for q=0.97.Stabilizing unstable periodic orbit of periodT=9 for q=0.97.
(a)(b)(c)Forq=1.1 we choose an unstable periodic orbit with period T=16.05, localized in the interval [71.45,87.5] as shown in Figure 7, the control is started at t=20, Figure 8 displays the results of control. Although the control is executing at t=20, it does not give effect rapidly, and the orbit is stabilized at t=63, when the control is activated the error begins to diminish, and becomes close to zero after t=63.Figure 7
Selecting an unstable periodic orbit in the chaotic attractor of periodT=16.05 for q=1.1.Stabilization an unstable periodic orbit of periodT=16.05 for q=1.1.
(a)(b)(c)The stabilization of the periodic orbits is very important, because it permits, on the one hand to make some predictions, and secondly, it is more realistic than the stabilization of the stationary points in the financial circle, where one cannot generally fix the interest rate and the investment demand as well as the price index, for a long period.


## 3.1. Integer Case

A general approach to control a nonlinear dynamical system via feedback control can be formulated as follows:(3.1)ẋ(t)=f(x,u,t),
where x(t) is the system state vector, and u(t) the control input vector. Given a reference signal x̃(t), the problem is to design a controller in the state feedback form: (3.2)u(t)=g(x,t),
where g is vector-valued function, so that the controlled system(3.3)ẋ(t)=f(x,g(x,t),t)
can be driven by the feedback control g(x,t) to achieve the goal of target tracking so we must have(3.4)limt→tf∥x(t)-x̃(t)∥=0.Proposition 3.1.
Let us consider the nonlinear system(3.5)ė=F(e,t).
where e=x-x̃,x̃(t) is a periodic orbit (or fixed point) of the given system (3.1) with u=0, and F(e,t)=f(x,g(x,t),t)-f(x̃,0,t).
If0 is a fixed point of system (3.5) and all eigenvalues of the jacobian matrix A=∂F/∂x|0 have negative real parts then the trajectory x(t) of system (3.3) converge to x̃(t)Proof.
Sincex̃(t) is a periodic orbit (or fixed point) of the given system (3.1) with u=0, so it satisfies
(3.6)x̃̇(t)=f(x̃,0,t),
a subtraction of (3.6) from (3.1) gives
(3.7)ẋ(t)-x̃̇(t)=f(x,g(x,t),t)-f(x̃,0,t),
so
(3.8)ė=F(e,t).
Since all eigenvalues of the jacobian matrix A have negative real parts, it follows from Theorem 2.1 that 0 is asymptotically stable, so we have limt→+∞∥e(t)∥=0 then limt→+∞∥x(t)-x̃(t)∥=0, finally x(t)→t→tfx̃(t).


## 3.2. Fractional Case

Let us consider the fractional system(3.9)Dαx(t)=f(x,u,t).
We proceed as in the integer case. the controlled system can be written as(3.10)Dαx(t)=f(x,g(x,t),t).
Let x̃(t) be a periodic orbit (or fixed point) of the given system (3.9) with u=0, then we obtain the system error(3.11)Dαe(t)=F(e,t)Proposition 3.2.
If0 is a fixed point of system (3.11) and the eigenvalues of the jacobian matrix A=∂F/∂x|0 satisfies the condition
(3.12)mini|arg(λi)|>απ2,i=1,2,…,n
then the trajectory x(t) of system (3.10) converge to x̃(t).Proof.
It follows directly from Proposition2.3.


## 3.3. Application to the Fractional Financial System

Let us consider the fractional financial system (2.13), we propose to stabilize unstable periodic orbit (or fixed point) (x̃,ỹ,z̃), the controlled system is as follows:(3.13)Dq1x=z+(y-a)x+u1(t),Dq2y=1-by-x2+u2(t),Dq3z=-x-cz+u3(t).
Since (x̃,ỹ,z̃) is solution of (2.13), then we have:(3.14)Dq1x̃=z̃+(ỹ-a)x̃,Dq2ỹ=1-bỹ-x̃2,Dq3z̃=-x̃-cz̃.
Subtracting (3.14) from (3.13) with notation, e1=x-x̃,e2=y-ỹ,e3=z-z̃, we obtain the system error:
(3.15)Dq1e1=e3-ae1+xy-x̃ỹ+u1(t),Dq2e2=-be2-e1(x+x̃)+u2(t),Dq3e3=-e1-ce3+u3(t).
We define the control functions as follow: (3.16)u1(t)=-(xy-x̃ỹ),u2(t)=e1(x+x̃),u3(t)=e1.
So the system error (3.15) becomes(3.17)Dq1e1=e3-ae1,Dq2e2=-be2,Dq3e3=-ce3.
The Jacobian matrix is  [-a010-b000-c] and its characteristic polynomial is: (3.18)p(x)=x3+(a+b+c)x2+(ab+c(a+b))x+abc
so we have the eigenvalues λ1=-a,λ2=-b,λ3=-c. Since all eigenvalues are real negatives one has arg(λi)=π, therefore |arg(λi)|>q(π/2), for all q satisfies 0<q<2, it follows from Proposition 3.2 that the trajectory x(t) of system (3.13) converges to x̃(t) and the control is completed.


## 3.4. Simulation Results

In this section we give numerical results which prove the performance of the proposed scheme. As mentioned in Section2.3 we have implemented the improved Adams-Bashforth-Moulton algorithm for numerical simulation.The initial states are taken asx(0)=2,y(0)=3,z(0)=2.


### 3.4.1. Stabilizing the Unstable Fixed Points

The control can be started at any time according to our needs, so we choose to activate the control whent≥20, in order to make a comparison between the behavior before activation of control and after it.Forq=0.9 unstable point p1 has been stabilized, as shown in Figure 3(a), note that u1(t)=-(x(t)y(t)-0×10)=-x(t)y(t), so the control is activated when t≥20 and |x(t)y(t)|≤0.2 (more precisely t=22.5) in order to make the perturbation u1(t) smaller. firstly the evolution of x(t),y(t),z(t) is chaotic, then when the control is started at t=22.5 we see that p1 is rapidly stabilized.(a) Stabilizing the equilibrium pointp1 for q=0.9. (b) Evolution of the perturbation u(t).
(a)(b)In Figure3(b) we observe the evolution of the perturbation u(t), when the control is started we see that u2(t) and u3(t) are very small but u1(t) is a bit larger, after that the perturbation u(t) becomes close to zero rapidly.Forq=0.95, the unstable point p2 has been stabilized, as shown in Figure 4(a).(a) Stabilizing the equilibrium pointp2 for q=0.95. (b) Stabilizing the equilibrium point p3 for q=1.4.
(a)(b)Forq=1.4 the fixed point p3 was stabilized, Figure 4(b) shows the results of control.Whent is less than 20, there is a chaotic behavior, but when the control is activated at t=20, the two points p2 and p3 are rapidly stabilized.In the real world of finance if we want to have a good investment demand we can choose to stabilizep1, and in this case the interest rate and price index will be near zero. During the recent financial crisis in 2009 many banks decided to reduce interest rates to nearly zero in order to control this situation.


### 3.4.2. Stabilizing Unstable Periodic Orbit

Although the unstable periodic orbits are dense in the chaotic attractor, we can choose one of them (which represent the performance of the system), by analyzing data experimental, after that we stabilize it. In this paper the close-return (CR) method [19] is used for the detection of UPO embedded in the attractor.Forq=0.97 we choose an unstable periodic orbit with period T=9, localized in the interval [78.2,87.2] as shown in Figure 5, then the control is started at t=87.2, when the trajectory x(t) begins to emerge from the unstable orbit, Figure 6 displays the results of control, if t is less then 78.2 there is chaotic behavior (the error e(t) is large), after the activation of control, this chaotic behavior is replaced by a periodic behavior and we note that the error e(t) becomes very close to zero.Figure 5
Selecting an unstable periodic orbit in the chaotic attractor of periodT=9 for q=0.97.Stabilizing unstable periodic orbit of periodT=9 for q=0.97.
(a)(b)(c)Forq=1.1 we choose an unstable periodic orbit with period T=16.05, localized in the interval [71.45,87.5] as shown in Figure 7, the control is started at t=20, Figure 8 displays the results of control. Although the control is executing at t=20, it does not give effect rapidly, and the orbit is stabilized at t=63, when the control is activated the error begins to diminish, and becomes close to zero after t=63.Figure 7
Selecting an unstable periodic orbit in the chaotic attractor of periodT=16.05 for q=1.1.Stabilization an unstable periodic orbit of periodT=16.05 for q=1.1.
(a)(b)(c)The stabilization of the periodic orbits is very important, because it permits, on the one hand to make some predictions, and secondly, it is more realistic than the stabilization of the stationary points in the financial circle, where one cannot generally fix the interest rate and the investment demand as well as the price index, for a long period.


## 3.4.1. Stabilizing the Unstable Fixed Points

The control can be started at any time according to our needs, so we choose to activate the control whent≥20, in order to make a comparison between the behavior before activation of control and after it.Forq=0.9 unstable point p1 has been stabilized, as shown in Figure 3(a), note that u1(t)=-(x(t)y(t)-0×10)=-x(t)y(t), so the control is activated when t≥20 and |x(t)y(t)|≤0.2 (more precisely t=22.5) in order to make the perturbation u1(t) smaller. firstly the evolution of x(t),y(t),z(t) is chaotic, then when the control is started at t=22.5 we see that p1 is rapidly stabilized.(a) Stabilizing the equilibrium pointp1 for q=0.9. (b) Evolution of the perturbation u(t).
(a)(b)In Figure3(b) we observe the evolution of the perturbation u(t), when the control is started we see that u2(t) and u3(t) are very small but u1(t) is a bit larger, after that the perturbation u(t) becomes close to zero rapidly.Forq=0.95, the unstable point p2 has been stabilized, as shown in Figure 4(a).(a) Stabilizing the equilibrium pointp2 for q=0.95. (b) Stabilizing the equilibrium point p3 for q=1.4.
(a)(b)Forq=1.4 the fixed point p3 was stabilized, Figure 4(b) shows the results of control.Whent is less than 20, there is a chaotic behavior, but when the control is activated at t=20, the two points p2 and p3 are rapidly stabilized.In the real world of finance if we want to have a good investment demand we can choose to stabilizep1, and in this case the interest rate and price index will be near zero. During the recent financial crisis in 2009 many banks decided to reduce interest rates to nearly zero in order to control this situation.


## 3.4.2. Stabilizing Unstable Periodic Orbit

Although the unstable periodic orbits are dense in the chaotic attractor, we can choose one of them (which represent the performance of the system), by analyzing data experimental, after that we stabilize it. In this paper the close-return (CR) method [19] is used for the detection of UPO embedded in the attractor.Forq=0.97 we choose an unstable periodic orbit with period T=9, localized in the interval [78.2,87.2] as shown in Figure 5, then the control is started at t=87.2, when the trajectory x(t) begins to emerge from the unstable orbit, Figure 6 displays the results of control, if t is less then 78.2 there is chaotic behavior (the error e(t) is large), after the activation of control, this chaotic behavior is replaced by a periodic behavior and we note that the error e(t) becomes very close to zero.Figure 5
Selecting an unstable periodic orbit in the chaotic attractor of periodT=9 for q=0.97.Stabilizing unstable periodic orbit of periodT=9 for q=0.97.
(a)(b)(c)Forq=1.1 we choose an unstable periodic orbit with period T=16.05, localized in the interval [71.45,87.5] as shown in Figure 7, the control is started at t=20, Figure 8 displays the results of control. Although the control is executing at t=20, it does not give effect rapidly, and the orbit is stabilized at t=63, when the control is activated the error begins to diminish, and becomes close to zero after t=63.Figure 7
Selecting an unstable periodic orbit in the chaotic attractor of periodT=16.05 for q=1.1.Stabilization an unstable periodic orbit of periodT=16.05 for q=1.1.
(a)(b)(c)The stabilization of the periodic orbits is very important, because it permits, on the one hand to make some predictions, and secondly, it is more realistic than the stabilization of the stationary points in the financial circle, where one cannot generally fix the interest rate and the investment demand as well as the price index, for a long period.


## 4. Conclusions

Chaotic phenomenon makes prediction impossible in the financial world; then the deletion of this phenomenon from fractional financial system is very useful, the main contribution of this paper is to this end.Nonlinear feedback control scheme has been extended to control fractional financial system. The results are proved analytically by applying the Lyapunov linearization method and stability condition for fractional system. Numerically the unstable fixed pointsp1,2,3 have been successively stabilized for different values of q; moreover unstable periodic orbit has stabilized. This proves the performance of the proposed scheme.



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*Source: 270646-2010-07-13.xml*