Control and Synchronization of Fractional-Order Financial System Based on Linear Control

Abstract

Control and synchronization of the financial systems with fractional-order are discussed in this paper. Based on the stability theory of fractional-order differential equations, Routh-Hurwitz stability condition, and by using linear control, simpler controllers are designed to achieve control and synchronization of the fractional-order financial systems. The proposed controllers are linear and easy to implement, which have improved the existing results. Theoretical analysis and numerical simulations are shown to demonstrate the validity and feasibility of the proposed method.

1. Introduction

Chaos, as a very interesting nonlinear phenomenon, has been intensively investigated in many fields over the last four decades. Since the pioneering work [1] that chaotic dynamics could exist in an economical model, research on the dynamical behavior of economical model has become one of the most interesting and important topics which have received increasing attention. Many continuous chaotic models have been proposed to study complex economic dynamics, such as the forced Vander-Pol model [2], the IS-ML model [3], Behrens-Feichtinger model [4], and Cournot-Puu model [5]. Just as all the other chaotic systems in engineering, financial chaotic system has complex dynamical behaviors and possess some special features, such as excessive sensitivity to initial conditions, the complex patterns of phase portraits, positive Lyapunov exponents, and bounded and fractal properties of the motion in the phase space. These features are inherent properties of the system itself, rather than caused by external disturbances, which denote some economic behaviors in the fields of finance, stocks, and social economics. In addition, as a nonlinear system, there exist many attractors with different topology, such as fixed points, limit cycles, quasiperiodic attractors, and chaotic attractors in chaotic financial systems, which make the contradiction in economic operation process complex and changeful. Chaotic behavior in financial systems is undesirable due to threatening the safety of investment. Therefore, to study the chaotic behaviors in nonlinear economical systems plays a very important role in decision-making by policy makers. Once the instability of a periodic solution, bifurcation, multiperiodic bifurcation, or similar phenomena appear in the economic chaotic systems, decision makers should take some measures to intervene. Some researches suggest that controlling chaos should improve the performance of a chaotic economy and stabilizing periodic solutions of a chaotic market model may increase economic efficiency [6–10]. In recent years, some authors have paid much attention to chaos control in economic models and obtained some results. For example, Ahmed et al. and Agiza used OGY method to control chaos in economic systems [11, 12], respectively. Pyragas [7] and Holyst and Urbanowicz [9] proposed the delayed feedback control, which has been widely used for controlling chaos in economic models. Sun et al. adopted impulsive control to control a financial model [13]. Du et al. applied phase space compression to control chaos in economic systems [14].

On the other hand, it has been found that many systems in interdisciplinary fields can be described by fractional differential equations, for example, dielectric polarization, electrode-electrolyte polarization, electromagnetic waves, viscoelastic systems, quantitative finance, and diffusion wave [15–17]. Recently, fractional-order systems have become an active research area, particularly in control and synchronization of chaotic systems. Interest has been growing in fractional calculus not only from physicists and engineers but also from researchers in life science and economics [18–20]. In fact, financial variables possessing long memories make fractional models more appropriate for dynamic behaviors in finance. Research on fractional-order financial models has a wider range of applications. In 2008, Chen studied nonlinear dynamics and chaos in a fractional-order financial system [21]. As he pointed out, one of the major differences between fractional-order and integer-order models is that fractional-order models possess memory; that is, the fractional-order model depends on the history of the system. The magnitude of the financial variables such as foreign exchange rates, gross domestic product, interest rates, production, and stock market prices can have very long memory; the reason for describing financial systems using a fractional nonlinear model is that it simultaneously possesses memory and chaos. From then on, researchers set out to investigate the fractional financial models, for instance, in [22] a sliding mode controller was designed for a fractional-order chaotic financial system and in [23] control of a fractional-order chaotic financial system by nonlinear feedback control was discussed. As we know, linear feedback control is especially attractive and has been successfully applied to practical implementation, which was adopted in [24–26] to realize control and synchronization of integer-order chaotic systems. However, there exists substantial difference between fractional-order differential systems and integer-order differential ones. Most of the properties, conclusions, and methods to deal with integer-order systems cannot be simply extended to the case of fractional-order ones. Therefore, results about fractional-order chaotic systems are much less than those of integer-order systems. To the best of our knowledge, there are few results about control and synchronization of fractional-order systems via linear feedback control [27, 28].

In this paper, based on the stability theory of fractional-order differential equations, Routh-Hurwitz stability condition, we investigate control and synchronization of fractional-order financial systems proposed by Chen via linear control. Sufficient conditions are established and easy to verify. Compared with sliding mode control and nonlinear feedback control used for discussing fractional-order financial systems in [22, 23], the linear control is economic and easy to implement, through which control and synchronization of fractional-order financial systems will be obtained only by choosing suitable feedback gains. The main job of this paper lies in two aspects. One is to remove chaotic phenomenon from fractional financial system by controlling, which makes prediction impossible in the financial world. The other is to realize harmonious and sustainable development between drive financial systems and response ones by investigating synchronization. The obtained results have a certain value to the theoretical guidance and application.

The remainder of this paper is organized as follows. In Section 2, preliminary results are presented and fractional-order financial system is described. In Section 3, some sufficient criteria for control of the fractional-order financial system are given. In Section 4, we discussed synchronization of the fractional-order financial system via linear error feedback. In Section 5, numerical simulations are given to illustrate the effectiveness of the main results. Finally, conclusions are drawn in Section 6.

2. Preliminaries and System Description

Fractional calculus is a generalization of integration and differentiation to a noninteger-order integrodifferential operator 𝐷^𝛼_𝑡 defined by 𝐷^𝛼_𝑡=𝑑^𝛼𝑑𝑡^𝛼(𝛼)(𝛼),𝑅>0,1,𝑅=0,_𝑡𝑎(𝑑𝜏)^−𝛼,𝑅(𝛼)<0. (2.1)

Fractional calculus is being used in various fields gradually, such as biophysics, nonlinear dynamics, informatics, and control engineering [29]. There are some definitions for fractional derivatives. The commonly used definitions are Grunwald-Letnikov (GL), Riemann-Liouville (RL), and Caputo (C) definitions.

The Grunwald-Letnikov (GL) derivative with fractional-order 𝛼 is given by ^𝐺_𝛼𝐷^𝛼_𝑡𝑓(𝑡)=lim_ℎ→0𝑓_ℎ^(𝛼)(𝑡)=lim_ℎ→0ℎ^{−𝛼[(𝑡−𝛼)/ℎ]}_𝑖=0(−1)^𝑖𝛼_𝑖𝑓(𝑡−𝑖ℎ), (2.2) where [⋅] means the integer part.

The Riemann-Liouvill (RL) fractional derivatives are defined by ^𝑅_𝛼𝐷^𝛼_𝑡𝑓(𝑡)=𝑑^𝑛𝑑𝑡^𝑛1Γ(𝑛−𝛼)^𝑡_𝑎𝑓(𝜏)(𝑡−𝜏)^{(𝛼−𝑛+1)}𝑑𝜏,𝑛−1<𝛼<𝑛, (2.3) where Γ(⋅) is the gamma function, Γ(𝜏)=∫^∞₀𝑡^𝜏−1𝑒^−𝑡𝑑𝑡.

The Caputo (C) fractional derivative is defined as follows: ^𝐶_𝛼𝐷^𝛼_𝑡𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡_𝑎𝑓^(𝑛)(𝜏)(𝑡−𝜏)^{(𝑛−𝛼−1)}𝑑𝜏,𝑛−1<𝛼<𝑛. (2.4)

It should be noted that the advantage of Caputo approach is that the initial conditions for fractional differential equations with Caputo derivatives take on the same form as those for integer-order differential, which have well-understood physical meanings. Therefore, in the rest of this paper, the notation 𝐷^𝛼_∗ is chosen as the Caputo fractional derivative operator ^𝐶_𝛼𝐷.

The fractional-order chaotic financial system can be described by 𝐷^𝛼1_∗𝐷𝑥=𝑧+𝑦−𝑎𝑥,_𝛼2∗𝑦=1−𝑏𝑦−𝑥²,𝐷^𝛼3_∗𝑧=−𝑥−𝑐𝑧, (2.5) where 𝛼₁, 𝛼₂, and 𝛼₃ are the fractional-order, 𝛼₁,𝛼₂,𝛼₃∈(0,1). System (2.5) describes the time variation of three state variables: the interest rate 𝑥, the investment demand 𝑦, and the price index 𝑧; 𝑎 is the saving amount, 𝑏 is the cost per investment, 𝑐 is the elasticity of demand of commercial markets, parameters 𝑎, 𝑏, and 𝑐 are positive real constants. When 𝑎=3, 𝑏=0.1, 𝑐=1, fractional-order (𝛼₁,𝛼₂,𝛼₃) is taken as (0.86,0.92,0.95), the largest Lyapunov exponent of system (2.5) is greater than 0 [23]. Figure 1 displays chaotic attractors of system (2.5).

(a)

(b)

Figure 1 

Chaotic attractors of system (2.5) with order (0.86,0.92,0.95).

To obtain our results, the following lemma is presented.

Consider the following fractional-order system: 𝐷^𝛼1_∗,𝐷𝑥=𝑓𝑥,𝑦,𝑧_𝛼2∗,𝐷𝑦=𝑔𝑥,𝑦,𝑧_𝛼3∗𝑧=ℎ𝑥,𝑦,𝑧. (2.6)

Lemma 2.1 (see [30]). System (2.6) is asymptotically stable at the equilibrium points if |arg(𝜆_𝑖(𝐴))|>𝛼𝜋/2, 𝛼=max(𝛼₁,𝛼₂,𝛼₃), 𝑖=(1,2,3), for all eigenvalues 𝜆_𝑖 of the Jacobian matrix 𝐽: 𝐽=𝜕𝑓𝜕𝑥𝜕𝑓𝜕𝑦𝜕𝑓𝜕𝑧𝜕𝑔𝜕𝑥𝜕𝑔𝜕𝑦𝜕𝑔𝜕𝑧𝜕ℎ𝜕𝑥𝜕ℎ𝜕𝑦𝜕ℎ𝜕𝑧. (2.7)

3. Chaos Control

3.1. Analysis of Equilibrium Points

Let 𝑧+𝑦−𝑎𝑥=0,1−𝑏𝑦−𝑥²=0,−𝑥−𝑐𝑧=0. (3.1) It is easy to obtain that system (2.5) has three equilibrium points if 𝑃𝑐−𝑏−𝑎𝑏𝑐>0,₁=10,𝑏,𝑃,0₂=(𝑐−𝑏−𝑎𝑏𝑐)𝑐,(1+𝑎𝑐)𝑐,−1𝑐⋅(𝑐−𝑏−𝑎𝑏𝑐)𝑐,𝑃₃=−(𝑐−𝑏−𝑎𝑏𝑐)𝑐,(1+𝑎𝑐)𝑐,1𝑐⋅(𝑐−𝑏−𝑎𝑏𝑐)𝑐. (3.2) The Jacobian matrix of system (2.5) at equilibrium point (𝑥^∗,𝑦^∗,𝑧^∗) is 𝑦𝐽=^∗−𝑎𝑥^∗1−2𝑥^∗−𝑏0−10−𝑐. (3.3) The characteristic equation of the Jacobian matrix 𝐽 is given by 𝜆³+𝑎+𝑏+𝑐−𝑦^∗𝜆²+𝑎𝑏+𝑎𝑐+𝑏𝑐−𝑏𝑦^∗−𝑐𝑦^∗+2𝑥^∗2𝜆𝑥+1+𝑎𝑏𝑐−𝑏𝑐𝑦+𝑏+2𝑐^∗2=0. (3.4)

When 𝑎=3, 𝑏=0.1, 𝑐=1, 𝑃₁=(0,10,0). Substituting the parameters 𝑎, 𝑏, and 𝑐 into (3.4), one obtains 𝜆³−5.9𝜆²−6.6𝜆−0.6=0. (3.5) The roots of (3.5) are 𝜆₁=6.8730, 𝜆₂=−0.8730, and 𝜆₃=−0.1; obviously, 𝜆₁ is a positive real number, based on the lemma, arg(𝜆₁)=0, arg(𝜆₂)=arg(𝜆₃)=𝜋. So, the equilibrium point 𝑃₁ is unstable.

When 𝑎=3, 𝑏=0.1, 𝑐=1, 𝑃₂=(√0.6,4,−√0.6), 𝑃₃=(−√0.6,4,√0.6). Substituting the parameters 𝑎, 𝑏, and 𝑐 into (3.4), the characteristic equation of the Jacobian matrix 𝐽 at equilibrium points 𝑃₂,𝑃₃ is 𝜆³+0.1𝜆²+1.2𝜆+1.2=0. (3.6) The roots of (3.6) are 𝜆₁=−0.7256, 𝜆₂=0.3128+1.2474𝑖, and 𝜆₃=0.3128−1.2474𝑖. According to the lemma, 𝜆₁ is a negative real number, arg(𝜆₁)=𝜋, arg|(𝜆_2,3)|=1.3251. 𝛼=2|arg(𝜆_2,3)|/𝜋=0.8436. Therefore, we can conclude that if the maximum fractional-order 𝛼 among 𝛼₁, 𝛼₂, 𝛼₃ is less than 0.8436, the equilibrium points 𝑃₂, 𝑃₃ are stable. On the contrary, the equilibrium points 𝑃₂, 𝑃₃ are unstable.

According to the above analysis, when the maximum fractional-order 𝛼 among 𝛼₁,𝛼₂, and 𝛼₃ is less than 0.8436, there exist two stable equilibrium points; when 𝛼₁=𝛼₂=𝛼₃=0.8436, system (2.5) will admit a limit cycle; when fractional-order 𝛼₁, 𝛼₂, and 𝛼₃ are all greater than 0.8436, there are no stable equilibria, with all the equilibrium points being unstable, which implies that there may exist chaos for system (2.5). By calculating the values of Lyapunov exponents of system (2.5), it could be found that system (2.5) exhibits chaotic behaviors if maximum fractional-order 𝛼 among 𝛼₁, 𝛼₂, and𝛼₃ is greater than or equal to 0.86 [23].

3.2. Chaos Control

In this subsection, linear state feedback controller is designed to control fractional-order chaotic financial system to its equilibrium.

The controlled fractional-order chaotic financial system is given by 𝐷^𝛼1_∗𝑥=𝑧+𝑦−𝑎𝑥+𝑘₁𝑥−𝑥^∗,𝐷^𝛼2_∗𝑦=1−𝑏𝑦−𝑥²+𝑘₂𝑦−𝑦^∗,𝐷^𝛼3_∗𝑧=−𝑥−𝑐𝑧+𝑘₃𝑧−𝑧^∗, (3.7) where 𝑘₁, 𝑘₂, and 𝑘₃ denote feedback gains and (𝑥^∗,𝑦^∗,𝑧^∗) is the desired equilibrium point. Obviously, system (3.7) has one equilibrium point (𝑥^∗,𝑦^∗,𝑧^∗).

The Jacobian matrix of system (3.7) at equilibrium point (𝑥^∗,𝑦^∗,𝑧^∗) is 𝑦𝐽=^∗−𝑎+𝑘₁𝑥^∗1−2𝑥^∗𝑘₂−𝑏0−10𝑘₃−𝑐. (3.8) The characteristic equation of the Jacobian matrix (3.8) is 𝜆³+𝑎+𝑏+𝑐−𝑘₁−𝑘₂−𝑘₃−𝑦^∗𝜆²+𝑎𝑏+𝑎𝑐+𝑏𝑐+𝑘₁𝑘₂+𝑘₂𝑘₃+𝑘₁𝑘₃−(𝑏+𝑐)𝑘₁+𝑦−𝑎−𝑐𝑘₂−𝑏𝑐𝑦^∗+𝑦−𝑎−𝑏𝑘₃+1−𝑏𝑦^∗+2𝑥^∗2𝜆−𝑏𝑐𝑘₁+𝑐𝑘₁𝑘₂+𝑐𝑦^∗−𝑎𝑐−1𝑘₂−𝑏𝑦^∗+𝑎𝑏+2𝑥^∗2𝑘₃+𝑏𝑘₁𝑘₃+𝑎−𝑦𝑘₂𝑘₃−𝑘₁𝑘₂𝑘₃+𝑎𝑏𝑐+𝑏+𝑐−𝑦^∗+2𝑐𝑥^∗2=0. (3.9)

Our goal is to find suitable feedback gains such that all the state trajectories of system (3.7) are controlled to its equilibrium point, that is to say, roots of (3.9) should satisfy the conditions in the lemma.

Theorem 3.1. When 𝑎=3, 𝑏=0.1, 𝑐=1, 𝑃₁=(0,10,0); system (3.7) stabilizes to equilibrium point 𝑃₁, if state feedback gains 𝑘₁, 𝑘₂, and 𝑘₃ satisfy the following conditions: −10≤𝑘₁−𝑘₃<−6,𝑘₁+𝑘₃<−6,𝑘₂<0.1. (3.10)

Proof. Substituting the parameters 𝑎, 𝑏, and 𝑐 into (3.9), one obtains 10𝜆+1−10𝑘₂𝜆²−𝑘₁+𝑘₃+6𝜆+𝑘₁𝑘₃−𝑘₁+7𝑘₃−6=0. (3.11) It is very easy to obtain the roots of (3.11): 𝜆₁=𝑘₂−0.1,𝜆_2,3=12𝑘₁+𝑘₃+6±𝑘₁−𝑘₃+10𝑘₁−𝑘₃+6. (3.12) Note that 𝜆₁ is a negative real number, 𝜆_2,3 are a pair of conjugate imaginary roots, and the real parts of imaginary root are negative, that is, arg(𝜆₁)=𝜋, arg(𝜆_2,3)>𝜋/2. Therefore, the trajectory of the controlled fractional-order system (3.7) is asymptotically stable at equilibrium point 𝑃₁.

In Theorem 3.1, we designed three simple linear feedback controllers to ensure controlled system (3.7) stabilized to 𝑃₁. In practice, two linear feedback controllers or single linear feedback controller in controlled system (3.7) will do the same thing. Then we have the following corollaries.

Corollary 3.2. Controlled system (3.7) will approach asymptotically to 𝑃₁ with one of the following conditions about feedback gains: −10≤𝑘₁<−6,𝑘₂<0.1,𝑘₃𝑘=0,₁+𝑘₃<−6,−10≤𝑘₁−𝑘₃<−6,𝑘₂=0. (3.13)

Corollary 3.3. Controlled system (3.7) will approach asymptotically to 𝑃₁ if feedback gains satisfy −10≤𝑘₁<−6,𝑘₂=0,𝑘₃=0. (3.14)

Theorem 3.4. When 𝑎=3, 𝑏=0.1, 𝑐=1, 𝑃₂=(√0.6,4,−√0.6); if feedback gains 𝑘₁, 𝑘₂, and 𝑘₃ satisfy one of the following conditions, system (3.7) will approach and stabilizes to equilibrium point 𝑃₂ asymptotically: 𝑘₁=0,𝑘₂<−0.9,𝑘₃𝑘=0,₁<121−√62161220,𝑘₂=0,𝑘₃=0. (3.15)

Proof. (1) Substitute the parameters 𝑎=3, 𝑏=0.1, 𝑐=1, and 𝑘₁=𝑘₃=0 into (3.9), one obtains 𝜆³+110−𝑘₂𝜆²+656𝜆+5=0. (3.16) According to the Routh-Hurwitz criterion, real parts of these eigenvalues 𝜆_1,2,3 of (3.16) are all negative if 110−𝑘₂1>0,10−𝑘₂65>65, (3.17) that is, 𝑘₂<−0.9. That implies that the trajectory of the controlled fractional-order system (3.7) is asymptotically stable at equilibrium point 𝑃₂.
(2) Substituting the parameters 𝑎=3, 𝑏=0.1, 𝑐=1, and 𝑘₂=𝑘₃=0 into (3.9), one obtains 𝜆³+110−𝑘₁𝜆²+65−11𝑘10¹6𝜆+5−1𝑘10¹=0. (3.18) By applying the Routh-Hurwitz criterion, if the following conditions for feedback gains are met, 110−𝑘₁6>0,5−11𝑘10¹1>0,10−𝑘₁65−11𝑘10¹>65−1𝑘10¹, (3.19) then real parts of these eigenvalues 𝜆_1,2,3 are all negative. It follows that 𝑘₁<(1/220)(121−√62161). Thus, the trajectory of the controlled fractional-order system (3.7) is asymptotically stable at equilibrium point 𝑃₂.

Remark 3.5. Actually, we adopt single linear feedback controller to stabilize 𝑃₂ of controlled system in Theorem 3.4, namely, stabilizing 𝑃₂ by adding single linear feedback controller on the first state or the second state, but we cannot do it via adding single linear feedback controller on the third state; the reasons are described as follows.
Substituting the parameters 𝑎=3, 𝑏=0.1, 𝑐=1, and 𝑘₁=𝑘₂=0 into (3.9), one obtains 𝜆³+110−𝑘₃𝜆²+65+9𝑘10³𝜆−11𝑘10³+65=0. (3.20) By using the Routh-Hurwitz criterion, real parts of these eigenvalues 𝜆_1,2,3 are all negative if 110−𝑘₃>0,−11𝑘10³+651>0,10−𝑘₃65+9𝑘10³>−11𝑘10³+65. (3.21) By using simple calculation, such 𝑘₃ is obviously absent. So the trajectory of the controlled fractional-order system (3.7) is not asymptotically stable at equilibrium point 𝑃₂.

Theorem 3.6. When 𝑎=3, 𝑏=0.1, 𝑐=1, 𝑃₂=(−√0.6,4,√0.6); if feedback gains 𝑘₁, 𝑘₂, and 𝑘₃ satisfy one of the following conditions, system (3.7) will approach and stabilize to equilibrium point 𝑃₂ asymptotically: 𝑘₁=0,𝑘₂<−0.9,𝑘₃𝑘=0,₁<121−√62161220,𝑘₂=0,𝑘₃=0. (3.22)

Proof. The proof is the same as that of Theorem 3.4 and so we omit it here.

4. Chaos Synchronization

In this section, we will investigate synchronization of fractional-order financial system (2.5). Three and two simple linear feedback controllers are designed to achieve synchronization, which simplify the existing synchronization schemes and reduce the synchronization cost. Drive system and response system are described as follows, respectively: 𝐷^𝛼1_∗𝑥_𝑚=𝑧_𝑚+𝑦_𝑚−𝑎𝑥_𝑚,𝐷^𝛼2_∗𝑦_𝑚=1−𝑏𝑦_𝑚−𝑥²_𝑚,𝐷^𝛼3_∗𝑧_𝑚=−𝑥_𝑚−𝑐𝑧_𝑚,𝐷^𝛼1_∗𝑥_𝑠=𝑧_𝑠+𝑦_𝑠−𝑎𝑥_𝑠+𝑢₁,𝐷^𝛼2_∗𝑦_𝑠=1−𝑏𝑦_𝑠−𝑥²_𝑠+𝑢₂,𝐷^𝛼3_∗𝑧_𝑠=−𝑥_𝑠−𝑐𝑧_𝑠+𝑢₃, (4.1)(4.2) where 𝑢₁, 𝑢₂, and 𝑢₃ denote the external control inputs, to be designed later. It follows from systems (4.1) and (4.2) that the following error dynamical system is 𝐷^𝑞1_∗𝑒₁=𝑦_𝑚−𝑎𝑒₁+𝑥_𝑠𝑒₂+𝑒₃−𝑢₁,𝐷^𝑞2_∗𝑒₂=𝑥_𝑚+𝑥_𝑠𝑒₁−𝑏𝑒₂−𝑢₂,𝐷^𝑞3_∗𝑒₃=−𝑒₁−𝑐𝑒₃−𝑢₃, (4.3) where 𝑒₁=𝑥_𝑚−𝑥_𝑠, 𝑒₂=𝑦_𝑚−𝑦_𝑠, and 𝑒₃=𝑧_𝑚−𝑧_𝑠. Our aim is to find suitable control laws 𝑢_𝑖(𝑖=1,2,3) for stabilizing the error dynamics system (4.3). To this end, the following theorem is proposed.

Theorem 4.1. For any given initial conditions, synchronization between systems (4.1) and (4.2) will occur if control schemes are defined as follows: 𝑢₁=𝑘₁𝑒₁,𝑢₂=𝑘₂𝑒₂,𝑢₃=𝑘₃𝑒₃, (4.4) where 𝑘₁, 𝑘₂, and 𝑘₃ are feedback gains and satisfy the following conditions: 𝑘₁𝑘>𝐿−𝑎,₂<−𝑏−9𝐿²4𝐿−𝑎−𝑘₁,𝑘₃<−𝑐. (4.5)

Proof. Combining (4.3) with (4.4), the error system (4.3) is given by 𝐷^𝑞1_∗𝑒₁=𝑦_𝑚−𝑎−𝑘₁𝑒₁+𝑥_𝑠𝑒₂+𝑒₃,𝐷^𝑞2_∗𝑒₂=𝑥_𝑚+𝑥_𝑠𝑒₁−𝑏+𝑘₂𝑒₂,𝐷^𝑞3_∗𝑒₃=−𝑒₁−𝑐+𝑘₃𝑒₃. (4.6) Error dynamical system (4.6) can be rewritten as the following matrix form: 𝐷^𝑞1_∗𝑒₁𝐷^𝑞2_∗𝑒₂𝐷^𝑞3_∗𝑒₃𝑒=𝐴𝑒¹₂𝑒₃, (4.7) where 𝑦𝐴=^𝑚−𝑎−𝑘₁𝑥_𝑠1𝑥_𝑚+𝑥_𝑠−𝑏+𝑘₂0−10−𝑐+𝑘₃. (4.8)
Suppose that 𝜆 is one of the eigenvalues of matrix 𝐴 and the corresponding noneigenvector is 𝜀=(𝜀₁,𝜀₂,𝜀₃)^𝑇, that is, 𝐴𝜀=𝜆𝜀. (4.9) Taking conjugate transposal on both sides of (4.9), one obtains (𝐴𝜀)^𝐻=𝜆𝜀^𝐻. (4.10) Equation (4.9) multiplied left by 1/2𝜀^𝐻 plus (4.10) multiplied right by 1/2𝜀, we have 𝜀^𝐻121𝐴+2𝐴^𝐻1𝜀=2𝜆+𝜆𝜀^𝐻𝜀. (4.11) Because a chaotic system has bounded trajectories, there exists a positive constant 𝐿, such that |𝑥|<𝐿, |𝑦|<𝐿. Thus, 𝜀^𝐻12𝐴+𝐴^𝐻𝜀=𝜀^𝐻₁,𝜀^𝐻₂,𝜀^𝐻₃𝑦_𝑚−𝑎−𝑘₁𝑥_𝑚+2𝑥_𝑠20𝑥_𝑚+2𝑥_𝑠2−𝑏+𝑘₂000−𝑐+𝑘₃𝜀₁𝜀₂𝜀₃≤𝜀^𝐻₁,𝜀^𝐻₂,𝜀^𝐻₃𝐿−𝑎−𝑘₁323𝐿02𝐿−𝑏+𝑘₂000−𝑐+𝑘₃𝜀₁𝜀₂𝜀₃. (4.12)
From (4.11), we have 12𝜆+𝜆=𝜀^𝐻(1/2)𝐴+(1/2)𝐴^𝐻𝜀𝜀^𝐻𝜀≤1𝜀^𝐻𝜀𝜀^𝐻₁,𝜀^𝐻₂,𝜀^𝐻₃𝑃𝜀₁,𝜀₂,𝜀₃^𝑇, (4.13) where 𝑃=𝐿−𝑎−𝑘₁323𝐿02𝐿−𝑏+𝑘₂000−𝑐+𝑘₃. (4.14)
It is obvious that real parts of all eigenvalues 𝜆 are negative and matrix 𝑃 should be negative definite, namely, the following inequalities hold: 𝐿−𝑎−𝑘₁−<0,𝑏+𝑘₂𝐿−𝑎−𝑘₁−94𝐿²<0,𝑏+𝑘₂𝑐+𝑘₃𝐿−𝑎−𝑘₁+94𝐿²𝑐+𝑘₃<0. (4.15) Simplifing the above inequalities, one has 𝑘₁𝑘>𝐿−𝑎,₂<−𝑏−9𝐿²4𝐿−𝑎−𝑘₁,𝑘₃<−𝑐. (4.16) Therefore, based on stability theorem of fractional-order systems, error system (4.6) is asymptotically stable at the origin, which implies that synchronization between systems (4.1) and (4.2) will be achieved.

Based on the above analysis, it is easy to obtain that two linear feedback controllers could also achieve synchronization between systems (4.1) and (4.2). Then, we have the following corollary.

Corollary 4.2. For any given initial condition, if control schemes are described as 𝑢₁=𝑘₁𝑒₁, 𝑢₃=𝑘₃𝑒₃ and feedback gains satisfy 𝐿−𝑎<𝑘₁9<𝐿−𝑎+𝐿4𝑏²,𝑘₃<−𝑐, (4.17) then the response system (4.2) can synchronize the drive system (4.1).

Proof. The proof is similar to that of Theorem 4.1. After some computations, we have 12𝜆+𝜆=𝜀^𝐻(1/2)𝐴+1/2𝐴^𝐻𝜀𝜀^𝐻𝜀≤1𝜀^𝐻𝜀𝜀^𝐻₁,𝜀^𝐻₂,𝜀^𝐻₃𝑃𝜀₁,𝜀₂,𝜀₃^𝑇, (4.18) where 𝑃=𝐿−𝑎−𝑘₁323𝐿02𝐿−𝑏000−𝑐+𝑘₃. (4.19) Matrix 𝑃 must be negative definite; if the following inequalities hold: 𝐿−𝑎−𝑘₁<0,−𝑏𝐿−𝑎−𝑘₁−94𝐿²𝑏<0,𝑐+𝑘₃𝐿−𝑎−𝑘₁+94𝐿²𝑐+𝑘₃<0, (4.20) then one obtains 𝐿−𝑎<𝑘₁9<𝐿−𝑎+𝐿4𝑏²,𝑘₃<−𝑐. (4.21) Therefore, real parts of all eigenvalues 𝜆 are negative; according to the stability theorem of fractional-order systems, error system (4.6) is asymptotically stable. This means that the slave system (4.2) can asymptotically synchronize the master system (4.1).

Remark 4.3. References [22, 23] presented control of the model by using sliding mode control and nonlinear control, respectively, but with no consideration of synchronization. In contrast, control cost of linear control is low.

Remark 4.4. Response time of control and synchronization could be adjusted with suitable state feedback gains and error feedback gains in allowed limits.

5. Numerical Simulations

In this section, to verify theoretical results obtained in the previous section, the corresponding numerical simulations will be performed and an improved predictor-corrector algorithm is applied (see the appendix). In all simulations, fractional-order (𝛼₁, 𝛼₂, and 𝛼₃) is chosen as (0.86, 0.92, and 0.95) to ensure the existence of chaos in system (2.5).

Chaos Control
The parameters of system (2.5) are selected as 𝑎=3, 𝑏=0.1, and 𝑐=1. The initial values of controlled system (3.7) are chosen as (𝑥(0),𝑦(0),𝑧(0))=(3,1,4), respectively. Based on Theorem 3.1, we chose state feedback gain (𝑘₁,𝑘₂,𝑘₃) as (−9,−5,−2), under these conditions, roots of (3.11) are 𝜆₁=−5.1, 𝜆₁=−2.5+0.866𝑖, and 𝜆₁=−2.5−0.866𝑖. Controlled system (3.7) is asymptotically stable at 𝑃₁. Figure 2 shows that the controlled system (3.7) can be stabilized to 𝑃₁. When state feedback gain (𝑘₁,𝑘₂,𝑘₃)=(0,−3,0) and (−2,0,0), roots (𝜆₁,𝜆₂,𝜆₃) of (3.16) and (3.18) are (−2.8256,−0.1372−0.6371𝑖,−0.1372+0.6371𝑖) and (−2.8256,−0.1372+0.6371𝑖,−0.1372−0.6371𝑖), respectively. According to Theorem 3.4, controlled system (3.7) will stabilize at 𝑃₂. Figures 3 and 4 display the simulation results, respectively. When the parameters of the controlled system (3.7) are selected as above, based on Theorem 3.6, 𝑃₃ is stabilized. Figures 5 and 6 display the stabilization of the equilibrium point 𝑃₃ for state feedback gain (0, −8, 0) and (−2, 0, 0), respectively.

Figure 2 

The time response of the states for stabilizing system (3.7) to 𝑃1(𝑘1=−9,𝑘2=−5,𝑘3=−2).

Figure 3 

The time response of the states for stabilizing system (3.7) to 𝑃2(𝑘1=0,𝑘2=−3,𝑘3=0).

Figure 4 

The time response of the states for stabilizing system (3.7) to 𝑃2(𝑘1=−2,𝑘2=0,𝑘3=0).

Figure 5 

The time response of the states for stabilizing system (3.7) to 𝑃3(𝑘1=0,𝑘2=−8,𝑘3=0).

Figure 6 

The time response of the states for stabilizing system (3.7) to 𝑃3(𝑘1=−2,𝑘2=0,𝑘3=0).

Chaos Synchronization
When the parameters of system (2.5) are chosen as 𝑎=3, 𝑏=0.1, and 𝑐=1, select the initial values of the drive and the response systems as (𝑥_𝑚(0),𝑦_𝑚(0),𝑧_𝑚(0))=(4,−3,2) and (𝑥_𝑠(0),𝑦_𝑠(0),𝑧_𝑠(0))=(−3,2,4), respectively. For error feedback gain (𝑘₁,𝑘₂,𝑘₃) = (12,9,−2), simulation result of the synchronization between systems (4.1) and (4.2) is shown in Figure 7. The synchronization error states between systems (4.1) and (4.2) and displayed in Figure 8. When error feedback gains 𝑘₁=8, 𝑘₂=0, and 𝑘₃=−2, it can be seen that the derive system (4.1) and the response system (4.2) achieve the synchronization in Figure 9. Figure 10 displays the error state time response between systems (4.1) and (4.2).

Figure 7 

Synchronization states of systems (4.1) and (4.2) (𝑘1=12,𝑘2=9,𝑘3=−2).

Figure 8 

Synchronization error states of systems (4.1) and (4.2) (𝑘1=12,𝑘2=9,𝑘3=−2).

Figure 9 

Synchronization states of systems (4.1) and (4.2) (𝑘1=8,𝑘2=0,𝑘3=−2).

Figure 10 

Synchronization error states of systems (4.1) and (4.2) (𝑘1=8,𝑘2=0,𝑘3=−2).

6. Conclusion

In this paper, based on the stability theory of fractional-order systems and Routh-Hurwitz stability condition, some sufficient conditions for control and synchronization of the fractional-order chaotic financial system by linear feedback control have been derived. Finally, numerical simulations are provided to verify the effectiveness of the results obtained. The results are economical, reliable and efficient. It is noted that the method applied in the paper can also be extended to other fractional-order chaotic systems.

Appendix

An improved predictor-corrector algorithm [31] for fractional-order differential equations is presented in brief. In comparison with the classical one-step Adams-Bashforth-Moulton algorithm, the numerical approximation of the improved algorithm is more accurate and the computational cost is lower.

The following differential equation: 𝑑^𝛼𝑥𝑑𝑡^𝛼()𝑥=𝑓𝑡,𝑥,0≤𝑡≤𝑇,_(𝑘)=𝑥₀^(𝑘),𝑘=0,1,2,…,[𝛼]−1, (A.1) is equivalent to the Volterra integral equation [32] 𝑥=^[𝛼]−1_𝑘=0𝑥₀^(𝑘)𝑡^𝑘+1𝑘!Γ(𝛼)^𝑡₀𝑓(𝜏,𝑥)(𝑡−𝜏)^1−𝛼𝑑𝜏. (A.2) Set ℎ=𝑇/𝑁, 𝑡_𝑛=𝑛ℎ(𝑛=0,1,2,…,𝑁). Then the above equation can be discretized as follows: 𝑥_ℎ𝑡_𝑛+1=^[𝛼]−1_𝑘=0𝑥₀^(𝑘)𝑡^𝑘+ℎ𝑘!Γ(^𝛼)𝑓𝑡𝛼+2^𝑛+1,𝑥^𝜌_ℎ𝑡_𝑛+1+ℎ^𝛼Γ(𝛼+2)^𝑛𝑗=0𝑎_𝑗,𝑛+1𝑓𝑡_𝑗,𝑥_ℎ𝑡_𝑗, (A.3) where 𝑎_𝑗,𝑛+1=𝑛^𝛼+1−(𝑛−𝛼)(𝑛+1)^𝛼+1,𝑗=0,𝑛−𝑗+2^𝛼+1+𝑛−𝑗^𝛼+1−2𝑛−𝑗+1^𝑎+1𝑥,1≤𝑗≤𝑛,1,𝑗=𝑛+1,_𝜌ℎ𝑡_𝑛+1=^[𝛼]−1_𝑘=0𝑥₀^(𝑘)𝑡^𝑘_𝑛+1+1𝑘!Γ(𝛼)^𝑛𝑗=0𝑏_𝑗,𝑛+1𝑓𝑡_𝑗,𝑥_ℎ𝑡_𝑗, (A.4) in which, 𝑏_𝑗,𝑛+1=ℎ^𝛼/𝛼((𝑛−𝑗+1)^𝛼−(𝑛−𝑗)^𝛼).

The error estimate 𝑒 is Max|𝑥(𝑡_𝑗)−𝑥_ℎ(𝑡_𝑗)|=𝑂(ℎ^𝜌)(𝑗=0,1,…,𝑁), where 𝜌=Min(2,1+𝛼).

Acknowledgments

The authors would thank the referee and the editor for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (no. 60974090), the Fundamental Research Funds for the Central Universities (no. CDJXS11172237), the Specialized Research Fund for the Doctoral Program of Higher Education of China (no. 20093401120001; no. 102063720090013), the Natural Science Foundation of Anhui Province (no. 11040606M12), and the Natural Science Foundation of Anhui Education Bureau (no. KJ2010A035).