Modified Function Projective Synchronization for a Partially Linear and Fractional-Order Financial Chaotic System with Uncertain Parameters

This paper investigates the modified function projective synchronization between fractional-order chaotic systems, which are partially linear financial systems with uncertain parameters. Based on the stability theory of fractional-order systems and the Lyapunov matrix equation, a controller is obtained for the synchronization between fractional-order financial chaotic systems. Using the controller, the error systems converged to zero as time tends to infinity, and the uncertain parameters were also estimated so that the phenomenon of parameter distortion was effectively avoided. Numerical simulations demonstrate the validity and feasibility of the proposed method.


Introduction
In recent years, study on chaos is one of the most interesting research topics in real and physical systems [1][2][3][4].In 1985, chaotic behavior was discovered in financial systems [5].In fact, chaotic phenomena often appear in financial systems [6], such as the supernormal shock of the international oil prices, stock market crash, and financial crisis.Owing to its randomness and unpredictability, these chaotic phenomena usually lead to financial crisis and instability in the financial system.Since financial risks come from uncertainties, it is important to introduce unknown parameters in financial systems [7,8].Therefore, chaotic behaviors and uncertainties in financial systems must be taken into account [9].It is necessary to investigate chaos control strategies for financial systems to address financial crisis and other related problems.Hence, it is important to achieve a synchronized and healthy development of financial markets.
Since Pecora and Carroll [10] proposed synchronization between two chaotic systems in 1990, chaotic synchronization has been extensively and intensively studied in various fields [11,12].Different types of chaotic synchronization have been reported, including generalized synchronization [13], lag synchronization [14], function projective synchronization [15], modified projective synchronization [16], and modified function projective synchronization (MFPS) [17,18].The modified function projective synchronization (MFPS) means that the drive and response systems could be synchronized up to a scaling function matrix, but not a constant matrix.Recent studies on complicated financial systems have shown satisfactory results using the nonlinear method [19].Chaos synchronization in fractional-order financial systems has also been studied in recent years [20][21][22].
The synchronization of financial systems indicates that the financial systems in two different areas are to maintain synchronized development by the appropriate control conditions.That is, the drive system and the response system can be interpreted as a virtual economic target and a controlled objective, respectively.Hence, the goal of synchronization is to control a specific objective to a virtual economic target using the proposed controller.The dynamical behaviors of a financial system are more complex.Therefore, financial systems are not always completely synchronized, and some complex synchronization methods [20][21][22], such as function projective synchronization and MFPS, should be considered.Meanwhile, because of the complexity of the financial environment, this study highlights its theoretical value and ensures the comprehensiveness of the theory.
Also, in practice, the parameter distortion phenomenon frequently appears in fractional-order chaotic systems.In other words, parameters may be uncertain or drift with time in the chaotic synchronization between the response and drive systems because of various kinds of interferences.Many projective synchronization methods of chaotic systems with unknown parameters have been proposed in recent years [23].Hence, achieving synchronization and identifying parameters are very important in financial chaotic systems.
In this study, we investigated fractional-order financial systems proposed by Chen [24].A new controller is proposed to achieve synchronization.The controller makes the drive and response systems remain asymptotically stable.Based on the stability theory of fractional-order systems and the Lyapunov matrix equation, the MFPS was realized for partially linear and fractional-order (PLFO) financial chaotic systems, and the unknown parameters were also estimated so that the phenomenon of parameter distortion was avoided.Numerical simulation results showed that the proposed method effectively eliminated chaos and stabilized two financial systems.
The rest of the paper is organized as follows.In Section 2, the definition of MFPS, the PLFO chaotic system, and a useful lemma are presented.In Section 3, the scheme of MFPS is presented for PLFO chaotic systems with uncertain parameters.Numerical simulations are presented in Section 4. Finally, a conclusion is drawn in Section 5.

Preliminaries
The definitions of PLFO chaotic system and coupled PLFO chaotic system were given in [25].We will give the definition of PLFO chaotic system with uncertain parameters and coupled PLFO chaotic system with uncertain parameters as follows.
Definition 1.The fractional-order chaotic systems with uncertain parameters are defined as follows: It is called an uncertain PLFO chaotic system, where the state vector is  = ( 1 ,  2 , . . .,   )  ∈   ,  ∈  is a variable, and  :  ×   →  is a differentiable function.Its order is subject to 0 <  ≤ 1.  ∈   represents the vector of uncertain parameters.The coefficient matrix (, ) ∈  × is dependent on the variable  and the uncertain parameter .
Definition 2. Consider two uncertain PLFO chaotic systems (1), which can be described as The system is coupled by the variable .The subscripts  and  in system (4) represent the drive and response systems, respectively. ∈   is a controller.θ is the estimated value of unknown parameters.The coefficient matrix M(, θ) ∈  × is dependent on  and θ.System (4) is the coupled PLFO chaotic system with uncertain parameters [26].
Definition 3. System ( 4) is said to be MFPS with respect to the scaling function matrix ℎ() if there exists a vector controller where ‖ ⋅ ‖ is the Euclidean norm.It implies that the MFPS between the drive and response system (4) can be achieved or the error dynamic systems are globally asymptotically stable.
. ., ℎ  () represent identical or different constants, the MFPS is simplified to modified projective synchronization (such as antisynchronization and complete synchronization).If the synchronization problem is transformed into a control problem.
Lemma 5 (see [27]).Presume that the fractional-order autonomous system is where   ∈ (0, 1] ( = 1, 2, . . ., ) are the orders of the fractional derivative and  = ( Then, the MFPS between the drive and response system (4) is transformed into the analysis of the asymptotical stability of zero solution of the error system (7).
Theorem 6 (see [28]).Given the autonomous system (7), there exist a real symmetric positive definite matrix  and a positive definite matrix  such that the MFPS between the drive and response system (4) can be achieved if () + (()) H  = −, where H is the conjugate transpose of a matrix and () + (()) H  = − is called a continuous Lyapunov matrix equation.
According to Lemma 5, the equilibrium point of system ( 7) is asymptotically stable.That is, which indicates that the MFPS between the drive and response system (4) can be achieved.The proof is completed.
Remark 7. Since () is a polynomial matrix, it is difficult to solve for the eigenvalues.However, based on Theorem 6, the equilibrium point in system ( 7) is asymptotically stable if and only if () + (()) H  = − is a continuous Lyapunov matrix equation.

Designing the Lyapunov Matrix Equation
Next, we introduce a scheme of MFPS for coupled PLFO chaotic systems with uncertain parameters (7).
Proof.Assume that () =   − ℎ() ⋅   ; we have We select  = −ℎ() M(, θ)  −(, )++d  (ℎ()  )/d  ; then it has Hence, we obtain a controller ( 16) and a synchronization error system (17).From ( 18) and ( 19), the error system and the estimate of unknown parameters can be simply written as where  = ( Hence, we can obtain  +  H  = −, where  is a real symmetric positive definite matrix and  is a positive definite matrix.We usually choose  = , where  is an identity matrix.Then, According to Theorem 6,  +  H  = − is a continuous Lyapunov matrix equation.Meanwhile, we can realize the MFPS of system (7) through controller (16).Therefore, lim →∞ ‖()‖ = 0, which indicates that the MFPS within system (17) can be achieved.The proof is completed.
Remark 9.In the proof of Theorem 8, it is easy to see that the scaling function matrix ℎ() has no direct effect on d  /d  and d    /d  .The results verify the validity and feasibility of the proposed control strategy.
Remark 11.This method has universal applicability if and only if the chaotic system is a PLFO chaotic system (1).

Numerical Simulations
Huang and Li [29] proposed a dynamic model of finance comprising three first-order differential equations.Then, following the dynamic model, Chen [24] proposed a fractionalorder financial system to describe the running of a financial system as follows: It has three nonlinear equations of fractional-order chaotic systems.The state variables  1 ,  2 ,  3 represent the interest rate, the investment demand, and the price index, respectively.Changes in  1 are mostly influenced by two factors: contradictions from the investment market, which indicate a surplus between investment and savings, and structural adjustment from prices of goods.The changing rate of  2 is proportional to the rate of investment but inversely proportional to the cost of investment and interest rates.Changes in  3 are controlled by a contradiction between supply and demand in commercial markets and are affected by inflation rates.Particularly, the negative interest rate  1 , negative investment demand  2 , and negative price index  3 mean that the banks' deposit rates are below the level of inflation, the investment market presents overinvestment, and the rate of deflation is increasing, respectively.The constants , ,  are representatives of the saving amount, the per-investment cost, and the elasticity of demand of the commercial markets.The fractional-order financial system (25) will exhibit chaotic behavior when   =  ≥ 0.85,  = 1, 2, 3, and  = 1,  = 1,  = −1,  = 3,  = 0.1, and  = 1.When  = 0.9, the three-dimensional phase diagram of system ( 25) is illustrated in Figure 1 and the largest Lyapunov exponent is  = 0.14.The results highlight the chaotic attractor of fractional-order financial system (25).
It is obvious that system (25) is a PLFO chaotic system.Therefore, the drive and response systems can be constructed as follows: where   ( = 1, 2) is the controller.In the coupled system (26), the drive system evolves independently, while the response system is governed by the drive system through the variable  2 .That is, the drive system is and the response system is System ( 26) becomes a coupled PLFO chaotic system with uncertain parameters.Based on Theorem 8 and   ( = 1, 2, 3) = , we have From ( 17) to (19), we can obtain the error systems as It shows that  11 =  2  1 ,  12 = − 1 ,  13 =  2 ,  24 =  1 , and  25 = − 2 .Therefore, the unknown parameters systems are given by where √; then the simulation results are shown in Figures 2 and 3. Obviously, in Figure 2, the errors converge to zero as time goes to infinity, which implies that the MFPS of system ( 26) is achieved with the scaling function factors.Figure 3 shows that the estimated values of unknown parameters ñ, m, ã, p, and c converge to  = 1,  = 1,  = 3,  = −1, and  = 1, respectively, as  → ∞.Therefore, the uncertain parameters , , , , and  are identified.Additionally, with the scaling function factors ℎ 1 () = 0.1 cos(2) and ℎ 2 () = 3 √, phase portrait of the drive and response systems is illustrated in Figure 4.

Conclusion
In this paper, the MFPS was investigated for the coupled uncertain fractional-order financial system.Based on the stability criterion of the fractional-order system and the continuous Lyapunov matrix equation, a synchronization controller was proposed.Then, using an uncertain financial system, we verified the validity of the proposed controller.The unknown parameters were also estimated so that the phenomenon of parameter distortion was effectively avoided.The results showed that the controller was feasible and effective.
In theory, the proposed approach can realize a synchronized and healthy development of financial markets.In the case of possessing homogeneous investment demand, we can keep profits with price index synchronous development so that interference factors can be avoided.