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

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.


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][2][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 Section 2, 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.
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.

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: where  ∈   denotes the state vector of the system, Φ() :   →  × , and Θ ∈   represent the vector of unknown parameters.Take system (2) as the drive system, and introduce the response system with a controller  ∈   as the following equation: where  ∈   is the state vector, ( +  ) = lim  →  +  (), ( −  ) = lim  →  −  (), Ψ() :   →  × , Ω ∈   composes the unknown parameter vector of response system,  = [ 1 ,  2 , . . .,   ] Τ ∈   is the control input vector, and   denotes the impulsive control gains.The impulsive instant sequence 3) represents the external disturbance; that is, there exists positive bounded function   making the following inequalities hold: Then the synchronization error dynamic system can be obtained as follows: where  From (3), we know that the error system between (2) and (3) is where  is the identical matrix with the same dimension as   .
For given synchronization scaling function ℎ  () and scaling matrix Λ, the adaptive impulsive controller is as the following: where Θ, Ω are the estimations of the parameters of the two systems, sgn(⋅) denotes the sign function, and  is a negative constant, which would influence the rate of the convergence.
The larger , the faster rate.So we can adjust the value of  according to the desirable convergent rate.
The parameter update laws are as follows: 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.

Theorem 7. System (2) and system (3) can realize MFPS under adaptive impulse controller
where  max (⋅) is the maximum eigenvalue.
Proof.Let us choose the following Lyapunov function: With the controllers and the parameter update laws, here noting  = [ 1 ,  2 , . . .,   ] Τ , the time derivation of the Lyapunov function along the trajectory of error system is given in two cases.In the case when  ̸ =   , we have In the case when  =   , we have where According to the definition and assumption of  and , it is guaranteed that  1 −  2 ≤ 0; therefore it can be obtained that where 2 = ; then V() ≤ −().
Then, we can obtain the following comparison system: When  = 0,   =  −1 < 1, and (− + / −1 )(  − −1 ) < − ln  −1 ,  ∈  + , 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.

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: where  1 ,  2 ,  3 , and  4 are controllers to be designed and , , , , and  are unknown parameters.System (18) can be presented in the form of We choose the novel hyperchaotic system [22] as the response system: supposing all the parameters in the two systems are unknown.Then the above system can be described in the form of According to Theorem 7, the controllers can be obtained as And the parameters estimation update laws are given as Through the above results, error system can be asymptotically stable.
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 Figure 6.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.
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 Figure 7.The time responses of MFPS errors without impulses are shown in Figure 8.
From Figures 7 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.

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.

Figure 7 :
Figure 7: Time responses of FPS errors without impulses.

Figure 8 :
Figure 8: Time responses of MFPS errors without impulses.
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 ℎ  () ̸ = 0 and a vector controller  such that lim