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.

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 [

Among those synchronization methods, in recent years, function projective synchronization (FPS) of special form is introduced by some researches [

As we all know, financial systems are concerned with our daily life [

Impulsive phenomena exist in many biological science and mechanics fields actually. Consequently, the impulsive control becomes an interesting and useful synchronization approach [

The rest of this paper is organized as follows. In Section

In this section, we will give some description of our system studied. The novel nonlinear hyperchaotic finance system [

Lyapunov exponents spectrum of system (

Phase portraits of hyperchaotic finance system (

3D view in the

3D view in the

3D view in the

3D view in the

Time series of

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.

In this section, we discuss the modified function projective synchronization of the uncertain dynamical system.

Consider a class of chaotic systems with unknown parameters:

Consider

The two systems described by system (

If

From (

For given synchronization scaling function

The parameter update laws are as follows:

In the control law,

Suppose the

Then error system (

Now, the goal is to find some conditions on the control gains. Because the impulsive interval

System (

Let us choose the following Lyapunov function:

In the case when

Let

Then, we can obtain the following comparison system:

When

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.

In this section, new financial hyperchaotic system (

To further illustrate the effectiveness of the controller, we select the new financial hyperchaotic system as the drive system:

System (

We choose the novel hyperchaotic system [

Then the above system can be described in the form of

According to Theorem

To verify the results obtained in Section

Time responses of MFPS errors with impulses.

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 Figure

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 Figure

Time responses of FPS errors without impulses.

Time responses of MFPS errors without impulses.

From Figures

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.

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.

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.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Nature Science Foundation of China (Grants 51276081 and 11171135), the Society Science Foundation from Ministry of Education of China (Grants 12YJAZH002 and 08JA790057), the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Advanced Talents’ Foundation of Jiangsu University (Grants 07JDG054 and 10JDG140), and the Students’ Research Foundation of Jiangsu University (no. Y13A126). In particular, thanks are due to the support of Jiangsu University.