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The impulsive synchronization and adaptive-impulsive synchronization of a novel financial hyperchaotic system are investigated. Based on comparing principle for impulsive functional differential equations, several sufficient conditions for impulsive synchronization are derived, and the upper bounds of impulsive interval for stable synchronization are estimated. Furthermore, a nonlinear adaptive-impulsive control scheme is designed to synchronize the financial system using invariant principle of impulsive dynamical systems. Moreover, corresponding numerical simulations are presented to illustrate the effectiveness and feasibility of the proposed methods.

Since the seminal work of Pecora and Carroll [

The main idea of impulsive synchronization is that the response system received a sequence of synchronizing impulse signals from the drive system only at some discrete time instants. The synchronization velocity is rapid, and it has very strong advantage in practice due to reduced control cost. So, impulsive synchronization has received a great deal of interest from various fields. Yang and Chua [

Recently, some researchers synchronize the chaotic system through combining adaptive control and impulsive control, and they name it adaptive-impulsive control [

Since chaos phenomenon in financial field is founded in 1985, it has huge impacts on Chinese and western economics. There is chaos in economic and financial systems; this means that the system itself has intrinsic instability, and generally it is harmful to systems. So, control and synchronization of the financial chaotic or hyperchaotic system have more significance. Recently, Cai et al. [

Motivated by the aforementioned comments, in the paper, we will discuss impulsive synchronization and adaptive-impulsive synchronization of the novel financial system. Firstly, based on comparing principle, several sufficient conditions for impulsive synchronization are presented, and the upper bounds of impulsive interval for stable synchronization are defined. Furthermore, we will design a nonlinear adaptive-impulsive control scheme to synchronize the financial system using invariant principle of impulsive dynamical systems. Besides, corresponding numerical simulation results are illustrated to verify the effectiveness and feasibility of the theoretical results.

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

In general, the impulsive differential system is described by

The function

For

Let

Consider

A function

Assume that the following three conditions are satisfied.

Consider

There exists a

Consider

Then the stability properties of the trivial solution of the comparison system (

Let

The novel financial dynamical system [

Hyperchaotic attractors of the financial hyperchaotic system.

Equation (

Subtracting (

Our aim is to find some conditions on the control gains

Let

The conditions are

Let the Lyapunov function be in the form of

Hence, condition (1) of Theorem

It follows from Theorem

We assume that the impulses are equidistant and separated by

Let the impulses be equidistant and separated by interval

In order to verify the effectiveness of the impulsive synchronization method, some simulation results are illustrated. For simplicity, we assume

The boundaries of the stable region for different values of

We choose impulsive control gain matrix as

The time evolution of impulsive synchronization errors.

The time response of states of the drive system (

In this section, we consider complete synchronization of the financial hyperchaotic system under the adaptive-impulsive control.

The response system under the adaptive-impulsive control is described by the following equation:

We define the error vector

For

Suppose that

Choose a Lyapunov function as

In the case

In the case

Define the set

The designed adaptive-impulsive controllers have no specific requirements on impulsive interval. An important research topic of conventional impulsive control is how to get larger impulsive interval [

The control gain strengths of adaptive controller can be defined adaptively. The adaptive control gain strengths are always fixed [

In the adaptive-impulsive control process, impulsive controller wastes less energy than general impulsive control; nevertheless, continuous adaptive controller needs continuous energy. But is the energy wasted by adaptive-impulsive controller less than that of impulsive control? The answer is maybe. So, from the view of energy saving, how to constitute the more effective adaptive-impulsive controller is essential and needs us to investigate in the future.

Adaptive controllers have the merit of simple design, but continuous control needs more energy. Impulsive controllers can save much energy. The new adaptive-impulsive controller which the paper constituted integrates the advantages of adaptive controller and impulsive controller; it is designed simply and wastes less energy. Therefore, the proposed method in this paper can be applied to many fields, such as secure communication and commercial systems.

Numerical simulations are given in this subsection to verify the effectiveness and feasibility of the theoretical results obtained. We also assume

The time evolution of the synchronization errors between systems (

The time response of states of the drive system (

The time evolution of the control gain strengths.

Figure

Comparing the above results of impulsive synchronization and adaptive-impulsive synchronization, we can find that the synchronization time using nonlinear adaptive-impulsive control scheme is shorter than that using impulsive control. From this point of view, adaptive-impulsive control is more effective than impulsive control.

In this paper, we investigated impulsive synchronization and adaptive-impulsive synchronization of a novel financial hyperchaotic system theoretically and numerically at the first time. We have proposed an impulsive synchronization scheme for the financial system, obtained some synchronization criteria by means of comparing system principle, estimated the upper bounds of impulsive interval for stable synchronization, and provided numerical simulation results to show the effectiveness of the synchronization criteria. Furthermore, an adaptive-impulsive synchronization method for the financial system has been introduced, some synchronization conditions have been given, and corresponding numerical simulations have been presented to verify the effectiveness of the theoretical results. The results are helpful for synchronization development of financial systems and financial markets.

Complete synchronization is achieved for chaotic systems with well-matched parameters. However, parameter mismatch is inevitable in practical implementations of chaos synchronization because of noise or other artificial factors. And very small parameter mismatch might induce loss of perfect synchronization but might reserve quasisynchronization for the given allowable error. So we will investigate the effects of parameter mismatch of synchronization and derive some applicable synchronization criteria in a near future study.

The authors do not have a direct financial relation with any commercial identity mentioned in their paper that might lead to a conflict of interests for any of the authors.

This work is supported by the National Natural Science Foundation of China (Grants nos. 61004006 and 61203094), Natural Science Foundation of Educational Committee of Henan Province of China (Grants nos. 12A460001 and 2011A520004), Foundation of Science and Technology Committee of Henan Province of China (Grant no. 122102210053), the Joint Funds between Henan Provincial Government and Ministry of Education of China (Grant no. SBGJ090603), China Postdoctoral Science Foundation of China (Grant no. 2013M530181), Shanghai Postdoctoral Scientific Program of China (Grant no. 13R21410600), Research Foundation of Henan University of China (Grant no. 2012YBZR009), and the Eleventh Batch of Teaching Reform Project of Henan University of China.