Impulsive Synchronization and Adaptive-Impulsive Synchronization of a Novel Financial Hyperchaotic System

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.


Introduction
Since the seminal work of Pecora and Carroll [1], chaos synchronization has been an active topic in nonlinear science, due to its potential applications in secure communication, control theory, telecommunications, biological networks and artificial neural networks, and so forth.So far, many effective approaches have been presented to synchronize chaotic systems such as adaptive control [2,3], fuzzy control [4], static feedback control [5], variable structure control [6], stochastic control [7], impulsive control [8][9][10], and others.Impulsive control, as a discontinuous control method, has attracted more interest recently due to its easy implementation in engineering control.In some cases [11,12], it may be impossible to use synchronization at all times, and to use impulsive control may prove to be more efficient.
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 [13] presented a theory of impulsive synchronization of two chaotic systems and a promising application of impulsive synchronization of chaotic systems to a secure digital communication scheme.Sun and Zhang [14] investigated the impulsive synchronization of a Chua oscillator, and the experimental results show that the accuracy of the synchronization depends on the period and the width of the impulse.Luo [15] gave the sufficient condition for impulsive synchronization of a new chaotic system.Ma and Wang [16] introduced the impulsive control and synchronization of a new unified hyperchaotic system; the control gains and impulsive intervals are both variable and analyzed by the impulsive synchronization of a class of fractional-order hyperchaotic systems [17].These works on impulsive synchronization were based on the theory of comparison systems, and it is easy to define the impulsive interval and impulsive control gain.
Recently, some researchers synchronize the chaotic system through combining adaptive control and impulsive control, and they name it adaptive-impulsive control [18][19][20].In [18], Li et al. discussed adaptive-impulsive synchronization and parameter identification of a class of chaotic and hyperchaotic systems, and their controllers and identifiers have a limit that the system state variable function independent of parameters must be Lipchitz.Chen and Chang [19] derived an adaptive impulse control with only one restriction criterion to achieve synchronization of nonlinear chaotic systems in the exponential rate of convergence, and they assumed that the system satisfied the local Lipchitz condition.But not every chaotic or hyperchaotic system satisfies the Lipchitz condition; simultaneously the Lipchitz coefficient is hard to estimate.Wan and Sun [20] investigated the nonlinear adaptive-impulsive synchronization of chaotic systems, applied it to quantum cellular neural network (Quantum-CNN), and found adaptive-impulsive controllers more effective than the adaptive control scheme.
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. [21] studied the modified function lag projective synchronization of a novel financial hyperchaotic system by continuous adaptive control method.To the best of our knowledge, the impulsive synchronization and adaptive-impulsive synchronization of this novel financial system have not been studied.
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 2, some basic theories of impulsive differential equations are called.In Section 3, the novel financial system is given, and its dynamics equations and attractors diagrams are illustrated.In Section 4, several sufficient conditions for impulsive synchronization are introduced, the upper bound of impulsive interval is presented, and numerical simulation results are provided to show the effectiveness of the synchronization criteria.In Section 5, a nonlinear adaptive-impulsive control scheme is constituted to synchronize the financial system, and corresponding numerical simulations are presented to verify the effectiveness of the theoretical results.Finally, the conclusions are drawn in Section 6.

Basic Theories of Impulsive Differential Equations
In general, the impulsive differential system is described by where state variable x ∈   is left continuous at  =   , discrete set {  } of time instants denotes the time instants at which impulses are sent to the system and satisfies 0 <  ( Definition 3 (comparison system).Let  ∈  0 , and assume that where  :  + ×  + →  is continuous and   :  + →  + is nondecreasing.Then the system is called the comparison system of (1).
Theorem 6. Assume that the following three conditions are satisfied.
Then the stability properties of the trivial solution of the comparison system (4) imply the corresponding stability properties of the trivial solution of (1).

Impulsive Synchronization of the Financial Hyperchaotic System
Equation ( 7) is taken as the drive system; then the response system under impulsive control is characterized by where state variable  ∈   is left continuous at  =   and   denotes the moment when impulsive control occurs; discrete set {  } of time instants satisfies Subtracting ( 8) from (7), one obtains the dynamical system of synchronization error: Our aim is to find some conditions on the control gains   and the impulsive intervals ( +1 −   ) such that the impulsive controlled response system ( 8) is globally asymptotically synchronous with the drive system (7) for any initial states.Theorem 8. Let  2 be the largest eigenvalue of (I+B   )(I+B  ), impulsive control gain matrix   = diag( 1 ,  2 ,  3 ,  4 ), and the spectral radius of (I + B  )(I + B  ) ≤ 1.

The conditions are
Proof.Let the Lyapunov function be in the form of (1) Case I ( ̸ =   ).The time derivative of (11) along the solution of ( 9) is So, condition (2) of Theorem 6 is satisfied with   () =  2 .Obviously, condition (3) of Theorem 6 is also satisfied.Then, the comparison system is given by It follows from Theorem 7 that if is satisfied, then the origin of system ( 9) is asymptotically stable.This completes the proof.
We assume that the impulses are equidistant and separated by ; that is, for any  ∈ ,  +1 −   = .Then we have the following result.Corollary 9. Let the impulses be equidistant and separated by interval .Then the origin of system (9) is uniformly asymptotically stable if the following conditions hold: Moreover, an estimate of the upper bound of  is given as Here, impulsive control gain matrix should satisfy (I+B  ) ≤ 1.
We choose impulsive control gain matrix as B  = diag(−0.8,−0.8, −0.8, −0.8), the constant  = 1.5, then we can get 0 <  ≤ 0.2607, and here we select  = 0.25.Simulation results are displayed in Figures 3 and 4. Figure 3 illustrates that the impulsive synchronization errors   () ( = 1, 2, 3, 4) asymptotically converge to zero. Figure 4 depicts that the state variables of the drive and response systems reach complete synchronization in a very short period of time.

Adaptive-Impulsive Synchronization of the Financial Hyperchaotic System
In this section, we consider complete synchronization of the financial hyperchaotic system under the adaptive-impulsive control.

Adaptive-Impulsive Synchronization
Scheme.The response system under the adaptive-impulsive control is described by the following equation: where  1 , Our aim is to find the suitable adaptive-impulsive controller for stabilizing the error variables at the origin.To this end, we design the adaptive controllers as follows: For  = 1, 2, 3, 4,   > 0 are arbitrary constants and   are adaptive control gain strengths.The error dynamical system is obtained as follows: Theorem 10.Suppose that   =  max ((I+B   )(I+B  )) ≤ 1,  = 1, 2, 3, 4,  = 1, 2, . ... Then the drive system (7) will be globally asymptotically synchronous with the response system (18) by using nonlinear adaptive-impulsive controller {;   ,   }.

Proof. Choose a Lyapunov function as
where  is a positive constant.Figure 4: The time response of states of the drive system (7) and the response system (8).
In the case  ̸ =   , the time derivative of (21) along the solution of ( 20) is where I is the four-dimensional identity matrix.When  > −0.17, the matrix Q is negative definite.Through choosing the proper , V ≤ −e  e is achieved.
is, () → 0 and  →  0 as  → ∞.That is to say, the drive system (7) and the response system (18) are synchronized.This completes the proof.
Remark 11.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 [13,25,26].Adaptive-impulsive control scheme which the paper constituted fundamentally eliminates the limit on impulsive interval of conventional impulsive control.When the impulsive gain is fixed, the larger the impulsive interval, the more flexible the impulsive controller, and simultaneously the less the energy used.
Remark 12.The control gain strengths of adaptive controller can be defined adaptively.The adaptive control gain strengths are always fixed [21,[27][28][29]; sometimes it may be the maximum value, and thus that can give a kind of energy waste.The method of our paper is different from them.The control gain strengths can be automatically adapted to a suitable value depending on constant   and their initial values.
Remark 13.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 adaptiveimpulsive 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.
Remark 14. Adaptive controllers have the merit of simple design, but continuous control needs more energy.Impulsive controllers can save much energy.The new adaptiveimpulsive 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.
Numerical simulations are given in this subsection to verify the effectiveness and feasibility of the theoretical results obtained.We also assume {  } is equidistant, and  +1 −   = .We select the parameters as  = 0.9,  = 0.2,  = 1.5,  1 = 0.2, and  2 = 0.17, so that the systems ( 7) and ( 18) are hyperchaotic when no control inputs are applied.The initial states of the drive system (7) and the response system (18) are taken as ( 1 (0),  2 (0),    Figure 5 shows that the error variables  1 ,  2 ,  3 , and  4 tend to zero, respectively.Figure 6 denotes the time response of the drive system (7) and the response system (18).Figure 7 presents the time evolution of the control gain strengths, which displays that they converge to  1 = −7.16, 2 = −13.54, 3 = −10.54,and  4 = −15.99 as  → ∞.As shown in Figures 5-7, complete synchronization between the drive system (7) and the response system (18) is obtained, and the control gain strengths are estimated adaptively by using the nonlinear adaptive-impulsive controllers {;   ,   }.
Comparing the above results of impulsive synchronization and adaptive-impulsive synchronization, we can find that the synchronization time using nonlinear adaptiveimpulsive control scheme is shorter than that using impulsive control.From this point of view, adaptive-impulsive control is more effective than impulsive control.

Conclusions
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.

Figure 2 :Figure 3 :
Figure 2: The boundaries of the stable region for different values of  and .

Figure 6 :
Figure6: The time response of states of the drive system (7) and the response system(18).

Figure 7 :
Figure 7: The time evolution of the control gain strengths.