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

Abstract

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.

1. 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 [8–10], 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 [18–20]. 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.

2. Basic Theories of Impulsive Differential Equations

In general, the impulsive differential system is described by𝐱̇=𝑓(𝑡,𝐱),𝑡≠𝜏_𝑘,Δ𝐱=𝐱𝑡⁺_𝑘−𝐱𝑡⁻_𝑘=𝑈(𝑘,𝐱),𝑡=𝜏_𝑘𝐱𝑡,𝑘=1,2,…,₊₀=𝐱₀, (1) where state variable 𝐱∈𝑅^𝑛 is left continuous at 𝑡=𝜏_𝑘, discrete set {𝜏_𝑘} of time instants denotes the time instants at which impulses are sent to the system and satisfies 0<𝜏₁<𝜏₂<⋯<𝜏_𝑘<𝜏_𝑘+1<⋯, 𝜏_𝑘→∞ as 𝑘→∞. 𝑓∶𝑅₊×𝑅^𝑛→𝑅^𝑛 is continuous, and 𝑈(𝑘,𝐱) is the state variable at instant 𝜏_𝑘. First, we call the following definitions and theorems [22].

Definition 1. The function 𝑉∶𝑅₊×𝑅^𝑛→𝑅₊ is said to belong to class 𝑉₀ if (1)𝑉 is continuous in (𝜏_𝑘−1,𝜏_𝑘]×𝑅^𝑛, and, for each 𝑥∈𝑅^𝑛, lim_{(𝑡,𝑦)→(𝜏}⁺_𝑘,𝑥)𝑉(𝑡,𝑦)=𝑉(𝜏⁺_𝑘,𝑥) exists for 𝑘=1,2,…;(2)𝑉 is locally Lipschitzian in 𝐱.

Definition 2. For (𝑡,𝐱)∈(𝜏_𝑘−1,𝜏_𝑘]×𝐑^𝑛, the right and upper Dini’s derivatives of 𝑉∈𝑉₀ are defined as 𝐷⁺𝑉(𝑡,𝐱)≜limsup_ℎ→01ℎ[𝑉(𝑡+ℎ,𝐱+ℎ𝐟(𝑡,𝐱))−𝑉(𝑡,𝐱)]. (2)

Definition 3 (comparison system). Let 𝑉∈𝑉₀, and assume that 𝐷⁺𝑉(𝑡,𝐱)≤𝑔(𝑡,𝑉(𝑡,𝐱)),𝑡≠𝜏_𝑘,𝑉(𝑡,𝐱+𝑈(𝑘,𝐱))≤𝜑_𝑘(𝑉(𝑡,𝐱)),𝑡=𝜏_𝑘, (3) where 𝑔∶𝑅₊×𝑅₊→𝑅 is continuous and 𝜑_𝑘∶𝑅₊→𝑅₊ is nondecreasing. Then the system 𝜔̇≤𝑔(𝑡,𝜔),𝑡≠𝜏_𝑘,𝜔𝜏⁺_𝑘≤𝜑_𝑘𝜔𝜏_𝑘,𝜔𝜏⁺₀=𝜔₀≥0 (4) is called the comparison system of (1).

Definition 4. Consider 𝑆_𝜌=𝐱∈𝑅^𝑛∣‖𝐱‖<𝜌, (5) where ‖⋅‖ denotes the Euclidean norm on 𝑅^𝑛.

Definition 5. A function 𝛼 is said to belong to class 𝜅 if 𝛼∈𝐶[𝑅₊,𝑅₊], 𝛼(0)=0, and 𝛼(𝑥) is strictly increasing in 𝑥. Assume that 𝑓(𝑡,0)=0, 𝑈(𝑘,0)=0, and 𝑔(𝑡,0)=0 for all 𝑘.

Theorem 6. Assume that the following three conditions are satisfied.(1)Consider 𝑉∶𝑅₊×𝑆_𝜌→𝑅₊, 𝜌>0, 𝑉∈𝑉₀, 𝐷⁺𝑉(𝑡,𝐱)≤𝑔(𝑡,𝑉(𝑡,𝐱)), and 𝑡≠𝜏_𝑘.(2)There exists a 𝜌₀>0 such that 𝑥∈𝑆_𝜌0 implies that 𝑥+𝑈(𝑘,𝑥)∈𝑆_𝜌0 for all 𝑘, and 𝑉(𝑡,𝐱+𝑈(𝑘,𝐱))≤𝜑_𝑘(𝑉(𝑡,𝐱)), 𝑡=𝜏_𝑘, and 𝑥∈𝑆_𝜌0.(3)Consider 𝛼(‖𝐱‖)≤𝑉(𝑘,𝐱)≤𝛽(‖𝐱‖) on 𝑅₊×𝑆_𝜌, where 𝛼,𝛽∈𝜅.

Then the stability properties of the trivial solution of the comparison system (4) imply the corresponding stability properties of the trivial solution of (1).

Theorem 7 (see [22]). Let 𝑔(𝑡,𝜔)=𝜆̇(𝑡)𝜔, 𝜆∈𝐶¹[𝑅^𝑛,𝑅^𝑛], 𝜑_𝑘(𝜔)=𝑑_𝑘𝜔, and 𝜔≥0 for all 𝑘; then the origin of system (1) is asymptotically stable if conditions 𝜆𝜏_𝑘+1+ln𝜀𝑑_𝑘≤𝜆𝜏_𝑘,∀𝑘,𝑤ℎ𝑒𝑟𝑒𝜀>1, (6) and 𝜆̇(𝑡)≥0 are satisfied.

3. System Descriptions

The novel financial dynamical system [21, 23] is described as follows: 𝑥̇₁=𝑥₃+𝑥₂−𝑎𝑥₁+𝑥₄,𝑥̇₂=1−𝑏𝑥₂−𝑥²₁,𝑥̇₃=−𝑥₁−𝑐𝑥₃,𝑥̇₄=−𝑑₁𝑥₁𝑥₂−𝑑₂𝑥₄, (7) where 𝑥₁, 𝑥₂, 𝑥₃, and 𝑥₄ are state variables. 𝑥₁ denotes the interest rate, 𝑥₂ is the investment demand, 𝑥₃ is the price exponent, and 𝑥₄ is the average profit margin. 𝑎, 𝑏, 𝑐, 𝑑₁, and 𝑑₂ are system parameters, and when 𝑎=0.9, 𝑏=0.2, 𝑐=1.5, 𝑑₁=0.2, and 𝑑₂=0.17, system (7) is hyperchaotic as displayed in Figures 1(a)–1(d). It is easy to know that the state variables of system (7) are bounded.

(a)

(b)

(c)

(d)

Figure 1 

Hyperchaotic attractors of the financial hyperchaotic system.

4. 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 𝑦̇₁=𝑦₃+𝑦₂−𝑎𝑦₁+𝑦₄,𝑡≠𝜏_𝑘,𝑦̇₂=1−𝑏𝑦₂−𝑦²₁,𝑡≠𝜏_𝑘,𝑦̇₃=−𝑦₁−𝑐𝑦₃,𝑡≠𝜏_𝑘,𝑦̇₄=−𝑑₁𝑦₁𝑦₂−𝑑₂𝑦₄,𝑡≠𝜏_𝑘,Δ𝑦_𝑖=𝑦_𝑖𝑡⁺_𝑘−𝑦_𝑖𝑡⁻_𝑘=𝐵_𝑖𝑘𝑒_𝑖,𝑖=1,2,3,4;𝑡=𝜏_𝑘,𝑘=1,2,…, (8) where state variable 𝑦∈𝑅^𝑛 is left continuous at 𝑡=𝜏_𝑘 and 𝜏_𝑘 denotes the moment when impulsive control occurs; discrete set {𝜏_𝑘} of time instants satisfies 0<𝜏₁<𝜏₂<⋯<𝜏_𝑘<𝜏_𝑘+1<⋯, 𝜏_𝑘→∞ as 𝑘→∞; 𝐵_𝑘=diag(𝐵_1𝑘,𝐵_2𝑘,𝐵_3𝑘,𝐵_4𝑘) is impulsive control gain constant matrix, and 𝐵_𝑖𝑘 is the impulsive control gain, 𝑖=1,2,3,4. 𝐞=(𝑒₁,𝑒₂,𝑒₃,𝑒₄)^𝑇=(𝑦₁−𝑥₁,𝑦₂−𝑥₂,𝑦₃−𝑥₃,𝑦₄−𝑥₄)^𝑇 is the synchronization error.

Subtracting (8) from (7), one obtains the dynamical system of synchronization error: 𝑒₁̇=𝑒₃+𝑦₂𝑦₁−𝑥₂𝑥₁−𝑎𝑒₁+𝑒₄,𝑡≠𝜏_𝑘,𝑒₂̇=−𝑏𝑒₂−𝑦²₁+𝑥²₁,𝑡≠𝜏_𝑘,𝑒₃̇=−𝑒₁−𝑐𝑒₃,𝑡≠𝜏_𝑘,𝑒₄̇=−𝑑₁𝑦₁𝑦₂+𝑑₁𝑥₁𝑥₂−𝑑₂𝑒₄,𝑡≠𝜏_𝑘,Δ𝑒_𝑖=𝐵_𝑖𝑘𝑒_𝑖,𝑖=1,2,3,4;𝑡=𝜏_𝑘,𝑘=1,2,…. (9)

Our aim is to find some conditions on the control gains 𝐵_𝑖𝑘 and the impulsive intervals (𝜏_𝑘+1−𝜏_𝑘) such that the impulsive controlled response system (8) is globally asymptotically synchronous with the drive system (7) for any initial states.

Theorem 8. Let 𝜆₂ be the largest eigenvalue of (𝐈+𝐁^𝑇_𝑘)(𝐈+𝐁_𝑘), impulsive control gain matrix 𝐵_𝑘=diag(𝐵_1𝑘,𝐵_2𝑘,𝐵_3𝑘,𝐵_4𝑘), and the spectral radius of (𝐈+𝐁_𝑘)𝜌(𝐈+𝐁_𝑘)≤1.

The conditions are 𝑀=max1−𝑑₁𝑦₂,𝑥₁,𝑑₁𝑥₁,𝑥₂,𝜆₁=max(2𝑀−𝑎),(𝑀−𝑏),−𝑐,𝑀−𝑑₂,𝜆₁𝜏_𝑘+1−𝜏_𝑘≤−ln𝜀𝜆₂,𝜀>1,𝜆₁≥0. (10)

Proof. Let the Lyapunov function be in the form of 𝑉(𝑒)=12𝑒^𝑇𝑒. (11)
(1)  Case I (𝑡≠𝜏_𝑘). The time derivative of (11) along the solution of (9) is 𝑉̇(𝑒)=𝑒₁𝑒₁̇+𝑒₂𝑒₂̇+𝑒₃𝑒₃̇+𝑒₄𝑒₄̇=𝑒₃𝑒₁+𝑦₂𝑦₁𝑒₁−𝑥₂𝑥₁𝑒₁−𝑎𝑒²₁+𝑒₄𝑒₁−𝑏𝑒²₂−𝑒₂𝑦²₁+𝑒₂𝑥²₁−𝑒₁𝑒₃−𝑐𝑒²₃−𝑑𝑦₁𝑦₂𝑒₄+𝑑𝑥₁𝑥₂𝑒₄−𝑑₂𝑒²₄=−𝑎𝑒²₁−𝑏𝑒²₂−𝑐𝑒²₃−𝑑₂𝑒²₄+𝑒₄𝑒₁1−𝑑𝑦₂−𝑒₂𝑒₁𝑥₁−𝑑𝑥₁𝑒₂𝑒₄+𝑥₂𝑒²₁≤−𝑎𝑒²₁−𝑏𝑒²₂−𝑐𝑒²₃−𝑑₂𝑒²₄+𝑒₄𝑒₁1−𝑑𝑦₂+𝑒₂𝑒₁𝑥₁+𝑑𝑥₁𝑒₂𝑒₄+𝑥₂𝑒²₁=−𝑎𝑒²₁−𝑏𝑒²₂−𝑐𝑒²₃−𝑑₂𝑒²₄+𝑀𝑒₄𝑒₁+𝑒₂𝑒₁+𝑒₂𝑒₄+𝑒²₁≤(2𝑀−𝑎)𝑒²₁+(𝑀−𝑏)𝑒²₂−𝑐𝑒²₃+𝑀−𝑑₂𝑒²₄=𝜆₁𝑉(𝑒). (12)
Hence, condition (1) of Theorem 6 is satisfied with 𝑔(𝑡,𝜔)=𝜆₁𝜔.
(2)  Case II (𝑡=𝜏_𝑘). Since 𝐈+𝐁_𝑘 is symmetric, by employing Euclidean norm, we have 𝜌(𝐈+𝐁_𝑘)=‖𝐈+𝐁_𝑘‖. Then for any 𝜌₀>0 such that 𝑥∈𝑆_𝜌0, we have ‖𝐱+𝑈(𝑘,𝐱)‖≤‖𝐈+𝐁_𝑘‖‖𝐱‖=𝜌(𝐈+𝐁_𝑘)‖𝐱‖≤‖𝐱‖. Then, 𝑉(𝐞)=12𝐞^𝑇𝐈+𝐁_𝑘^𝑇𝐈+𝐁_𝑘𝐞≤𝜆₂𝑉(𝐞). (13) So, condition (2) of Theorem 6 is satisfied with 𝜑_𝑘(𝜔)=𝜆₂𝜔. Obviously, condition (3) of Theorem 6 is also satisfied. Then, the comparison system is given by 𝜔̇=𝜆₁𝜔,𝑡≠𝜏_𝑘,𝜔𝜏⁺_𝑘=𝜆₂𝜔𝜏_𝑘,𝜔𝜏⁺₀=𝜔₀≥0. (14)
It follows from Theorem 7 that if 𝜆₁𝜏_𝑘+1+ln𝜀𝜆₂≤𝜆₁𝜏_𝑘,∀𝑘,where𝜀>1, (15) 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 𝛿; that is, for any 𝑘∈𝑁, 𝜏_𝑘+1−𝜏_𝑘=𝛿. Then we have the following result.

Corollary 9. Let the impulses be equidistant and separated by interval 𝛿. Then the origin of system (9) is uniformly asymptotically stable if the following conditions hold: 0<𝛿≤−ln𝜀𝜆₂𝜆₁,𝜀>1. (16) Moreover, an estimate of the upper bound of 𝛿 is given as 𝛿_max=−ln𝜀𝜆₂𝜆₁. (17) Here, impulsive control gain matrix should satisfy 𝜌(𝐈+𝐁_𝑘)≤1.

In order to verify the effectiveness of the impulsive synchronization method, some simulation results are illustrated. For simplicity, we assume {𝜏_𝑘} is equidistant, and 𝜏_𝑘+1−𝜏_𝑘=𝛿>0. The parameters are selected as 𝑎=0.9, 𝑏=0.2, 𝑐=1.5, 𝑑₁=0.2, and 𝑑₂=0.17, such that the systems (7) and (8) exhibit hyperchaotic behavior if no control is applied. The initial values of the drive system (7) and the response system (8) are (𝑥₁(0),𝑥₂(0),𝑥₃(0),𝑥₄(0))=(−1,−2,−3,−4) and (𝑦₁(0),𝑦₂(0),𝑦₃(0),𝑦₄(0))=(1,2,3,4), respectively. By computing, we can get 𝑀=4, 𝜆₁=7.1. We suppose 𝐵_1𝑘=𝐵_2𝑘=𝐵_3𝑘=𝐵_4𝑘=𝐵; then −2≤𝐵≤0. The estimate of the stable region for different values of 𝐵 and 𝜀 is shown in Figure 2.

Figure 2 

The boundaries of the stable region for different values of 𝜀 and 𝐵.

We choose impulsive control gain matrix as 𝐁_𝑘=diag(−0.8,−0.8,−0.8,−0.8), the constant 𝜀=1.5, then we can get 0<𝛿≤0.2607, and here we select 𝛿=0.25. Simulation results are displayed in Figures 3 and 4. Figure 3 illustrates that the impulsive synchronization errors 𝑒_𝑖(𝑡) (𝑖=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.

(a)

(b)

(c)

(d)

Figure 3 

The time evolution of impulsive synchronization errors.

(a)

(b)

(c)

(d)

Figure 4 

The time response of states of the drive system (7) and the response system (8).

5. 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.

5.1. Adaptive-Impulsive Synchronization Scheme

The response system under the adaptive-impulsive control is described by the following equation: 𝑦̇₁=𝑦₃+𝑦₂−𝑎𝑦₁+𝑦₄+𝑢₁,𝑡≠𝜏_𝑘,𝑦̇₂=1−𝑏𝑦₂−𝑦²₁+𝑢₂,𝑡≠𝜏_𝑘,𝑦̇₃=−𝑦₁−𝑐𝑦₃+𝑢₃,𝑡≠𝜏_𝑘,𝑦̇₄=−𝑑₁𝑦₁𝑦₂−𝑑₂𝑦₄+𝑢₄,𝑡≠𝜏_𝑘,Δ𝑦_𝑖=𝑦_𝑖𝑡⁺−𝑦_𝑖(𝑡⁻)=𝐵_𝑖𝑘𝑒_𝑖,𝑖=1,2,3,4;𝑡=𝜏_𝑘,𝑘=1,2,…, (18) where 𝑢₁, 𝑢₂, 𝑢₃, and 𝑢₄ are adaptive controllers to be designed. And 𝐵_𝑘=diag(𝐵_1𝑘,𝐵_2𝑘,𝐵_3𝑘,𝐵_4𝑘) is impulsive control gain constant matrix, 𝐵_𝑖𝑘 (𝑖=1,2,3,4) is the impulsive control gain, and {𝑢;𝐵_𝑘,𝜏_𝑘} is nonlinear adaptive-impulsive controller to be constituted.

We define the error vector 𝐞=(𝑒₁,𝑒₂,𝑒₃,𝑒₄)^𝑇=(𝑦₁−𝑥₁,𝑦₂−𝑥₂,𝑦₃−𝑥₃,𝑦₄−𝑥₄)^𝑇. 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: 𝑢₁=𝑥₁𝑥₂−𝑦₁𝑦₂+𝑚₁𝑒₁,𝑢₂=𝑦²₁−𝑥²₁+𝑚₂𝑒₂,𝑢₃=𝑚₃𝑒₃,𝑢₄=−𝑑₁𝑥₁𝑥₂+𝑑₂𝑦₁𝑦₂+𝑚₄𝑒₄,𝑚̇_𝑖=−𝛾_𝑖𝑒²_𝑖,𝑖=1,2,3,4. (19)

For 𝑖=1,2,3,4, 𝛾_𝑖>0 are arbitrary constants and 𝑚_𝑖 are adaptive control gain strengths. The error dynamical system is obtained as follows: 𝑒₁̇=𝑒₃−𝑎𝑒₁+𝑒₄+𝑚₁𝑒₁,𝑡≠𝜏_𝑘,𝑒₂̇=−𝑏𝑒₂+𝑚₂𝑒₂,𝑡≠𝜏_𝑘,𝑒₃̇=−𝑒₁−𝑐𝑒₃+𝑚₃𝑒₃,𝑡≠𝜏_𝑘,𝑒₄̇=−𝑑₂𝑒₄+𝑚₄𝑒₄,𝑡≠𝜏_𝑘,Δ𝑒_𝑖=𝐵_𝑖𝑘𝑒_𝑖,𝑖=1,2,3,4;𝑡=𝜏_𝑘𝑚̇,𝑘=1,2,…,_𝑖=−𝛾_𝑖𝑒²_𝑖,𝑖=1,2,3,4. (20)

Theorem 10. Suppose that 𝜌_𝑖𝑘=𝜆_max((𝐈+𝐁^𝑇_𝑘)(𝐈+𝐁_𝑘))≤1, 𝑖=1,2,3,4, 𝑘=1,2,…. Then the drive system (7) will be globally asymptotically synchronous with the response system (18) by using nonlinear adaptive-impulsive controller {𝑢;𝐵_𝑘,𝜏_𝑘}.

Proof. Choose a Lyapunov function as 1𝑉=2𝐞^𝑇1𝐞+2⁴_𝑖=11𝛾_𝑖𝑚_𝑖+𝑙², (21) where 𝑙 is a positive constant.
In the case 𝑡≠𝜏_𝑘, the time derivative of (21) along the solution of (20) is 𝐷⁺𝑉=𝐞^𝑇𝐞̇−⁴_𝑖=1𝑚_𝑖+𝑙𝑒²_𝑖=𝑒₁𝑒₃−𝑎𝑒₁+𝑒₄+𝑚₁𝑒₁+𝑒₂−𝑏𝑒₂+𝑚₂𝑒₂+𝑒₃−𝑒₁−𝑐𝑒₃+𝑚₃𝑒₃+𝑒₄−𝑘𝑒₄+𝑚₄𝑒₄−⁴_𝑖=1𝑚_𝑖+𝑙𝑒²_𝑖=−𝑎𝑒²₁+𝑒₁𝑒₄−𝑏𝑒²₂−𝑐𝑒²₃−𝑑₂𝑒²₄−⁴_𝑖=1𝑙𝑒²_𝑖=𝐞^𝑇1−𝑎00210−𝑏0000−𝑐0200−𝑑₂−𝑙𝐈𝐞=𝐞^𝑇𝐐𝐞, (22) where 𝐈 is the four-dimensional identity matrix. When 𝑙>−0.17, the matrix 𝐐 is negative definite. Through choosing the proper 𝑙, 𝑉̇≤−𝐞^𝑇𝐞 is achieved.
In the case 𝑡=𝜏_𝑘, 𝑉𝜏⁺_𝑘=12𝑒^𝑇𝜏⁺_𝑘𝑒𝜏⁺_𝑘+12⁴_𝑖=11𝛾_𝑖𝑚_𝑖𝜏⁺_𝑘+𝑙²=12⁴_𝑖=11+𝐵_𝑖𝑘²𝑒²_𝑖𝜏_𝑘+12⁴_𝑖=11𝛾_𝑖𝑚_𝑖𝜏⁺_𝑘+𝑙²=12𝜌⁴_{𝑖𝑘𝑖=1}𝑒²_𝑖𝜏_𝑘+12⁴_𝑖=11𝛾_𝑖𝑚_𝑖𝜏_𝑘+𝑙²<12⁴_𝑖=1𝑒²_𝑖𝜏_𝑘+12⁴_𝑖=11𝛾_𝑖𝑚_𝑖𝜏_𝑘+𝑙²=𝑉𝜏_𝑘. (23)
Define the set ℜ≜{𝑉̇=0,𝑡≠𝜏_𝑘,𝑘=1,2,…}∪{𝑉(𝜏⁺_𝑘)=𝑉(𝜏_𝑘),𝑘=1,2,…}, and the set Ω={(𝑒,𝑚)∈ℝ^2𝑛∶𝑒(𝑡)=0,𝑚=𝑚₀∈ℝ^𝑛} is the largest invariant set contained in ℜ for error dynamical equations (21). According to corollary 5.1 [24], the orbit of the system (20) converges to the set Ω, that is, 𝑒(𝑡)→0 and 𝑚→𝑚₀ as 𝑡→∞. 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, 27–29]; 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 𝛾_𝑖 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 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.

Remark 14. 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.

5.2. Numerical Simulations

Numerical simulations are given in this subsection to verify the effectiveness and feasibility of the theoretical results obtained. We also assume {𝜏_𝑘} is equidistant, and 𝜏_𝑘+1−𝜏_𝑘=𝛿. We select the parameters as 𝑎=0.9, 𝑏=0.2, 𝑐=1.5, 𝑑₁=0.2, and 𝑑₂=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 (𝑥₁(0),𝑥₂(0),𝑥₃(0),𝑥₄(0))=(−1,−2,−3,−4) and (𝑦₁(0),𝑦₂(0),𝑦₃(0),𝑦₄(0))=(1,2,3,4), respectively. The nonlinear adaptive-impulsive controllers {𝑢;𝐵_𝑘,𝜏_𝑘} are designed as 𝐁_𝑘=diag{−0.8,−0.8,−0.8,−0.8}, 𝛾₁=𝛾₂=𝛾₃=𝛾₄=−2, and 𝛿=1, and the initial values of the control gain strengths are set as 𝑚₁(0)=𝑚₂(0)=𝑚₃(0)=𝑚₄(0)=8. The corresponding simulation results are illustrated in Figures 5–7.

(a)

(b)

(c)

(d)

Figure 5 

The time evolution of the synchronization errors between systems (7) and (18).

(a)

(b)

(c)

(d)

Figure 6 

The time response of states of the drive system (7) and the response system (18).

Figure 7 

The time evolution of the control gain strengths.

Figure 5 shows that the error variables 𝑒₁, 𝑒₂, 𝑒₃, and 𝑒₄ 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 𝑚₁=−7.16, 𝑚₂=−13.54, 𝑚₃=−10.54, and 𝑚₄=−15.99 as 𝑡→∞. 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 {𝑢;𝐵_𝑘,𝜏_𝑘}.

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.

6. 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.

Conflict of Interests

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.

Acknowledgments

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.