Journal of Control Science and Engineering

Volume 2018 (2018), Article ID 4976380, 12 pages

https://doi.org/10.1155/2018/4976380

## A Four-Dimensional Hyperchaotic Finance System and Its Control Problems

School of Business and Administration, Qilu University of Technology, Jinan 250353, China

Correspondence should be addressed to Lin Cao

Received 29 August 2017; Revised 11 December 2017; Accepted 3 January 2018; Published 1 February 2018

Academic Editor: Sundarapandian Vaidyanathan

Copyright © 2018 Lin Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The construction and several control problems of a new hyperchaotic finance system are investigated in this paper. Firstly, a new four-dimensional hyperchaotic finance system is introduced, based on which a new hyperchaos is then generated by setting parameters. And the qualitative analysis is numerically studied to confirm the dynamical processes, for example, the bifurcation diagram, Poincaré sections, Lyapunov exponents, and phase portraits. Interestingly, the obtained results show that this new system can display complex characteristics: chaotic, hyperchaotic, and quasiperiodic phenomena occur alternately versus parameters. Secondly, three single input adaptive controllers are designed to realize the control problems of such system: stabilization, synchronization, and coexistence of antisynchronization and complete synchronization, respectively. It is noted that the designed controllers are simpler than the existing ones. Finally, numerical simulations are provided to demonstrate the validity and the effectiveness of the proposed theoretical results.

#### 1. Introduction

It is well known that the classical chaotic attractor was firstly found by Lorenz in 1963 [1]. As a most significant system in nonlinear dynamical systems, chaos systems and their relative problems have attracted a lot of consideration from all kinds of researchers in many fields of science, and many significant results have been obtained in the past few decades; for details, see [2–6] and the references therein. From then on, chaos becomes a hot topic for a broad class of applications in physics, electrical engineering, secure communication, and many other fields [4–6]. As we know that the positive Lyapunov exponent of the chaotic systems is a critical prerequisite in these applications. The three-dimensional chaotic system only has one positive Lyapunov exponent; thus it is typically difficult to meet some certain requirements. For instance, the messages which are masked by the simple chaotic systems are easy to decipher in secure communication [4, 5]. Different from the ordinary chaos, the hyperchaotic system has at least two positive Lyapunov exponents and thus has more prominent advantages due to its higher dimensions and more unpredictable behaviors. Accordingly, the hyperchaotic systems have been investigated extensively, and various control techniques and approaches have been developed and utilized [7–9]. It should be pointed out that the above-mentioned works are based on the hyperchaotic models with complex structure. However, such models are difficult to be verified in reality. This motivates the author to construct a simple system that can display rich dynamical characteristics.

It is well known that the OGY method [10] for chaos control was firstly observed in 1990. Meanwhile, the PC method [11] for chaos synchronization firstly achieved chaos synchronization in 1990. Since then, a lot of researchers who are from all kinds of scientific fields had to pay an increasing interest to investigate chaos systems and their relative problems in the past years, and many significant theoretical results and experiments conclusions have been published; for details please see [12–27] and the references therein. In recent years, several typical kinds of chaos synchronization have been identified, for example, complete synchronization (or synchronization) [14, 15], phase synchronization [16, 17], lag synchronization [18, 19], generalized synchronization [20, 21], antisynchronization [22, 23], projective synchronization [24, 25], coexistence of antisynchronization and complete synchronization [26, 27], and simultaneous synchronization and antisynchronization [27]. However, for the chaotic or hyperchaotic systems, there are still some important questions needed to be further investigated, such as the complexity of the controllers and the existence of some control problems, which partly motivates the present work.

For the economic systems, the chaotic behavior in those systems was first found in 1985 [28]. It is noted that chaotic behavior in economics implies that the considered economic system has an inherent indefiniteness. Thus, it is of important value to study the chaotic finance system in order to achieve a stable economic growth. A novel chaotic finance system [3] was presented in 2001. There are four subblocks which construct the system model, that is, money, production, labor force, and stock. This system model is described by three state variables of the time variations: stands for the interest rate; and are the investment demand and the price index, respectively. Since this chaotic finance system is proposed, many works have been done for this finance system [29, 30]. Recently, a novel hyperchaotic finance system [31] is presented in 2010 and some important results have been obtained. However, there are some limitations in the existing results. For example, in the stabilization problem and the complete synchronization problem of such system, the designed controllers are too complicated to be used in applications. On the other hand, the coexistence of antisynchronization and complete synchronization problem of this hyperchaotic finance system has not been investigated so far. Therefore, this new hyperchaotic finance system needs to be further investigated.

Motivated by the above discussions, a four-dimensional hyperchaotic finance system is presented, which can generate double-wing chaotic and hyperchaotic attractors with three equilibrium points. In comparison with the most existing results, this model has simple structure and can display complex dynamics: chaotic, hyperchaotic, and quasiperiodic phenomena occur alternately. Furthermore, some basic dynamic properties of the new hyperchaotic finance system regarding equilibria, dissipation, Lyapunov exponent, Lyapunov dimension, bifurcation diagram, and Poincaré sections are investigated. Then, three control problems: stabilization, synchronization, and coexistence of antisynchronization and complete synchronization are derived with simple yet physically implementable controllers. Finally, numerical simulations are provided to demonstrate effectiveness and the validity of the proposed theoretical results. In conclusion, the main contributions of this paper are given as follows:(1)A new hyperchaotic finance system is firstly introduced, based on which a new hyperchaos is then generated by setting the parameters.(2)Dynamic properties of the new generated hyperchaotic finance system are investigated extensively in Section 3, which are important and interesting. It should be pointed out that these dynamic properties results are different from the existing results.(3)Three control problems of the hyperchaotic finance system are investigated extensively in Section 4. It is noted that the obtained results in this paper have some advantages over the exiting results. In particular, it should be pointed out that the coexistence of antisynchronization and complete synchronization in the two hyperchaotic finance systems is studied, which is a new result.

The rest of this paper is organized as follows. In Section 2, a new hyperchaotic model is introduced, based on which a new hyperchaos is generated by setting the parameters. In Section 3, the dynamic properties of such hyperchaotic finance system are investigated. In Section 4, several control problems of such hyperchaotic finance system are studied extensively, followed by the conclusions in Section 5.

#### 2. Model Formulation

A dynamic model of finance has been reported in [29–31], which is composed of four subblocks, production, money, stock, and labor force, and expressed by four first-order differential equations. The model describes the time variations of four state variables: the interest rate , the investment demand , the price exponent , and the average profit margin . It is well known that the factors affecting the interest rates are related not only to investment demand and price index, but also to the average profit margin. And the average profit margin and interest rate are proportional. Some important results have been obtained. In the next, the four-dimensional hyperchaotic finance system is expressed as follows:where is the state vector, is the saving, is the per investment cost, is the elasticity of demands of commercials, and are positive constant parameters.

In order to generate a new hyperchaos, we set , , , , and select as a governing parameter according to the two criterions in [7]. The dynamical properties of the hyperchaotic system (1) are firstly studied in Section 3, including symmetry, dissipation, equilibrium point, Lyapunov exponent, Lyapunov dimension, bifurcation diagram, and Poincaré sections. Then, three control problems of such system are investigated in Section 4, that is, stabilization, complete synchronization, and coexistence of antisynchronization and complete synchronization.

#### 3. Dynamic Properties

##### 3.1. Symmetry

Since the hyperchaotic system (1) is invariant under the coordinate transformation: , the system (1) is symmetric with respect to -axis, which implies that all values of such system parameters are under reflection about -axis.

##### 3.2. Dissipation

The divergence of this four-dimensional hyperchaotic system (1) is described asWhen , that is, , thus the system (1) is a dissipative system. It results in that volume element as . Therefore, all the trajectories of this four-dimensional hyperchaotic system (1) are ultimately embedded into an chaotic attractor.

##### 3.3. Equilibria

The equilibria of the system (1) are found by solving the following nonlinear equations:After computation, it results in where , , . As an example, set , the equilibria are given as follows: Linearizing the system (1), it results inFor , the eigenvalues can be calculated as , , , , which implies that is a saddle-focus, thus it is unstable. About and , the eigenvalues are , , , , , , , , respectively, which imply that and are stable points.

##### 3.4. Bifurcation Diagram

Bifurcation diagram is a useful method to show the dynamical processes of a system with respect to a parameter. In this subsection, the parameter is chosen as a control parameter. Figure 1 displays the bifurcation diagram of the variable versus the parameter which is from to . It is interesting that the evolution procedure of the system (1) through chaotic, hyperchaotic, and periodic orbit, as well as period doubling route to chaos, which displays complex dynamic properties with the increasing of the parament . In order to show the dynamics of the system (1) clearly, we test the time course of the system (1) by selecting several parameters . It can be seen from Figure 2, the state trajectory of the system (1) shows chaos and quasperiodic routes to chaos with the increase of parameter , which is consistent with above results.