Qualitative Study of a 4D Chaos Financial System

Zhang, Fuchen; Yang, Gaoxiang; Zhang, Yong; Liao, Xiaofeng; Zhang, Guangyun

doi:https://doi.org/10.1155/2018/3789873

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 3789873 | https://doi.org/10.1155/2018/3789873

Qualitative Study of a 4D Chaos Financial System

Fuchen Zhang,^1,2Gaoxiang Yang,³Yong Zhang,⁴Xiaofeng Liao,⁵and Guangyun Zhang¹

Academic Editor: Roberto Natella

Received11 Feb 2018

Accepted17 May 2018

Published16 Jul 2018

Abstract

Some dynamics of a new 4D chaotic system describing the dynamical behavior of the finance are considered. Ultimate boundedness and global attraction domain are obtained according to Lyapunov stability theory. These results are useful in estimating the Lyapunov dimension of attractors, Hausdorff dimension of attractors, chaos control, and chaos synchronization. We will also present some simulation results. Furthermore, the volumes of the ultimate bound set and the global exponential attractive set are obtained.

1. Introduction

Chaotic systems are characterized by their extreme sensitivity both to initial conditions and to system parameters. The study of various aspects of chaotic systems has received great interest among scientists from various fields due to their numerous potential applications in science and engineering [1–30]. Among such aspects, the problem of investigating the ultimate boundedness of chaotic systems and hyperchaotic systems is an important subject. Since the discovery of the famous Lorenz chaotic attractor [1], ultimate boundedness of the Lorenz attractor has been investigated by Leonov et al. in a series of articles [7, 10]. Since then many papers have studied ultimate boundedness of other chaotic systems [25–32]. However, the approach taken in each is only suitable for that particular chaotic system. It is very difficult to propose a universal approach to estimate the bounds for an arbitrary chaotic system. Zhang et al. studied the ultimate boundedness of the Lü system and the Chen system by using the Lyapunov stability theory and optimization method [25–27]. Particularly, Liao et al. studied the ultimate boundedness of the Yang-Chen system by using geometric and algebraic methods [31]. Zhang et al. studied the ultimate boundedness of a novel finance chaotic system by using the Lyapunov stability theory and iterative method [32]. To this end, it is necessary to study the bounds of the new chaotic systems.

Recently, a financial chaos dynamical system is reported as follows [32, 33]:

System (1) models a financial dynamical system composed of product, money, bond, and labor force. The variables , , and denote the interest rate, the investment demand, and the price index, respectively. The positive parameters , , and denote the saving amount, the per investment cost, and the demand elasticity of commercials, respectively. The factors that induce the change of the interest rate mainly come from two aspects: the price index and the surplus between the investment demand and the savings amount. The changing rate of is determined by the benefit rate of investment (we assume that the rate is constant during a certain period), the feedback of the investment demand, and the interest rate. The change of the price index is controlled by the real interest rate and price index.

According to the financial dynamical system (1), by adding a variable , one gets a new 4D financial chaos dynamical system as follows: where the variable denotes control input and economically state intervention to balance the economic environment and are positive real parameters of system (2).

The Lyapunov exponents of the financial system (2) are calculated numerically for the parameter values , , , , , and with the initial state . System (2) has the Lyapunov exponents as , , , and for the above parameters (see [8, 9] for a detailed discussion of Lyapunov exponents of strange attractors in chaos dynamical systems).

The chaotic attractor of system (2) in space for the positive parameters , , , , , and is shown in Figure 1.

Remark 1. The chaotic attractor is one of the important concepts of dynamical systems. Although system (2) has a chaotic attractor for the positive parameters , , , , , and , the type of the chaotic attractor of system (2) is still unknown. It is interesting to discuss whether the chaotic attractor of system (2) is hidden or self-excited in the future (see the excellent papers [17, 19] for a detailed discussion of the chaotic attractor of dynamical systems).
This paper is organized as follows. In the next section, we will study the bounds for the chaotic attractors (2) using the Lyapunov stability theory and optimization method. To validate the ultimate bound estimation, numerical simulations are also provided. Finally, we will give some concluding remarks in Section 3.

2. Main Properties

Theorem 1. Suppose that , .
Let be an arbitrary solution of system (2). Then, the following set is the ultimate bound set and positively invariant set of system (2), where

Proof 1. Define the Lyapunov-like function where . Computing the derivative of along the trajectory of (2), we have Obviously, the surface , that is, defined by is an ellipsoid in , , , , , and . Outside , while inside . Thus, the maximum value of can only be reached on . Since the is a continuous function and is a bounded closed set, then the function (5) can reach its maximum value on the surface defined in (7).
Obviously, contains the solutions of system (2). By solving the following conditional extremum problem, one can get the maximum value of the function (5) as follows: That is to say, Let us take as the new variables, then (9) transforms into the following form: According to the Lagrange multiplier method, we can easily get the above conditional extremum problem (10) as Finally, it is easy to show that (3) is the ultimate bound set and positively invariant set of system (2).

This completes the proof.

Remark 2. (i)When , and , then we can obtain that which is the ultimate bound set and positively invariant set of system (2) according to Theorem 1. Figure 2 shows the chaotic attractor of system (2) in the space defined by .(ii)We can figure out that the volume of the set in (3) is according to Theorem 1.

Though Theorem 1 gives the ultimate bound set and positively invariant set of the financial chaos system (2), it does not give the global exponential attractive set of system (2). The global exponential attractive set of system (2) is described by the following Theorem 2.

Theorem 2. Suppose that , and , and let Then, the estimation, holds for system (2), and thus which is the globally exponential attractive set of system (2), that is, .

Proof 2. Define the following function of a variable then we can get Define the Lyapunov function Differentiating the Lyapunov function in (18) with respect to time along the trajectory of system (2), when and , we have Thus, we have and which clearly shows that is the globally exponential attractive set of system (2).

The proof is complete.

Remark 3. (i)We can figure out that the volume of the set in (15) is according to Theorem 2.

3. Conclusions

In this paper, we considered some dynamics of a new 4D financial chaos system. We obtained the ultimate boundedness and global exponential attractive sets of this system according to Lyapunov stability theory. MATLAB simulations show that the proposed method is effective. Characteristic time scales of the 4D financial system, the homoclinic orbits, and heteroclinic orbits of the 4D financial system will be taken into consideration in the future.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors have read and approved the final manuscript.

Acknowledgments

This paper is supported by National Natural Science Foundation of China (Grant nos. 11501064 and 11426047), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant no. KJ1500605), the Research Fund of Chongqing Technology and Business University (Grant nos. 2014-56-11, KFJJ2017066, and 2018106), China Postdoctoral Science Foundation (Grant no. 2016M590850), Chongqing Postdoctoral Science Foundation Special Funded Project (Grant no. Xm2017174), 2016 Chongqing Social Science Planning Major Entrustment Project “Frontier theory research on census quality assessment” (Grant no. 2016WT03), the National Key Research and Development Program of China (Grant no. 2016YFB0800601), in part by the National Natural Science Foundation of China (Grant no. 61472331), and Natural Science Foundation of Shaanxi Provincial Education Department (Grant no. 15JK1016). The authors thank Professors Jinhu Lü in Chinese Academy of Sciences, Lennart Stenflo in the Royal Swedish Academy of Sciences, Gennady A. Leonov in the Saint-Petersburg State University, and Min Xiao in Nanjing University of Posts and Telecommunications for their assistance.

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Copyright © 2018 Fuchen Zhang et al. 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.

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