Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 2706719, 15 pages

http://dx.doi.org/10.1155/2016/2706719

## Stability and Hopf Bifurcation Analysis on a Nonlinear Business Cycle Model

Department of Management and Economics, Tianjin University, Tianjin 300072, China

Received 27 February 2016; Revised 5 June 2016; Accepted 11 July 2016

Academic Editor: Sergey A. Suslov

Copyright © 2016 Liming Zhao and Zhipei Zhao. 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

This study begins with the establishment of a three-dimension business cycle model based on the condition of a fixed exchange rate. Using the established model, the reported study proceeds to describe and discuss the existence of the equilibrium and stability of the economic system near the equilibrium point as a function of the speed of market regulation and the degree of capital liquidity and a stable region is defined. In addition, the condition of Hopf bifurcation is discussed and the stability of a periodic solution, which is generated by the Hopf bifurcation and the direction of the Hopf bifurcation, is provided. Finally, a numerical simulation is provided to confirm the theoretical results. This study plays an important role in theoretical understanding of business cycle models and it is crucial for decision makers in formulating macroeconomic policies as detailed in the conclusions of this report.

#### 1. Introduction

Accompanied with the development of an economy, increasingly, mainstream economic research has maintained a watchful eye on nonlinear dynamics theory, because its influence is spreading over both the microeconomic and macroeconomic fields. Economists are devoted to analyzing every crucial phenomenon of an economic system using economic data mining, such as irregular microeconomic fluctuation, erratic macroeconomic fluctuations, irregular economic growth, structural changes, and overlapping waves of economic development. However, to account for the limitations of these data, quantitative analysis techniques such as data mining and data analysis just scratch the surface of an economic system making it difficult for economists to conduct meaningful discussions or theoretical analysis of an economic system. Therefore, the qualitative theory of the ordinary differential equation plays an important role in analyzing macroeconomic operational mechanisms.

Among the various macroeconomic theories, economic cycles have always been an interesting field that has attracted most economists. The fluctuations of macroeconomic variables can reflect the degree of stability of the whole economic system [1–4]. Due to the historic trends of economic globalization, today there is no complete closed economy. Therefore, it is far more practical today for researchers to conduct studies on open economies. An open economy system includes both domestic and foreign economic activities, which produces the greatest influence on each part of a national economy [5–8].

This study focused on a macroeconomic dynamics model of the Kaldorian economic cycle in an open economic system based on a forecasted capital condition. Recently, many conflicting macroeconomic dynamic models have been generated. The classical model was proposed by Kaldor and the mathematical structure of the business cycle based on the Kaldorian concept has been researched extensively [9–11]. However, it must be noted that most of this research was based on closed economic systems with the exception of a few studies such as those of Lorenz, Asada, and a few others. Lorenz [12] considered the model in an open economic situation and expanded its capabilities. His research played an important role in models of business cycles in open economic situations. Lorenz proposed a multinational model (three nations) where these nations were connected by international trade. Using a numerical simulation, Lorenz found that his model would generate chaos. But this study research considered only fixed exchange rate conditions without capital movement and the physical capital stock of each nation was considered to be constant although external investment was allowed. In a sense, the model proposed by Lorenz is a short period model.

Asada [13] took another approach to study the Kaldorian business cycle model in an open economy. In Asada’s study, the movement of capital was considered and the study investigated an economic system with both fixed and flexible exchange rate conditions. He also considered the physical capital stock variations resulting from corporate investment, which is a fundamental feature of a Kaldorian business cycle model. To avoid a complicated analysis, Asada restricted his analysis to a small open economic system and the degree of capital mobility was an important parameter used to determine the kinetic properties of the system. Based on the work cited, this current study was intended to investigate the Kaldorian business cycle model with a fixed exchange condition, where the speed of market regulation and the degree of capital movement were considered as system parameters to obtain the salient dynamic properties of the model:where represents the speed of market regulation and (1) is the Kaldorian adjustment equation for a commodity market. Here, , where (, ) represents the normal Keynes consumption function, and , where (, ) represents the revenue function. In (1), (, ) represents the balance of current account and it is inversely correlated to the net real national income and positively correlated to the exchange rate :

Equation (2) is the capital accumulation equation. The rate of change of the actual physical capital stock is equal to net actual private investment in physical capital stock.

In formula (1) and (2), , where , , . represents the Kaldorian investment function and its meaning is that net investment expenditure can be expressed by the net real national income, the real physical capital stock, and the nominal interest rate, where is the nominal interest rate.

The balance function of international payments can be defined as follows:

In (3), , , represents the capital account equation. This equation illustrates the relationship as follows: (1) the balance of capital account and the degree of capital mobility; (2) the differences between domestic interest rate and others; (3) the proportional relation between the differences of expectant exchange rate and the exchange rate. Here represents the degree of capital movement, represents the expectant exchange rate, and represents the foreign nominal interest rate.

To investigate the Kaldorian model in a fixed exchange rate condition, this study assumes that

Equation (4) represents the institutional arrangement of the system in a condition of the fixed exchange rates. In other words, the exchange rate is a special constant. Equation (5) assumes that the expectant exchange rate will remain constant. Equation (6) shows that the money supply is constantly in flux accompanied by the total international balance of payments. From (1), (2), and (6), a 3-dimensional nonlinear dynamic system can be obtained which is expressed as follows:According to the research of Asada [13], the nominal interest rate function can be used in the investment functions and yieldingThen system (7) is transformed into the formwhere is actual gross domestic product, is actual consume expenditure, is the physical capital stock, is the actual income tax, is government expenditures, is nominal currency supply, is the market adjustment coefficient, and is the degree of capital movement.

Furthermore, if we denote , , , , then the model can be transferred to

It is well known that represents the gross domestic product and it is always positive. Hence, we can assume that , and the function is a smooth function in . Model (10) has right-hand sides of class .

The paper is organized as follows: in Section 2, the local stability of the equilibrium point of system is described (10), and the market adjustment coefficient and the degree of capital movement are the system parameters to determine the stable region of the system. In Section 3, the existence of Hopf bifurcation and the stability of the periodic solution generated by Hopf bifurcation based on parameters and are discussed. Finally, some conclusions are given.

#### 2. Analysis on Equilibrium State of the System

##### 2.1. Existence of Equilibrium Point and the Stability of the System

To find the stable economic growth path and obtain the relationship between economic growth, capital accumulation, and nominal currency supply, the equilibrium point of system (10) must be determined as follows:

Therefore, we can obtain the equilibrium point as , where

If the actual conditions of the system are considered, the physical capital stock , the nominal currency supply , and the gross domestic product should be positive values. Hence, the following hypothesis () is required:(*H*_{1})When all of parameters meet the hypothesis , the equilibrium point of system (10) is positive.

To determine the stability of system near the equilibrium point , according to the theory of nonlinear dynamics, the linear transformation should be considered as follows to move the equilibrium point to the original point:

Then, near original point, system (10) can be transferred into the form that

The Jacobian matrix of system (15) isAnd its characteristic polynomial iswhere According to the Routh-Hurwitz criterion, we can obtain the hypothesis:(*H*_{2})According to the theory of nonlinear dynamics, when the system meets hypotheses , , the equilibrium point is stable.

##### 2.2. Numerical Simulation of Equilibrium Point Stability

To attain the dynamic evolution behaviors and the complexity of system (10), system (10) should be considered with the following parameters based on the former theoretical analysis. According to relevant research [13] we take , , , , , , , , , , . And model (10) can be transferred into

System (20) has unique equilibrium point :From the matrix (16), the Jacobian matrix of system (20) at origin point is And its characteristic polynomial isAccording to (17), Thus, according to the Routh-Hurwitz criterion and hypothesis (19), we know that the equilibrium point of the system needs to meet ; that is,

From the plane -, we can obtain the stable boundary curve and the stable region , whose parameters, and , are shown in Figure 1. At the same time, the stable critical surface in three-dimensional space -- can be obtained. The area under the stable critical surface represents the stable region, and inversely, the area above it represents the unstable region as shown in Figure 2.