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.

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 [

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 [

Asada [

Equation (

In formula (

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

In (

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

Equation (

Furthermore, if we denote

It is well known that

The paper is organized as follows: in Section

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 (

Therefore, we can obtain the equilibrium point as

If the actual conditions of the system are considered, the physical capital stock

To determine the stability of system near the equilibrium point

Then, near original point, system (

The Jacobian matrix of system (

To attain the dynamic evolution behaviors and the complexity of system (

System (

From the plane

Stable region

Stable critical surface

The solid line in Figure

However, when the parameter pair

History and phase diagram of system with

History and phase diagram of system with

As is previously mentioned, as the parameters

According to the theory of nonlinear dynamics, if the characteristic equation of system (

Substitute

If system (

In a similar way, with a fixed

If system (

A continuous time nonlinear dynamic system is shown as follows:

If matrix

Thus, we obtain the unique expression of the first Lyapunov coefficient:

Accordingly, the Jacobian matrix

Assuming that matrixes

In a similar way, we can obtain the characteristic

According to the definition of the linear function we can obtain the bilinear function

Then we acquire the real part of the first Lyapunov coefficient of system (

Continuing the investigation of system (

Bifurcation curve diagram.

We will investigate the existence of Hopf bifurcation of the system in the two aspects below.

(1) Fix

Let

History and phase diagram of the system when

History and phase diagram of the system when

Let

When

History and phase diagram of the system when

History and phase diagram of the system when

Let

(2) Fix

Let

Let

History and phase diagram of the system when

History and phase diagram of the system when

History and phase diagram of the system when

History and phase diagram of the system when

We continued to analyze the properties of the Hopf bifurcation generated by the system. In the analysis of the stability, the critical value was found, where the system varies from a stable status to an unstable status. In addition, the bifurcation equation showed that the system could generate Hopf bifurcation. To acquire the stability of the limited cycle generated by the Hopf bifurcation, the first Lyapunov coefficient must be calculated.

Figure

First Lyapunov coefficient diagram with parameters

Parameter

First Lyapunov coefficient diagram with parameters

First Lyapunov coefficient diagram with parameters

Fixing the parameter as a constant to observe the variations of the first Lyapunov coefficient, as is shown in Figure

First Lyapunov coefficient diagram with parameters

First Lyapunov coefficient diagram with parameters

Fixing the parameter

Nonlinear dynamic finance and economic models offer rich dynamic behavior. The results of the analysis of models are important from theoretical and practical perspectives whether the viewpoint is of a nonlinear system or implementation in macroeconomic policy.

In this reported study, a three-dimension nonlinear Kaldorian business cycle model was established based on a condition of fixed exchange rates. First, by solving a specific model, the resulting system had a unique equilibrium point. Second, using the speed of market regulation and the degree of capital movement as the variables in the system, the stability of the model is discussed and obtained when

The authors declare that there are no competing interests regarding the publication of this paper.