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The dynamics behaviors of Kaldor–Kalecki business cycle model with diffusion effect and time delay under the Neumann boundary conditions are investigated. First the conditions of time-independent and time-dependent stability are investigated. Then, we find that the time delay can give rise to the Hopf bifurcation when the time delay passes a critical value. Moreover, the normal form of Hopf bifurcations is obtained by using the center manifold theorem and normal form theory of the partial differential equation, which can determine the bifurcation direction and the stability of the periodic solutions. Finally, numerical results not only validate the obtained theorems, but also show that the diffusion coefficients play a key role in the spatial pattern. With the diffusion coefficients increasing, different patterns appear.

Recently, business cycle, as one of the important economic phenomena, has received attractive attentions due its widely application in many fields such as economic decisions, macroeconomic regulation, and market regulation [

It is well known that diffusion effects of economic activities are widespread phenomenon that existed all over the world. As a result of the impact of the growth pole, the diffusion effects are the main interactions in economic activities. So, the diffusion effect should be considered in the business cycle model. However, to the best of our knowledge, there are very few works on this field. Inspired by the observation, in this paper, based on the Kaldor–Kalecki model, we propose a novel business cycle with diffusion effect and time delay under the Neumann boundary conditions, which is as follows:

Based on the Kaldor–Kalecki model, we propose a novel business cycle under the Neumann boundary conditions. Our model is a spatial-temporal model, which is more general than the existing models.

The time-independent and time-dependent stability are investigated. Moreover, the conditions of the Hopf bifurcation are obtained.

It is found that the diffusion coefficients play a key role in the spatial pattern. With the diffusion coefficients increasing, different patterns appear.

The rest paper is organized as follows. In Section

Let

The linear parts of (

If

For

Substituting

If

In the following, one investigates the conditions of Hopf bifurcation of (

Consider the exponential polynomial

According to Lemmas

If

By incorporate diffusion effect into the Kaldor–Kalecki model of business cycle, a novel Kaldor–Kalecki model of business cycle with diffusion effect and time delay is proposed. Our model is a spatial-temporal model, which is more general than the existing business cycle [

In this section, we give the normal form of Hopf bifurcation of (

Following the method of [

Let

It is not hard to see that

Define

Let

In the following, we define

As

Following [

By calculation, from (

It is easy to see

Because of

In order to seek appropriate

By Theorem

The sign of

The sign of

The sign of

In this section, two simulations are given to validate the obtained theorems. Let

By simple calculations, we can obtain the equilibrium is

The temporal solution of system (

The temporal solution of system (

Consider

The periodic solutions diagram of system (

The periodic solutions diagram of system (

In the following, we investigate the effect of diffusion on the dynamics of system (

The spatial pattern of system (

The temporal solution of system (

The spatial pattern of system (

It is well known that diffusion effects of economic activities are widespread phenomenon that existed all over the world. As a result of the impact of the growth pole, the diffusion effects are the main interactions in economic activities. So, the diffusion effect should be considered in the business cycle model. In this paper, we consider a Kaldor–Kalecki business cycle model with diffusion effect and time delay under the Neumann boundary conditions. First the time-independent and time-dependent stability are investigated. Then, we find that the time delay can give rise to the Hopf bifurcation when the time delay passes a critical value. Moreover, the normal form of Hopf bifurcation is obtained. Finally, numerical results not only validate the obtained theorems, but also show that the diffusion coefficients play a key role in the spatial pattern. With the diffusion coefficients increasing, different patterns appear.

The authors declare that they have no conflicts of interest.

This work was supported in part by the National Natural Science Foundation of China under Grant 61503310, in part by the Fundamental Research Funds for the Central Universities under Grants 2017CDJSK02XK24, 2017CDJKY, and 106112015CDJSK02JD06, in part by China Postdoctoral Foundation 2016M600720, and in part by Chongqing Postdoctoral Project under Grant Xm2016003.