Complexity
Volume 2018 (2018), Article ID 1263602, 11 pages
https://doi.org/10.1155/2018/1263602

Research Article

Dynamic Analysis for a Kaldor–Kalecki Model of Business Cycle with Time Delay and Diffusion Effect

Wenjie Hu,1,2 Hua Zhao,1,3 and Tao Dong⁴

¹College of Economics and Business Administration, Chongqing University, Chongqing 400030, China
²College of Economics and Management, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
³School of Management, Chongqing Technology and Business University, Chongqing 4000067, China
⁴College of Electronics and Information Engineering, Southwest University, Chongqing 400715, China

Correspondence should be addressed to Hua Zhao; zhaohua@cqu.edu.cn

Received 5 April 2017; Accepted 6 December 2017; Published 9 January 2018

Academic Editor: Dimitri Volchenkov

Copyright © 2018 Wenjie Hu 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.

Abstract.

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.

1. Introduction

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 [1–8]. In order to understand the mechanisms of business cycle, many models are proposed. One of the most famous business cycle models is the Kaldor–Kalecki business cycle [9, 10], which is described as

where is the gross product, is the capital stock at time, is the adjustment coefficient in the goods market, is the depreciation rate of the capital stock, represents the propensity to save, and is the investment. Under this model, the dynamic behaviors are widely studied such as stability, Hopf bifurcation, codimension-two bifurcation, and chaos [9–15].

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:

with its initial and boundary conditions given as follows:

where , , , , and is the market capacity. There are three contributions of this paper:

(1)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.(2)The time-independent and time-dependent stability are investigated. Moreover, the conditions of the Hopf bifurcation are obtained.(3)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 2, the time-independent stability, time-dependent stability, and the existence of Hopf bifurcation are obtained. In Section 3, the normal form of Hopf bifurcation is obtained. In Section 4, numerical results are given to validate the obtained theorems.

2. Local Stability and Hopf Bifurcation Analysis

Let be an equilibrium of (2) and , , , and ; (2) can be rewrite as

Taking the Taylor expansion of at 0 yields

where , , and .

The linear parts of (4) are as follows:

Then, the characteristic of (6) is

Following the method of [16], we define as the eigenvalue of under the Neumann boundary conditions on the . Let , be the corresponding eigenvectors, where , . We use to construct a basis of the phase space . Then, can be expanded in the following form of Fourier on :

Then, we have

By (7)–(9), we can obtain

The characteristic equation of (10) is as follows:

The characteristic equation of (11) is

where

Lemma 1. If holds, the positive equilibrium of system (2) is locally stable at .

Proof. For , (12) can be rewrite as . As , , , , and , one can obtain , and which means the roots of (12) have negative part. Therefore, the positive equilibrium of system (2) is locally stable at . The proof is completed.
Substituting into (12) and separating real parts and imaginary parts yield

According to , from (14), we have

By simple calculation, we have

As , one can obtain

It is easy to see if , (15) has no positive roots. Combining with the Lemma 1, we have the following theorem.

Theorem 2. If and , the positive equilibrium of system (2) is locally stable for any .

In the following, one investigates the conditions of Hopf bifurcation of (2). By (14), one can obtain

By simple calculation, one has

where

If , by simple calculation, one can obtain , and then one has . From (18), one has

Taking the derivative of in (21), one can obtain

where

If , one can obtain ; by simple calculation, one can get , . Combining with , one has

which means increase with the increasing of . So, must exist in . Now, one considers the case . Let ; (18) with is as follows:

Since , (25) has only one positive root of (25) if . Let be the positive root of (25); then one can obtain

By (26), one can obtain .

Lemma 3 (see [16, 17]). Consider the exponential polynomial

where and are constants. As vary, the sum of the order of the zeroes of in the open right half plane can change only if a zero appears on or crosses the imaginary axis.

Theorem 4. According to Lemmas 1 and 3, one has the following.
If and holds, system (2) is asymptotically stable for . System (2) undergoes a Hopf bifurcation at the origin when ; that is, system (2) has a branch of periodic solutions bifurcating from the trivial solution near .

Remark 5. 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 [8, 9].

3. Direction and Stability of the Hopf Bifurcation

In this section, we give the normal form of Hopf bifurcation of (2) by using the method of [16, 18]. Let and normalize by ; (2) can be rewritten as

where

For , from Theorem 2, we know are the eigenvalues of the linear part of (28):

where is one parameter family of bounded linear operator in into .

Following the method of [16], we define for , such that

where

is Drac-delta function.

Let be the infinitesimal generator of the semigroup induce by the solution of (28) and denote the formal adjoint of under the bilinear pairing; we have

for , . Then and are a pair of adjoint operators.

It is not hard to see that is an eigenvalue of and is an eigenvalue of . Define and ; then, one can obtain

Then, we have

where

Define and ; then, one can obtain

Then, we have

where

Let and with

for and

for .

In the following, we define and construct a new basis

Let and for ; the center space of linear equation (30) is given by

And , where denotes the complement subspace of in .

As is the infinitesimal generator induced by the solution of (32), then (28) can be rewritten as

where

By using the decomposition and (45), (30) can be written as

where , , and . By using center manifold, we can obtain

Let and , from (48), we can obtain

where

Following [18], we have

where

for .

By calculation, from (52), we have

where

It is easy to see when . Let; compare the coefficients with (52); we can obtain

Because of and in the , we need to determine them. It follows from (52) that

According to [18], we have

where

By comparing (56), (58), and (59), we have

Expanding (59), we can obtain

According to (60), the following can be obtained:

According to (60), (62), and , we can obtain

where , , and are constant vector.

In order to seek appropriate , we can obtain the following by the definition of and (60)

where

Then, we can obtain

Now, we can calculate and ; is also expressed, and then the following important parameter can be obtained

Theorem 6. By Theorem 4, one has the following results:

(i)The sign of can determine the direction of Hopf bifurcation: if , the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solution exists for .(ii)The sign of determines the stability of the bifurcating periodic solutions: if , the bifurcation periodic solutions are stable (unstable).(iii)The sign of determines the period of the bifurcating periodic solutions: if the period increases (decreases).

4. Numerical Simulations

In this section, two simulations are given to validate the obtained theorems. Let , , , , , , and system (2) can be rewrite as follows:

the Neumann boundary conditions of system (2) is as follows:

and the initial conditions is as follows:

By simple calculations, we can obtain the equilibrium is ,  ,  , and . Consider , by Theorem 4, system (2) is asymptotically stable, which is verified in Figures 1 and 2.

(a)

(b)

Figure 1: The temporal solution of system (2) with .

(a)

(b)

(c)

Figure 2: The temporal solution of system (2) with , .

Consider ; by Theorem 4, system (2) is unstable and a Hopf bifurcation occurs. By simple calculation, we can obtain the parameter 20,  , and which means Hopf bifurcation is supercritical since . The simulations are shown in Figures 3 and 4, which indicate that there is a stable limit cycle.

(a)

(b)

Figure 3: The periodic solutions diagram of system (2) with .

(a)

(b)

(c)

Figure 4: The periodic solutions diagram of system (2) with , .

In the following, we investigate the effect of diffusion on the dynamics of system (2). Let and ; the diagrams are shown in Figures 4–7. From Figure 4, we can see that system (2) have a periodic solution when . With increasing to 0.5, the periodic solution disappear, system (2) becomes divergent, which is shown in Figure 6. Moreover, from Figures 5–7, we can see that the spatial pattern with is different from the spatial pattern with . To summarize, it can be seen the diffusion coefficient affects the pattern formation of system (2).

Figure 5: The spatial pattern of system (2) with , .

(a)

(b)

Figure 6: The temporal solution of system (2) with , .

Figure 7: The spatial pattern of system (2) with , .

5. Conclusions

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.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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.

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