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 aswhere 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 asTaking the Taylor expansion of at 0 yieldswhere , , and .
The linear parts of (4) are as follows:Then, the characteristic of (6) isFollowing 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 haveBy (7)–(9), we can obtainThe characteristic equation of (10) is as follows:The characteristic equation of (11) iswhere
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 yieldAccording to , from (14), we haveBy simple calculation, we haveAs , 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 obtainBy simple calculation, one haswhereIf , by simple calculation, one can obtain , and then one has . From (18), one hasTaking the derivative of in (21), one can obtainwhereIf , one can obtain ; by simple calculation, one can get , . Combining with , one haswhich 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 polynomialwhere 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 aswhere 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 thatwhere 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 havefor , . 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 obtainThen, we havewhere
Define and ; then, one can obtainThen, we havewhere
Let and withfor andfor .
In the following, we define and construct a new basis Let and for ; the center space of linear equation (30) is given byAnd , where denotes the complement subspace of in .
As is the infinitesimal generator induced by the solution of (32), then (28) can be rewritten aswhereBy using the decomposition and (45), (30) can be written aswhere , , and . By using center manifold, we can obtainLet and , from (48), we can obtainwhere
Following [18], we havewherefor .
By calculation, from (52), we havewhere
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) thatAccording to [18], we havewhereBy comparing (56), (58), and (59), we haveExpanding (59), we can obtainAccording to (60), the following can be obtained:According to (60), (62), and , we can obtainwhere , , and are constant vector.
In order to seek appropriate , we can obtain the following by the definition of and (60)whereThen, we can obtainNow, 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.
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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.
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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).
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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.