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

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


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][2][3][4][5][6][7][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][10][11][12][13][14][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  > 0,  > 0,  ∈ (0, 1),  ∈ (0, 1), and Ω 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.

Theorem 4. According to Lemmas 1 and 3, one has the following.
If  1 <  and ( −  1 )( + ) − ( 1 ) < 0 holds, system (2) is asymptotically stable for  ∈ [0,  0 ).System (2) undergoes a Hopf bifurcation at the origin when  =  0 ; that is, system (2) has a branch of periodic solutions bifurcating from the trivial solution near  =  0 .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].
In order to seek appropriate  1 ,  2 , we can obtain the following by the definition of   0 and (60) where Then, we can obtain Now, we can calculate  20 () and  11 ();  21 is also expressed, and then the following important parameter can be obtained (67) Theorem 6.By Theorem 4, one has the following results: (i) The sign of  2 can determine the direction of Hopf bifurcation: if  2 > 0 ( 2 < 0), the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solution exists for  >  0 ( <  0 ).
(ii) The sign of  2 determines the stability of the bifurcating periodic solutions: if  2 < 0, ( 2 > 0) the bifurcation periodic solutions are stable (unstable).
(iii) The sign of  2 determines the period of the bifurcating periodic solutions: if  2 > 0 ( 2 < 0) the period increases (decreases).
In the following, we investigate the effect of diffusion on the dynamics of system (2).Let   = 0.1, 0.5 and  = 2.8 >  0 ; the diagrams are shown in Figures 4-7.From Figure 4, we can see that system (2) have a periodic solution when   = 0.1.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   = 0.1 is different from the spatial pattern with   = 0.5.To summarize, it can be seen the diffusion coefficient affects the pattern formation of system (2).

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.