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.
Highlights
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]
We find that the time delay can give rise to the Hopf bifurcation when the time delay passes a critical value
Numerical results validate the obtained theorems, and show that the diffusion coefficients play a key role in the spatial pattern
Summary
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]. Where Y(t) is the gross product, K(t) is the capital stock at time, α is the adjustment coefficient in the goods market, q is the depreciation rate of the capital stock, γ represents the propensity to save, and I(Y(t), K(t)) 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]. 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:. (1) Based on the Kaldor–Kalecki model, we propose a novel business cycle under the Neumann boundary conditions.
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