Semi‐Analytical Solutions for the Diffusive Kaldor–Kalecki Business Cycle Model with a Time Delay for Gross Product and Capital Stock

Abstract

This paper discusses the stability and Hopf bifurcation analysis of the diffusive Kaldor–Kalecki model with a delay included in both gross product and capital stock functions. The reaction-diffusion domain is considered, and the Galerkin analytical method is used to derive the system of ordinary differential equations. The methodology used to determine the Hopf bifurcation points is discussed in detail. Furthermore, full diagrams of the Hopf bifurcation regions considered in the stability analysis are shown, and some numerical simulations of the limit cycle are used to confirm the theoretical outcomes. The delay investment parameter and diffusion coefficient can have great impacts on the Hopf bifurcations and stability of the business cycle model. The investment parameters for the gross product and capital stock as well as the adjustment coefficient of the production market are also studied. These parameters can cause instability in, and the stabilization of, the business cycle model. In addition, we point out that, as the delay investment parameter increases, the Hopf bifurcation points for the diffusion coefficient values decrease considerably. When the delay investment parameter has a very small value, the solution of the business cycle model tends to become steady.

Stability analysis and Hopf bifurcation for two-species reaction-diffusion-advection competition systems with delays

“…The Galerkin technique has been proven on several models, such as the viral infection model [7], Nicholson's blowfly model [14], the business cycle system [11], classes of delay logistic equations [8,12,13], the limited-food model [6], neural network model [9], Gray and Scott's system [48], and the Belousov-Zhabotinsky model [15]. Overall, the outcomes demonstrate strong agreement between analytical and numerical solutions.…”

“…Stability analysis and bifurcation diagrams were investigated and compared with numerical simulations. [11] considered analytical solutions of the Kaldor-Kalecki model in one dimension, with delay in the gross product and capital stock functions. The diffusion and delay values were found to exert several influences on the business cycle's stability, and the Hopf lines for the diffusion ratios decreased as investment was increasingly delayed.…”