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

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

Generalised diffusive delay logistic equations: Semi-analytical solutions

This paper considers semi-analytical solutions for a class of generalised logis- tic partial dierential equations with both point and distributed delays. Both one and two-dimensional geometries are considered. The Galerkin method is used to approximate the governing equations by a system of ordinary dierential delay equations. This method involves assuming a spatial structure for the solution and averaging to obtain the ordinary dierential delay equation models. Semi-analytical results for the stability of the system are derived with the critical parameter value, at which a Hopf bifurcation occurs, found. The results show that diusion acts to stabilise the system, compared to equivalent non- diusive systems and that large delays, which represent feedback from the distant past, act to destabilize the system. Comparisons between semi-analytical and numerical solutions show excellent agreement for steady state and transient solutions, and for the parameter values at which the Hopf bifurcations occur