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.

For a long period of time, many significant nonlinear phenomena have been modelled and described via ordinary or partial differential equations (ODEs or PDEs). For example, these equations have been used to model population ecology [

One of the important economic models was created by Kaldor [

In 1999, Kaddar and Szydlowski [

The dynamic ODE system in (

The presence of the diffusion coefficient in both equations in system (

Based on the previous studies on system (

This paper is arranged as follows. Section

The dynamic behaviour of the Kaldor–Kalecki business cycle model is considered by using the following replacement equations for the investment

Here, parameters

In this section, we identify a reliable technique for solving nonlinear PDE models. The Galerkin analytical technique [

In order to use the Galerkin technique, we consider the following trial functions:

Note that the series in (

We can utilize the equations in system (

Hence, the expressions

This section provides the Hopf bifurcation maps and the regions of stability for both the numerical simulations for the PDEs in (

Figure

The regions produced by the Hopf bifurcation curves for the delay parameter

Figures

An exploration of the Hopf bifurcation maps in the

Figures

The two regions created by the Hopf bifurcation in the

Figure

The Hopf bifurcation curves for

Figure

The Hopf bifurcation maps on the

Figure

(a) The Hopf bifurcation maps in the

Figure

The Hopf bifurcation curve on the

Figure

The Hopf bifurcation curves in the

Lastly, a comparison is provided for the special parameter values

This section focuses on the steady-state results as well as the bifurcation diagrams, periodic results, and 2D phase-plane map. In addition, the map of the bifurcation diagrams is considered of the domain in the centre

Figures

(a, b) The bifurcation diagrams for the capital

Figure

The business cycle of gross product

This paper has provided a semianalytical outcome for the diffusive Kaldor–Kalecki model in the 1D geometry. The delay parameter was shown to exist for both gross product and capital stock functions. A system of ODEs was devolved using the Galerkin technique. The Hopf bifurcation points were found for dividing the graphs into two stability regions. Furthermore, we displayed full-map graphs for the delay parameter and diffusion coefficient values versus the parameters

The data used to support the findings of this study are available upon request to the author.

The author declares that there are no conflicts of interest regarding the publication of this paper.

The author carried out the proofs of the main results and approved the final manuscript.

The author wishes to thank the editor: Baogui Xin, for the useful comments.