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

(is 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.(e reaction-diffusion domain is considered, and the Galerkin analytical method is used to derive the system of ordinary differential equations. (e 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. (e delay investment parameter and diffusion coefficient can have great impacts on the Hopf bifurcations and stability of the business cycle model. (e investment parameters for the gross product and capital stock as well as the adjustment coefficient of the production market are also studied. (ese 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.


Introduction
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 [1][2][3][4], animals [5,6], health [7,8], chemicals [9,10], and business economics [11][12][13][14]. A business cycle model is utilized to explain the working of economic laws and can also be utilized to predict investment status, yield, costs, and other important factors in the business economic model. Also, it can help practitioners avoid fluctuations [15].
One of the important economic models was created by Kaldor [13] in 1940. It uses a nonlinear model constructed with a couple of ODEs, where the nonlinearity of the investment and saving functions lead to periodic limit cycle results. Furthermore, Kalecki [14] considered a time delay between an investment decision and its impact on capital accumulation in the business cycle model. It was shown that earnings are invested, and capital grows due to past investment decisions.
where Y(t) refers to the gross product; K(t) indicates the capital stock; δ and α are the depreciation value of the capital stock and the adjustment rate of the production market, respectively; I(Y, K) denotes the investment function; S(Y, K) describes the savings function; and τ > 0 is the time delay in investment due to the past investment decision. e dynamic ODE system in (1) has been studied and discussed by many researchers. Recently, Jianzhi and Hongyan [18] determined that the local stability of the positive equilibrium produces the corresponding characteristic equations. e existence of the Hopf bifurcation, the direction of this bifurcation, and the stability of the limit-cycle outcomes have been studied using numerical simulations. Wu [19,20] explored the simple-zero and double-zero singularities for the ODE equations presented in (1). ey constructed bifurcation diagrams and examined the doubleperiodic oscillation. Kaddar andTalibi Alaoui [17] proved that Hopf bifurcation points occur when the delay parameter τ is increased. Wu and Wang [21] considered the distribution of the roots of the characteristic equation of the system in (1) at the equilibrium point. ey also discussed the Hopf bifurcation and the stability of the limit cycle. Kaddar and Talibi Alaoui [12] illustrated the existence of a local Hopf bifurcation and also used an explicit algorithm to show the direction of the Hopf bifurcation and conduct a stability analysis of the system in (1). e presence of the diffusion coefficient in both equations in system (1) is extremely important as it can strongly affect the stability. It can also change the Hopf bifurcation points and therefore change the regions used for the stability analysis [1,22,23]. Blanke et al. [24] investigated a diffusive Kaldor-Kalecki business cycle system with a time delay under Neumann boundary conditions. ey conducted a stability analysis of the model and found that the time delay can give rise to the Hopf bifurcation when the delay stretches beyond a critical point. Furthermore, they showed that the diffusion coefficient played a key role in this model. Szydłowski and Krawiec [25] investigated the Kaldor-Kalecki system as a two-dimensional dynamic model. A time delay was considered for the capital accumulation equation. A qualitative analysis for the differential equations was considered for this model. Finally, they showed that there is a Hopf bifurcation leading to a limit-cycle result.
Based on the previous studies on system (1), it appears that there is a need for additional research on the effects of the diffusion value and the investment delay on the business cycle model. erefore, it is of importance to study the impacts of the diffusion and delay parameters on the stability and Hopf bifurcation. us, this paper will focus on the Kaldor-Kalecki business model with delays in the one-dimensional (1D) domain and has several significant aims. e first aim is to show what theoretical results can be obtained by utilizing the Galerkin technique. is helpful and reliable technique can help solve, and provide an excellent prediction for, the PDE system. Moreover, in order to discuss the effects of the diffusion coefficient d and the investment delay τ on the adjustment coefficient of the production market in detail, the investment parameters for the gross product and capital stock need to be examined. In addition, we also construct full-map diagrams of the Hopf bifurcation points and the stability analysis (stable and unstable regions) using examples of periods in limit cycle maps. All of these aims will help us explore the stability of the business cycle and predict whether policy makers' targets will be met. e knowledge gained may also help practitioners avoid large-scale business fluctuations.
is paper is arranged as follows. Section 2 explains the modelling of dynamic behaviour in the Kaldor-Kalecki business cycle model. Section 3 explains the methodology and the theoretical framework used to determine the Hopf bifurcation map. It also discusses how the Galerkin method can be used to create an ODE system from the PDE model. Section 4 constructs the maps of the Hopf bifurcation regions of stability for both the numerical simulations of the PDEs and the analytical outcome, thus showing the effects of the delay and diffusion parameters on the system. Finally, in Section 5, bifurcation diagrams and periodic oscillation limit-cycle maps showing both stable and unstable results are plotted to confirm the analytical outcomes.

Mathematical Model
e dynamic behaviour of the Kaldor-Kalecki business cycle model is considered by using the following replacement equations for the investment I(Y, K) and saving S(Y, K) variables: in the ODE equations in (1). For more details regarding the simple mathematical formulations used here, see [16,17,21,26] and the references therein. erefore, the nonlinear reaction-diffusion business cycle model with a time delay in the gross product and capital stock can be described with the following system: Here, parameters α, β, c, δ, and τ have the same meaning that they do in system (1). In addition, d denotes the diffusion coefficient of the system. is system is open, with a symmetrical pattern in the outcome in the middle of the domain for x � 0. In addition, Y s > 0 and K s > 0 represent the positive initial concentrations for times in the interval (−τ, 0). Note that Y s � K s � 1 in all the numerical simulations run in this model. e Runge-Kutta method [27,28] is used to determine the solutions of the ODE system. e Crank-Nicholson scheme [7,29] is considered according to the numerical simulation outcomes for PDE system (3). In the numerical simulation, spatial and temporal discretization are applied, i.e., (Δx, Δt) � (10 × 10 − 3 , 10 × 10 − 4 ).

Methodology and Theoretical Analytical Framework
In this section, we identify a reliable technique for solving nonlinear PDE models. e Galerkin analytical technique [30] is applied to a system of PDEs to produce ODE equations. is technique considers the orthogonality of the basic functions to convert the PDEs into an ODE system [10,27]. Several models have employed this technique, including the Gray and Scott cubic autocatalytic system [10], logistic equations with delays [1,2,29], and a viral infection model [7]. In general, all of the researchers who have considered this technique have obtained significant results and validated the method. In order to use the Galerkin technique, we consider the following trial functions: where (3). e free parameters in the system are then examined by computing the values for the delay PDEs. Next, the PDEs are weighted using two trial expansions: cos(η 1 ) and cos(η 2 ).

trial equation expansion (4) meets the boundary conditions in the PDE model in
e resulting system of four ODE equations is as follows: Note that the series in (4) has been abbreviated at the two-term equation point because the two-term outcome provides sufficient accuracy. It also presents excellent outcomes compared to the PDE numerical scheme system. Moreover, the one-term system can be expressed as a couple of equations by assuming that Y 2 � K 2 � 0 for the ODEs in system (5).
We can utilize the equations in system (5) to explore the existence of the Hopf bifurcation points theoretically. ese points are then used to determine a map showing the full stability analysis and any unstable and stable zones. e Hopf bifurcation denotes the periodic oscillation of the limit cycle in the neighbourhood of the steady state, which can cause a transition from a stable solution to an unstable solution. For more information, see [1,22,23,31]. Hence, the points of the Hopf bifurcation can be displayed by utilizing the Taylor series for the steady-state value points; see [7,29]. is result can be examined with Hence, the expressions Y i and K i in (6) are inserted into the ODE system in (5). Afterwards, the steady-state values are linearized. Next, the Jacobian matrix of the eigenvalues considers a small system perturbation, which demonstrates the typical growth value u by placing u � iω in the characteristic equation via dividing the characteristic equation by the real (RE) and imaginary (IM) equations. e next conditional equation then helps to determine the points in the Hopf bifurcation:

Stability and the Hopf Bifurcation Maps
is section provides the Hopf bifurcation maps and the regions of stability for both the numerical simulations for the PDEs in (3) and the theoretical outcome for the ODEs in (5). e delay parameter τ and diffusion coefficient d are studied in conjunction with the adjustment coefficient for the production market and the investment parameters for the gross product and capital stock. At the end of this section, we present some numerical examples in order to show the accuracy of the analytical outcomes. Figure 1 shows the maps of two different regions of the Hopf bifurcation for the delay parameter τ versus I (upper graph), β (middle graph), and α (lower graph). e analysis of the two-term solution (dashed line) and the numerical simulation for the PDEs (black crosses) are obtained in each case. Positive parameters are utilized in each graph, δ � 0.15, c � 0.10, and d � 0.05, where β � 1 and α � 5 (upper figure), α � 3 and I � 0.2 (middle figure), and I � 0.2 and β � 1 (lower figure). ere is a unique curve in each figure dividing the stability regions: the region above the curve indicates the unstable zone, whereas the region below the curve indicates the stable zone. Note that when the investment delay τ is increased, the critical values of the Hopf bifurcation points for the adjustment coefficient for the production market α increase steadily. Moreover, the investment parameters for the gross product I and the capital product β also increase with an increase in the value of τ. When the delay investment parameter has a very small value, the solution of the business cycle model tends to become stable and reaches a steady state. It appears that the analytical prediction corresponds to the numerical simulation of the PDEs, with less than 1% 3. e stability regions are shown. is figure shows that, as the adjustment coefficient for the production market α increases, the value of the investment in the capital product β decreases. Furthermore, Figure 2(b) provides the analytical results for the two-term solution for five different values of the delay parameter τ, namely, τ � 1, 2, 3, 4, and 5. At any selected fixed point of the adjustment coefficient for the production market α, the parameter for the investment in capital product β decreases as the delay investment parameter τ increases. Note that the resulting behaviour obtained in this figure is very similar to behaviours found in [1,7]. erefore, the investment delay term can also have a significant impact on Hopf bifurcation stability regions for this model. Figures 3(a) and 3(b) present maps of the Hopf bifurcation regions for α against the investment parameter for gross production (I). Figure 3(a) shows the theoretical outcomes (blue dashed line) as well as the numerical results (black crosses). e positive values used here are d � 0.05, δ � 0.15, c � 0.10, and β � 1. As the adjustment coefficient of production market α increases, the Hopf bifurcation points for the investment parameter for the production rate I decrease slowly. e matchup between the numerical scheme for the PDEs and the theoretical results for the ODE system is excellent. In Figure 3(b), the two-term analytical solutions are shown for five different values of the delay parameter, namely, τ � 1, 2, 3, 4, and 5. It can be seen that, as the adjustment coefficient for the production market parameter α increases, the unstable region becomes bigger than the stable area. Figure 4 presents the Hopf bifurcation maps for the diffusion coefficient in the d − I plane (upper graph), d − β plane (middle graph), and d − α plane (lower graph). In each figure, the two-term solution is shown with a red dashed line, while the black crosses indicate the numerical simulation results for the PDE model. e free parameters applied are β � 1 and α � 3 (upper graph), α � 3 and I � 0.2 (middle graph), and I � 0.2 and β � 1 (lower graph). e other parameters for all figures are τ � 1, δ � 0.15, and c � 0.10. As in Figure 1, the graphs indicate the stable and unstable regions. Each graph indicates the results of the increases in the diffusion coefficient values d. As a result, the Hopf bifurcation points for the rate of the adjustment coefficient for the production market α have also increased considerably. Furthermore, the investment parameter rates I and β (for production and capital) also increase steadily  It can be shown that, at any fixed point of α, the parameter for the investment in the capital product β increases as the diffusion coefficient d increases. Note that the resulting behaviour obtained in this figure is very similar to behaviours found in [7]. e diffusion term can also have a huge impact on the bifurcation regions in the business cycle model. Figure 6(a) determines the Hopf bifurcation maps in the τ − I plane, while the frequency of the periodic results for ω against τ is plotted in Figure 6    e parameters utilized here are δ � 0.15, c � 0.10, I � 0.2, α � 10, and β � 1. It can be seen that, as the delay in investment τ increases, the Hopf bifurcation points for the diffusion coefficient value d decrease considerably. Furthermore, the results indicate that the relationship between the diffusion value and investment delay has a very significant impact on the stability of the business cycle model in terms of investment activity. Figure 8 presents a map of the Hopf bifurcation regions in the plot of the β parameter against the investment parameter for gross production (I).

Effect of the Diffusion Coefficient Parameter d.
e two-term numerical solutions are shown. e positive values used are d � 0.05, τ � 1, δ � 0.15, c � 0.10, and α � 3. In this figure, it was found that, as the parameter for the investment in the capital product β increases, the Hopf bifurcation points for the investment in the gross product value I increase slowly up until β � 0.75. Beyond this value, the Hopf bifurcation points for the investment switch from a high-conversion state to a minimum-conversion state for the gross product I and go down until I � 0 at β ≈ 1.74. e comparisons in these figures show agreements between the analytical results for the ODEs and the simulations for the PDEs, with no more than 2% error for up to β � 2.
Lastly, a comparison is provided for the special parameter values τ � 15, α � 2, β � 1, and c � δ � d � 0.15. In this case, the points of the Hopf bifurcation of the investment parameter for gross production were examined for I c ≃2, 055, 2.101 for the analytical one-and two-term solutions, where I c ≃2.092 is used for the numerical simulation of the PDE model. e prediction for the analytical ODE system agrees with the numerical predictions for the PDE system, with less than 1% error between them at this point. Hence, the theoretical ODE system provides reliable predictions regarding the Hopf bifurcation map as well as the stability regions.

Bifurcation Diagrams and Periodic Oscillation Maps
is 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 x � 0.
Figures 9(a) and 9(b) plot the bifurcation diagrams for capital Y and gross product K with the adjustment coefficient for the production market α. e two-term solutions are shown as a blue dashed line, while the black dots indicated numerical simulation results. e parameters used are τ � 1, δ � 0.15, c � 0.10, β � 1, I � 0.2, and d � 0.05. is example shows the importance of the investment delay in changing from a steady state to an unstable one, inducing limited cyclic solutions. In both cases, the analytical Hopf bifurcation point is α c ≃4.33. All of the results for α > α c ≃4.33 are therefore unstable. After the Hopf bifurcation, the maximum amplitude over oscillation increases with growing α, while the minimum amplitude goes down. ere are good matchups between the numerical PDE results and the analytical two-term solutions over the domain of the adjustment coefficient for the production market α. Figure 10 presents the limits of the business cycle for the gross product Y(t) and capital product K(t) against time. e parameters used in Figures 10(a) Figure 7). In all these figures, the two-term solution is indicated by a black line, while the red dotted line refers to the numerical simulation. Note that the Hopf bifurcation point for the analytical outcomes in this example is d c ≃0.44 < 0.5, where d is considered to be the bifurcation parameter. When d c ≃0.44 < 0.5, the results become stable, as in Figures 10(a) and 10(b). However, at 0.4 < d c ≃0.44, the solution is unstable, as shown in Figures 10(c) and 10(d). e matchups between the analytical solutions and simulations in all of these figures are excellent.

Conclusions
is paper has provided a semianalytical outcome for the diffusive Kaldor-Kalecki model in the 1D geometry. e delay parameter was shown to exist for both gross product and capital stock functions. A system of ODEs was devolved using the Galerkin technique. e 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 α, β, and I for a stability analysis of the system. e effects of these values, which can influence the stability of the model, were studied fully. e diffusion and investment delay values have different impacts on the bifurcation maps for the business cycle model. We found that the Hopf bifurcation points for the diffusion coefficient values decreased as the investment delay parameter τ increased. e results were examined by exploring several different numerical examples of the limit cycle and could help in the development of policy maker expectations and avoiding economic fluctuations. is method is therefore an extremely helpful, significant, and effective analytical technique for examining PDE models with delays. e technique provides good outcomes for all of the scenarios used in this work. In the future, we are planning to apply this method to the same model with an added delay feedback control term.

Data Availability
e data used to support the findings of this study are available upon request to the author.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.