Modeling the Chaos and Bifurcation in Solow’s Business Trade Cycle Model by Using Delay Differential Equations

Dipesh; Pankaj Kumar

doi:10.53941/nacs.2026.100008

Abstract

This research extends the traditional Solow model by introducing a delay parameter via a delay differential equation to examine the dynamics of business cycles. Alongside the Solow farmwork, the Harrod-Domar model and a modified Solow model are considered to provide a broader perspective on growth dynamics and stability analysis. This study reveals that the inclusion of delays causes fluctuations in stability, leading to Hopf bifurcation, limit cycles, and chaotic behavior, thereby capturing the complex evolution of the trade cycle. The dynamics highlight how minor changes in the system parameters can reshape long-term economic trajectories. The analysis, conducted using MATLAB, underscores the significance of the Solow and Harrod-Domar models, as well as their variants, for understanding economic growth and industrial development. This approach helps stakeholders predict and mitigate economic changes by identifying key thresholds and dynamic patterns, thereby promoting resilience, stability, and long-term growth.

References

1.
Samuelson, P.A. Foundations of Economic Analysis; Harvard University Press: Cambridge, MA, USA, 1983; Volume 197.
2.
Gandolfo, G. Economic Dynamics: Study Edition; Springer Science & Business Media: Berlin/Heidelberg, Germany, 1997.
3.
Brock, W.A.; Malli, A.G. Differential Equations, Stability and Chaos in Dynamic Economics; North Holland: Amsterdam, The Netherlands, 1989.
4.
Acemoglu, D. Introduction to Modern Economic Growth; Princeton University Press: Princeton, NJ, USA, 2008.
5.
Acemoglu, D. Economic growth and development in the undergraduate curriculum. J. Econ. Educ. 2013, 44, 169–177.
6.
Chen, Q.; Dipesh, K.P.; Baskonus, H.M. On the equilibrium point and Hopf-Bifurcation analysis of GDP-national debt dynamics under the delayed external investment: A new DDE model. Alex. Eng. J. 2024, 91, 510–515.
7.
Chen, Q.; Dipesh, K.P.; Baskonus, H.M. Modeling and analysis of demand-supply dynamics with a collectability factor using delay differential equations in economic growth via the Caputo operator. AIMS Math. 2024, 9, 7471–7491.
8.
Dmitriev, V.I.; Kurkina, E.S. The Inverse Problem of Determining the Parameters of a Mathematical Model Linking GDP with National Debt. Comput. Math. Math. Phys. 2020, 31, 75–95.
9.
Fourie, D.J.; Blom, P.P.; Challenges, Strategies and Solutions to Manage Public Debt in South Africa. Afr. J. Public Aff. 2022, 13, 27–53.
10.
Străchinariu, A.V. The Impact of Macroeconomic Indicators on Public Debt Dynamics. Ann. Univ. Apulensis Ser. Oecon. 2021, 23, 1–9.
11.
Li, C.; Wan, J. The Influence of Enterprise Financing Structure in China on Business Performance—Simulation Analysis Based on the System Dynamics.In Proceedings of the 12th EAI International Conference, SIMUtools 2020, Guiyang, China, 28–29 August 2020; pp. 728–746.
12.
Srinivasa, K.; Mundewadi, R.A. Wavelets approach for the solution of nonlinear variable delay differential equations. Int. J. Math. Comput. Eng. 2023, 1, 139–148.
13.
Bilal, M.; Haris, H.; Waheed, A.; Faheem, M. The analysis of exact solitons solutions in monomode optical fibers to the generalized nonlinear Schrödinger system by the compatible techniques. Int. J. Math. Comput. Eng. 2023, 1, 149–170.
14.
Kumar, A.; Kumar, S. Dynamic nature of analytical soliton solutions of the (1+1)-dimensional Mikhailov-Novikov-Wang equation using the unified approach. Int. J. Math. Comput. Eng. 2023, 1, 217–228.
15.
Chen, Q.; Baskonus, H.M.; Gao, W.; Ilhan, E. Soliton theory and modulation instability analysis: The Ivancevic option pricing model in economy. Alex. Eng. J. 2022, 61, 7843–7851.
16.
Chen, Q.; Sabir, Z.; Raja, M.A.Z.; et al. A fractional study based on the economic and environmental mathematical model. Alex. Eng. J. 2023, 65, 761–770.
17.
Shananin, A.A.; Tarasenko, M.V.; Trusov, N.V. Mathematical modeling of household economy in Russia. Comput. Math. Math. Phys. 2021, 61, 1030–1051.
18.
Gimaltdinov, I.F. Research of the demand for consumer loans and money. Mat. Model. 2012, 24, 84–98.
19.
Mazloumfard, H.; Glantz, V. The influence of tax burden on the profit of banks in conditions of monopolistic competition: Economic-mathematical modeling. Financ. Mark. Inst. Risks 2017, 1, 28–36.
20.
Tadmon, C.; Njike Tchaptchet, E.R. Financial crisis spread, economic growth and unemployment: A mathematical model. Stud. Nonlinear Dyn. Econom. 2023, 27, 147–170.
21.
Arabov, N.; Nasimov, D.; Khuzhayorov, H.; et al. Modelling of commercial banks capitals competition dynamics. Int. J. Early Child. Spec. Educ. 2022, 14, 4124–4132.
22.
Wang, W.; Khan, M.A.; Kumam, P.; et al. A comparison study of bank data in fractional calculus. Chaos Solitons Fractals 2019, 126, 369–384.
23.
Selyutin, V.V.; Rudenko, M.A. Mathematical model of the banking firm as tool for analysis, management and learning. J. Inf. Technol. Educ. 2013, 16, 170–177.
24.
Comes, C.A. Banking system: Three level Lotka-Volterra model. Procedia Econ. Financ. 2012, 3, 251–255.
25.
Marasco, A.; Picucci, A.; Romano, A. Market share dynamics using Lotka–Volterra models. Technol. Forecast. Soc. Change 2016, 105, 49–62.
26.
Ruan, S. Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays. Q. Appl. Math. 2001, 59, 159–173.
27.
Segura, J.; Franco, D.; Perán, J. Long-run economic growth in the delay spatial Solow model. Spat. Econ. Anal. 2023, 18, 158–172.
28.
Kulikov, A.; Kulikov, D.; Radin, M. Periodic cycles in the Solow model with a delay effect. Math. Model. Anal. 2019, 24, 297–310.
29.
Borges, M.J.; Fabião, F.; Teixeira, J. Long Cycles Versus Time Delays in a Modified Solow Growth Model. In Eurasian Economic Perspectives; Springer: Cham, Switzerland, 2020; pp. 375–383.
30.
Tian, N.; Huang, D. Dynamical analysis of the Solow model with time delay and pollution. In Proceedings of the Second International Conference on Electrical, Electronics, and Information Engineering (EEIE 2023), Wuhan, China, 2–4 November 2023; pp. 651–656.
31.
Ferrara, M.; Guerrini, L.; Mavilia, R. Modified neoclassical growth models with delay: A critical survey and perspectives. Appl. Math. Sci. 2013, 7, 4249–4257.
32.
Harrod, R.F. An essay in dynamic theory. Econ. J. 1939, 49, 14–33.
33.
Domar, E. Capital Expansion, Rate of Growth, and Employment. Econometrica 1946, 14, 137–147.
34.
Samuelson, P.A. The stability of equilibrium: Comparative statics and dynamics. Econometrica 1941, 9, 97–120.
35.
Solow, R.M. A contribution to the theory of economic growth. Q. J. Econ. 1956, 70, 65–94.
36.
Inada, K.I. On a two-sector model of economic growth: Comments and a generalization. Rev. Econ. Stud. 1963, 30, 119–127.

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