International Journal of Differential Equations

Direction and Stability of Hopf Bifurcation in a Delayed Solow Model with Labor Demand

Sanaa ElFadily, Abdelilah Kaddar, and Khalid Najib

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This paper is concerned with a delayed model of mutual interactions between the economically active population and the economic growth. The main purpose is to investigate the direction and stability of the bifurcating branch resulting from the increase of delay. By using a second order approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points and we show that the system under consideration can undergo a supercritical or subcritical Hopf bifurcation and the bifurcating periodic solution is stable or unstable in a neighborhood of some bifurcation points, depending on the choice of parameters.

Article information

Int. J. Differ. Equ., Volume 2019 (2019), Article ID 7609828, 8 pages.

Received: 11 February 2019
Revised: 24 April 2019
Accepted: 8 May 2019
First available in Project Euclid: 24 July 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)


ElFadily, Sanaa; Kaddar, Abdelilah; Najib, Khalid. Direction and Stability of Hopf Bifurcation in a Delayed Solow Model with Labor Demand. Int. J. Differ. Equ. 2019 (2019), Article ID 7609828, 8 pages. doi:10.1155/2019/7609828.

Export citation


  • R. F. Harrod, “An essay in dynamic theory,” The Economic Journal, vol. 49, no. 193, pp. 14–33, 1939.
  • E. D. Domar, “Expansion and employment,” in The American Economic Review, vol. 37, pp. 34–55, American Economic Association, 1947.
  • C. Gustav, “Capital and income in the money economy,” in The Theory of Social Economy, p. 5117, Augustus M. Kelley, New York, NY, USA, 1967.
  • R. M. Solow, “A contribution to the theory of economic growth,” The Quarterly Journal of Economics, vol. 70, no. 1, pp. 65–94, 1956.
  • S. Hallegatte, M. Ghil, P. Dumas, and J.-C. Hourcade, “Business cycles, bifurcations and chaos in a neo-classical model with investment dynamics,” Journal of Economic Behavior & Organization, vol. 67, no. 1, pp. 57–77, 2008.
  • D. Cai, “Multiple equilibria and bifurcations in an economic growth model with endogenous carrying capacity,” International Journal of Bifurcation and Chaos, vol. 20, no. 11, pp. 3461–3472, 2010.MR2765073
  • D. Cai, “An economic growth model with endogenous carrying capacity and demographic transition,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 432–441, 2012.MR2887388
  • L. Guerrini and M. Sodini, “Nonlinear dynamics in the solow model with boundedpopulation growth and time-to-build technology,” Abstract and Applied Analysis, vol. 2013, Article ID 836537, 6 pages, 2013.MR3129348
  • S. ElFadily, A. Kaddar, and K. Najib, “Dynamics of a delayed solow model with effective labor demand,” Journal of Advances in Applied Mathematics, vol. 1, no. 3, pp. 175–182, 2016.
  • V. Jablanovic, “A chaotic economic growth model and the agricultural share of an output,” Journal of Agricultural Sciences, Belgrade, vol. 50, no. 2, pp. 207–216, 2005.
  • D. Guégan, “Chaos in economics and finance,” Annual Reviews in Control, vol. 33, no. 1, pp. 89–93, 2009.
  • M. Akhmet, Z. Akhmetova, and M. O. Fen, “Chaos in economic models with exogenous shocks,” Journal of Economic Behavior & Organization, vol. 106, pp. 95–108, 2014.
  • L. Zhao and Z. Zhao, “Stability and Hopf bifurcation analysis on a nonlinear business cycle model,” Mathematical Problems in Engineering, vol. 2016, Article ID 2706719, 15 pages, 2016.MR3541751
  • A. Sulaiman and M. Sadly, “Agricultural growth modeling based on nonlinear dynamical system,” in Proceedings of the International Conference on Advanced Computer Science and Information Systems, IEEE, Depok, Indonesia, 2013.
  • L. Guerrini, “Hopf bifurcation in a delayed Ramsey model with Von Bertalanffy population law,” International Journal of Differential Equations and Applications, vol. 11, no. 1, pp. 81–86, 2012.
  • D. Cai, H. Ye, and L. Gu, “A generalized solow-swan model,” Abstract and Applied Analysis, vol. 2014, Article ID 395089, 8 pages, 2014.MR3193506
  • M. Bohner, J. Heim, and A. Liu, “Qualitative analysis of a Solow model on time scales,” Journal of Concrete and Applicable Mathematics, vol. 13, no. 3-4, pp. 183–197, 2015.MR3307345
  • A. Akaev, “Nonlinear differential equation of macroeconomic dynamics for long-term forecasting of economic development,” Applied Mathematics, vol. 9, no. 5, pp. 512–535, 2018.
  • Y. Basnett and R. Sen, What Do Empirical Studies Say about Economic Growth and Job Creation in Developing Countries? London, UK, ODI, 2013.
  • Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, vol. 112 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1998.MR1711790
  • C. W. Cobb and P. H. Douglas, “A theory of production,” American Economic Review, vol. 18, no. 1, pp. 139–165, 1928.
  • J. K. Hale, Introduction to Functional-Differential Equations, Springer-Verlag, New York, NY, USA, 1993.MR1243878 \endinput