Center manifold reduction and perturbation method in a delayed model with a mound-shaped Cobb-Douglas production function. (English) Zbl 1470.34186

Summary: A. Matsumoto and F. Szidarovszky [“Delay differential neoclassical growth model”, J. Econ. Behav. Organ. 78, No. 3, 272–289 (2011; doi:10.1016/j.jebo.2011.01.014)] examined a delayed continuous-time growth model with a special mound-shaped production function and showed a Hopf bifurcation that occurs when time delay passes through a critical value. In this paper, by applying the center manifold theorem and the normal form theory, we obtain formulas for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Moreover, Lindstedt’s perturbation method is used to calculate the bifurcated periodic solution, the direction of the bifurcation, and the stability of the periodic motion resulting from the bifurcation.


34K18 Bifurcation theory of functional-differential equations
91B38 Production theory, theory of the firm
91B55 Economic dynamics
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[1] Asea, P. K.; Zak, P. J., Time-to-build and cycles, Journal of Economic Dynamics & Control, 23, 8, 1155-1175, (1999) · Zbl 1016.91068
[2] Zak, P. J., Kaleckian lags in general equilibrium, Review of Political Economy, 11, 321-330, (1999)
[3] Szydłowski, M., Time to build in dynamics of economic models. II. Models of economic growth, Chaos, Solitons and Fractals, 18, 2, 355-364, (2003) · Zbl 1056.91050
[4] Szydłowski, M.; Krawiec, A., A note on Kaleckian lags in the Solow model, Review of Political Economy, 16, 4, 501-506, (2004)
[5] Matsumoto, A.; Szidarovszky, F., Delay differential neoclassical growth model, Journal of Economic Behavior and Organization, 78, 3, 272-289, (2011)
[6] Matsumoto, A.; Szidarovszky, F.; Yoshida, H., Dynamics in linear cournot duopolies with two time delays, Computational Economics, 38, 3, 311-327, (2011) · Zbl 1282.91202
[7] d’Albis, H.; Augeraud-Veron, E.; Venditti, A., Business cycle fluctuations and learning-by-doing externalities in a one-sector model, Journal of Mathematical Economics, 48, 5, 295-308, (2012) · Zbl 1260.91149
[8] Bambi, M.; Fabbri, G.; Gozzi, F., Optimal policy and consumption smoothing effects in the time-to-build AK model, Economic Theory, 50, 3, 635-669, (2012) · Zbl 1246.91082
[9] Boucekkine, R.; Fabbri, G.; Pintus, P., On the optimal control of a linear neutral differential equation arising in economics, Optimal Control Applications & Methods, 33, 5, 511-530, (2012) · Zbl 1311.49064
[10] Matsumoto, A.; Szidarovszky, F., Nonlinear delay monopoly with bounded rationality, Chaos, Solitons and Fractals, 45, 4, 507-519, (2012)
[11] Ballestra, L. V.; Guerrini, L.; Pacelli, G., Stability switches and Hopf bifurcation in a Kaleckian model of business cycle, Abstract and Applied Analysis, 2013, (2013)
[12] Bianca, C.; Guerrini, L., On the Dalgaard-Strulik model with logistic population growth rate and delayed-carrying capacity, Acta Applicandae Mathematicae, 128, 39-48, (2013) · Zbl 1283.91124
[13] Bianca, C.; Ferrara, M.; Guerrini, L., Stability and bifurcation analysis in a Solow growth model with two time delays
[14] Guerrini, L.; Sodini, M., Nonlinear dynamics in the Solow model with bounded population growth and time-to-build technology, Abstract and Applied Analysis, 2013, (2013)
[15] Guerrini, L.; Sodini, M., Dynamic properties of the Solow model with increasing or decreasing population and time-to-build technology · Zbl 1298.34158
[16] Matsumoto, A.; Szidarovszky, F., Asymptotic behavior of a delay differential neoclassical growth model, Sustainability, 5, 2, 440-455, (2013)
[17] Day, R., Irregular growth cycles, American Economic Review, 72, 406-414, (1982)
[18] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and Applications of Hopf Bifurcation, 41, (1981), Cambridge, Mass, USA: Cambridge University Press, Cambridge, Mass, USA · Zbl 0474.34002
[19] Nayfeh, A. H., Introduction to Perturbation Techniques, (1981), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0449.34001
[20] MacDonald, N., Time Lags in Biological Models. Time Lags in Biological Models, Lecture Notes in Biomathematics, 27, (1978), Berlin, Germany: Springer, Berlin, Germany · Zbl 0403.92020
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