Periodic cycles in the Solow model with a delay effect

    Anatolij Kulikov Affiliation
    ; Dmitrij Kulikov Affiliation
    ; Michael Radin Affiliation


The three natural modifications of the known mathematical macroeconomics model of macroeconomics are studied in which a delay factor is presumed. This led to the replacement of the ordinary differential equation, which cannot exhibit periodic cycles on the equations with a deviating argument (functional-differential equations). It was possible to show the existence of periodic solutions that can and are intended to describe the periodic cycles in the market economy in two of the three variants of such changes in the classical form of the model.

The mathematical portion is based on the application of the modern theory of dynamical systems with an infinite-dimensional space of initial conditions. This will allow us to apply the Andronov-Hopf Theorem for equations with a deviating argument in such a form that the parameters of the cycles are located. We will also apply the well-known Krylov-Bogolyubov algorithm that is extended to infinite-dimensional dynamical systems that is used and reduces the problem to the analysis of the finite-dimensional system of ordinary differential equations-the normal Poincare-Dulac form.

Keyword : Solow model, functional-differential equations, stability, bifurcations, normal form, asymptotic formulas

How to Cite
Kulikov, A., Kulikov, D., & Radin, M. (2019). Periodic cycles in the Solow model with a delay effect. Mathematical Modelling and Analysis, 24(2), 297-310.
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Mar 18, 2019
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R. Bellman and K.L. Cooke. Differential-Difference Equations. Academic press, London, 1963.

C. Bianca and L. Guerrini. Existence of limit cycles in the Solow model with delayed-logistic population growth. The Scientific World Journal, 2014:1–8, 2014.

M. Ferrara, L. Guerrini and R. Mavilla. Modified neoclassical growth models with delay: a critical survey and perspectives. Applied Mathematical Sciences, 7(86):4249–4257, 2013.

M. Ferrara, L. Guerrini and M. Sodini. Nonlinear dynamics in a Solow model with delay and non-convex technology. Applied Mathematics and Computation, 228:1–12, 2014.

L. Gori, L. Guerrini and M. Sodini. A model of economic growth with physical and human capital: The role of time delays. Chaos, 16(4):093118, 2016.

J. Guckenheimer and P.J. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, Berlin-New-York, 1983.

J. Hale. Theory of functional differential equations. Springer-Verlag, Berlin-New-York, 1977.

M. Haragus and G. Iooss. Local Bifurcations, Center Manifolds and Normal Forms in Infinite Dimensional Dynamical Systems. EDP Sciences, Berlin, 2011.

G.E. Hutchinson. Circular causal systems in ecology. Annals of the New-York Academy of Sciences, 50(4):221–246, 1948.

A.Yu. Kolesov, A.N. Kulikov and N.Kh. Rozov. Invariant tori of a class of point mappings: the annulus principle. Differ. Equ., 39(5):614–631, 2003.

A.Yu. Kolesov, A.N. Kulikov and N.Kh. Rozov. Invariant tori of a class of point transformations: preservation of an invariant torus under perturbations. Differ. Equ., 39(6):775–790, 2003.

Yu.S. Kolesov. Harmonic self-oscillations of n-order differential equations with after effect. Vestnik Yaroslavl. univ., 7:3–88, 1974.

N.D. Kondratiev. Special opinion. Book 2. Nauka, Moscow, 1993.

A.N. Kulikov. On smooth invariant manifolds of nonlinear operator in the Banach space. Studies of stability and the theory of oscillation, pp. 67–85, 1976.

A.N. Kulikov and D.A. Kulikov. The effect of delay and the economic cycles. Taurida Journal of Computer Sciences Theory and Mathematics, 2(27):87–100, 2015.

A.N. Kulikov and D.A. Kulikov. The mathematical model of the market and the effect of delay. Mathematica v Yaroslavskom univ.Sbornik Obsor. Statey k 40-let. Matem. Faculty, pp. 132–150, 2016.

A.N. Kulikov and D. Shvitra. Bifurcations of small periodic solutions of differential equations with delay of neutral type. Differential equations and applications, 16:41–59, 1976.

D.A. Kulikov. About a mathematical model of market. IOP Conf. Series: Journal of Physics: Conference Series, 788:6p., 2017.

D.A. Kulikov. Stability and local bifurcations of the Solow model with delay. Zhurnal SVMO, 20(2):225–234, 2018.

V.V. Lebedev and K.V. Lebedev. Mathematical modelling of nonstationary processes. eTest, Moscow, 2011.

R.M. Solow. A contribution to the theory of economic growth. The Quarterly Journal of Economics, 70(1):65–94, 1956.

T.W. Swan. Economic growth and capital accumulation. Economic Record, 32:334–361, 1956.

M. Szydlowski and A. Krawiec. A mote on Kaleckian lags in the Solow model. Review of Political Economy, 16(4):501–506, 2004.

P.J. Zak. Kaleskian lags in general equilibrium. Review of Political Economy, 11(3):321–330, 1999.

W.B. Zhang. Synergetic Economics: Time and Change in Nonlinear Economics. Springer-Verlag, Berlin-New-York, 1983.