Geodesic
Goodwin's business cycle model and synchronization of oscillations of two interacting economies
Čelâbinskij fiziko-matematičeskij žurnal, Tome 6 (2021) no. 2, pp. 137-151.

Voir la notice de l'article provenant de la source Math-Net.Ru

Three well-known versions of the mathematical model of the business cycle proposed by R. Goodwin are considered. To analyze them, such methods of the theory of dynamical systems as the method of integral manifolds and normal forms of A. Poincare were used. It is shown that only one of the versions of this model can have a stable cycle in the neighborhood of the state of economic equilibrium. In the other two models, cycles exist, but they are unstable. For all three considered variants of R. Goodwin's model, asymptotic formulas are obtained. The question of the competitive interaction of the two economies is also considered. It is shown that the problem can be interpreted as the problem of synchronization of oscillations for two self-oscillating systems in the presence of weak coupling. Two types of such a coupling are considered. The analysis of such a problem was reduced to the analysis of the Poincare normal form. As a result, for one of the studied problems, oscillations of two types were identified: synchronous and antiphase oscillations. The issue of their stability has been investigated. For such periodic solutions, asymptotic formulas are obtained. When constructing normal forms, the correspondingly modified Krylov — Bogolyubov algorithm was used in all cases.
Keywords: Goodwin's model, synchronous and antiphase regimes, stability, bifurcation, normal form, asymptotic formula. 
@article{CHFMJ_2021_6_2_a0,
     author = {O. V. Baeva and D. A. Kulikov},
     title = {Goodwin's business cycle model and synchronization of oscillations of two interacting economies},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
     pages = {137--151},
     publisher = {mathdoc},
     volume = {6},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_2_a0/}
}
TY  - JOUR
AU  - O. V. Baeva
AU  - D. A. Kulikov
TI  - Goodwin's business cycle model and synchronization of oscillations of two interacting economies
JO  - Čelâbinskij fiziko-matematičeskij žurnal
PY  - 2021
SP  - 137
EP  - 151
VL  - 6
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_2_a0/
LA  - ru
ID  - CHFMJ_2021_6_2_a0
ER  - 
%0 Journal Article
%A O. V. Baeva
%A D. A. Kulikov
%T Goodwin's business cycle model and synchronization of oscillations of two interacting economies
%J Čelâbinskij fiziko-matematičeskij žurnal
%D 2021
%P 137-151
%V 6
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_2_a0/
%G ru
%F CHFMJ_2021_6_2_a0
O. V. Baeva; D. A. Kulikov. Goodwin's business cycle model and synchronization of oscillations of two interacting economies. Čelâbinskij fiziko-matematičeskij žurnal, Tome 6 (2021) no. 2, pp. 137-151. http://geodesic.mathdoc.fr/item/CHFMJ_2021_6_2_a0/

[1] Zhang W. B., Synergetic Economics: Time and Change in Nonlinear Economics, Springer-Verlag, Berlin, 1991 | MR | Zbl

[2] Keynes J. M., General Theory of Employment, Interest, and Money, Harcourt Brace, New York, 1936 | MR

[3] Gudwin R. M., “The nonlinear accelerator and the persistence of business cycles”, Econometrica, 19:1 (1951), 1–17 | DOI

[4] Lorenz H. W., “Goodwin's nonlinear acceselerator and chaotic motion”, Journal of Economics, 47:4 (1987), 413–418 | DOI | MR | Zbl

[5] Lorenz H. W., Nusse H. E., “Chaotic attractors, chaotic saddles, and fractal basin boundaries: Goodwin's nonlinear accelerator model reconsidered”, Chaos, Solitons Fractals, 13 (2002), 957–965 | DOI | MR | Zbl

[6] Bashkirtseva I., Ryazanova T., Ryashko L., “Confidence domains in the analysis of noise-unduced transition to chaos for Goodwin model of business cycles”, International Journal of Bifurcation and Chaos, 24:8 (2014), 1–10 | DOI | MR

[7] Ryazanova T., Jungeilges J., “Noise-induced transitions in a stochastic Goodwin-type business cycle model”, Structural Change and Economic Dynamics, 40 (2017), 103–115 | DOI

[8] Puu T., Nonlinear Economic Dynamics, Springer-Verlag, New York, 1993 | MR | Zbl

[9] Marsden J.E., McCrasken M., The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976 | MR | Zbl

[10] Arnold V.I., Additional chapters of the theory of ordinary differential equations, Nauka Publ., Moscow, 1978 (In Russ.)

[11] Guckenheimer J., Holmes P. J., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, Berlin, 1983 | MR | Zbl

[12] Demidovich B.P., Lectures on the mathematical theory of stability, Nauka Publ., Moscow, 1967 (In Russ.) | MR

[13] Kolesov A.Yu., Kulikov A.N., Rozov N.Kh., “Invariant tori of a class of point transformations: preservation of an invariant torus under perturbations”, Differential Equations, 39:6 (2003), 775–790 | DOI | MR | Zbl

[14] Kulikov D.A., “Self-similar periodic solutions and bifurcations from them in the problem of interaction of two weakly coupled oscillators”, News of universities. Applied nonlinear dynamics, 14:5 (2006), 120–132 (In Russ.) | Zbl

[15] Kulikov D. A., “Self-similar cycles and their local bifurcations in the problem of two weakly coupled oscillators”, Journal of Applied Mathematics and Mechanics, 74:4 (2010), 389–400 | DOI | MR | Zbl

[16] Huygens C., The Pendulum Clock, English translation, orig. publ. in 1673, Iowa State University Press, Iowa, 1986 | MR