Existence and stability of business cycles
Author:
Akitaka Dohtani
Journal:
Quart. Appl. Math. 54 (1996), 105-120
MSC:
Primary 90A16; Secondary 34C05
DOI:
https://doi.org/10.1090/qam/1373841
MathSciNet review:
MR1373841
Full-text PDF Free Access
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Abstract: We provide several sufficient conditions for the existence and the stability of limit cycles in two-dimensional differential equation systems. We also apply our results to business cycle models in the Keynesian tradition.
J.P. Benassy, A non-Walrasian model of the business cycle, J. Econom. Behavior Org. 5, 77–89 (1984)
L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Ergebnisse der Mathematik und ihrer Grenzgebiete, 2nd ed., Springer-Verlag, Berlin-Gottingen-Heidelberg, 1959
W. W. Chang and D. J. Smyth, The existence and persistence of cycles in a non-linear model: Kaldor’s 1940 model re-examined, Rev. Econom. Stud. 38, 37–44 (1971)
A. F. Filippov, A sufficient condition for the existence of a stable limit cycle for a second order equation, Mat. Sb. 30, 171–180 (1952) (Russian)
G. Gabisch and H.-W. Lorenz, Business Cycle Theory, Lecture Notes in Econom. Math. Systems, Vol. 283, Springer-Verlag, Berlin, 1987
M. Galeotti and F. Gori, Uniqueness of periodic orbits in Liénard-type business-cycle models, Metroeconomica 40, 135–146 (1989)
R. M. Goodwin, The nonlinear accelerator and the persistence of business cycles, Econometrica 19, 1–17 (1951)
S. Ichimura, Towards a general non-linear macrodynamic theory of economic fluctuations, Post-Keynesian Economics (K. K. Kurihara, ed.), Rutgers University Press, New Brunswick, 1955
N. Kaldor, A model of the trade cycle, Econom. J. 50, 78–92 (1940)
N. Levinson and O. K. Smith, A general equation for relaxation oscillations, Duke Math. J. 9, 382–403 (1942)
H. W. Lorenz, On the uniqueness of limit cycles in business cycle theory, Metroeconomica 38, 281–293 (1987)
H. W. Lorenz, Nonlinear Dynamical Economics and Chaotic Motion, Lecture Notes in Econom. Math. Systems, Vol. 334, Springer-Verlag, Berlin, 1989
S. Mizohata and M. Yamaguchi, On the existence of periodic solutions of the nonlinear differential equation $\ddot x + a\left ( x \right )\dot x + \Phi \left ( x \right ) = p\left ( t \right )$, Mem. Coll. Sci. Univ. Kyoto, Ser. A 27, 109–113 (1952)
T. Owase, Dynamical system theory and analysis of economic fluctuations, The Study of Dynamical Systems (N. Aoki, ed.), World Scientific, Singapore, 1989
T. Owase, Nonlinear dynamical systems and economic fluctuations: A brief historical survey, Trans. IEICE. E74, 1393–1400 (1991)
G. Sansone and R. Conti, Non-linear Differential Equations, Macmillan, New York, 1964
G. J. Schinasi, Fluctuations in a dynamic, intermediate-run IS-LM model: Applications of the Poincaré-Bendixon theorem, J. Econom. Theory 28, 369–375 (1982)
Ye Yanqian, Theory of Limit Cycles, Translations of Mathematical Monographs, Vol. 66, Amer. Math. Soc., Providence, R.I., 1986
T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Mathematical Society of Japan, Tokyo, 1966
Zhang Zhifen, On the uniqueness of the limit cycles of some nonlinear oscillation equations, Dokl. Akad. Nauk. SSSR 119, 659–662 (1958) (Russian)
J.P. Benassy, A non-Walrasian model of the business cycle, J. Econom. Behavior Org. 5, 77–89 (1984)
L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Ergebnisse der Mathematik und ihrer Grenzgebiete, 2nd ed., Springer-Verlag, Berlin-Gottingen-Heidelberg, 1959
W. W. Chang and D. J. Smyth, The existence and persistence of cycles in a non-linear model: Kaldor’s 1940 model re-examined, Rev. Econom. Stud. 38, 37–44 (1971)
A. F. Filippov, A sufficient condition for the existence of a stable limit cycle for a second order equation, Mat. Sb. 30, 171–180 (1952) (Russian)
G. Gabisch and H.-W. Lorenz, Business Cycle Theory, Lecture Notes in Econom. Math. Systems, Vol. 283, Springer-Verlag, Berlin, 1987
M. Galeotti and F. Gori, Uniqueness of periodic orbits in Liénard-type business-cycle models, Metroeconomica 40, 135–146 (1989)
R. M. Goodwin, The nonlinear accelerator and the persistence of business cycles, Econometrica 19, 1–17 (1951)
S. Ichimura, Towards a general non-linear macrodynamic theory of economic fluctuations, Post-Keynesian Economics (K. K. Kurihara, ed.), Rutgers University Press, New Brunswick, 1955
N. Kaldor, A model of the trade cycle, Econom. J. 50, 78–92 (1940)
N. Levinson and O. K. Smith, A general equation for relaxation oscillations, Duke Math. J. 9, 382–403 (1942)
H. W. Lorenz, On the uniqueness of limit cycles in business cycle theory, Metroeconomica 38, 281–293 (1987)
H. W. Lorenz, Nonlinear Dynamical Economics and Chaotic Motion, Lecture Notes in Econom. Math. Systems, Vol. 334, Springer-Verlag, Berlin, 1989
S. Mizohata and M. Yamaguchi, On the existence of periodic solutions of the nonlinear differential equation $\ddot x + a\left ( x \right )\dot x + \Phi \left ( x \right ) = p\left ( t \right )$, Mem. Coll. Sci. Univ. Kyoto, Ser. A 27, 109–113 (1952)
T. Owase, Dynamical system theory and analysis of economic fluctuations, The Study of Dynamical Systems (N. Aoki, ed.), World Scientific, Singapore, 1989
T. Owase, Nonlinear dynamical systems and economic fluctuations: A brief historical survey, Trans. IEICE. E74, 1393–1400 (1991)
G. Sansone and R. Conti, Non-linear Differential Equations, Macmillan, New York, 1964
G. J. Schinasi, Fluctuations in a dynamic, intermediate-run IS-LM model: Applications of the Poincaré-Bendixon theorem, J. Econom. Theory 28, 369–375 (1982)
Ye Yanqian, Theory of Limit Cycles, Translations of Mathematical Monographs, Vol. 66, Amer. Math. Soc., Providence, R.I., 1986
T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Mathematical Society of Japan, Tokyo, 1966
Zhang Zhifen, On the uniqueness of the limit cycles of some nonlinear oscillation equations, Dokl. Akad. Nauk. SSSR 119, 659–662 (1958) (Russian)
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Article copyright:
© Copyright 1996
American Mathematical Society
