Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 

 

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

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] J.P. Benassy, A non-Walrasian model of the business cycle, J. Econom. Behavior Org. 5, 77-89 (1984)
  • [2] 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
  • [3] 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)
  • [4] 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)
  • [5] G. Gabisch and H.-W. Lorenz, Business Cycle Theory, Lecture Notes in Econom. Math. Systems, Vol. 283, Springer-Verlag, Berlin, 1987
  • [6] M. Galeotti and F. Gori, Uniqueness of periodic orbits in Liénard-type business-cycle models, Metroeconomica 40, 135-146 (1989)
  • [7] R. M. Goodwin, The nonlinear accelerator and the persistence of business cycles, Econometrica 19, 1-17 (1951)
  • [8] 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
  • [9] N. Kaldor, A model of the trade cycle, Econom. J. 50, 78-92 (1940)
  • [10] N. Levinson and O. K. Smith, A general equation for relaxation oscillations, Duke Math. J. 9, 382-403 (1942)
  • [11] H. W. Lorenz, On the uniqueness of limit cycles in business cycle theory, Metroeconomica 38, 281-293 (1987)
  • [12] H. W. Lorenz, Nonlinear Dynamical Economics and Chaotic Motion, Lecture Notes in Econom. Math. Systems, Vol. 334, Springer-Verlag, Berlin, 1989
  • [13] 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)
  • [14] T. Owase, Dynamical system theory and analysis of economic fluctuations, The Study of Dynamical Systems (N. Aoki, ed.), World Scientific, Singapore, 1989
  • [15] T. Owase, Nonlinear dynamical systems and economic fluctuations: A brief historical survey, Trans. IEICE. E74, 1393-1400 (1991)
  • [16] G. Sansone and R. Conti, Non-linear Differential Equations, Macmillan, New York, 1964
  • [17] 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)
  • [18] Ye Yanqian, Theory of Limit Cycles, Translations of Mathematical Monographs, Vol. 66, Amer. Math. Soc., Providence, R.I., 1986
  • [19] T. Yoshizawa, Stability Theory by Liapunov's Second Method, Mathematical Society of Japan, Tokyo, 1966
  • [20] Zhang Zhifen, On the uniqueness of the limit cycles of some nonlinear oscillation equations, Dokl. Akad. Nauk. SSSR 119, 659-662 (1958) (Russian)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 90A16, 34C05

Retrieve articles in all journals with MSC: 90A16, 34C05


Additional Information

DOI: https://doi.org/10.1090/qam/1373841
Article copyright: © Copyright 1996 American Mathematical Society


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website