Document Type : Research Article

Authors

1 Alabama State University

2 Rust College

Abstract

In this paper, a three coupled Kaldor-Kalecki model with multiple delays is investigated. By means of the generalized Chafee's criterion, some sufficient conditions to guarantee the existence of oscillatory solution for the model are obtained. Computer simulations are provided to demonstrate the proposed results.

Keywords

[1] N. Kaldor, A model of the trade cycle, Economic Journal, 50 (1940), 78-92.
[2] N. Kalecki, A theory of the business cycle, Rev. Stud. 4 (1937), 77-97.
[3] W.W. Chang and D.J. Smith, The existence and persistence of cycles in a nonlinear model: Kaldor's 1940 model re-examined, Rev. Econ. Stud. 38 (1971), 37-44.
[4] H.R. Varian, Catastrophe theory and the business cycle, Econ. Inq. 17 (1979), 14-28.
[5] J. Grasman and J.J. Wentzel, Co-existence of a limit cycle and an equilibrium in Kaldor's business cycle model and its consequences, J. Econ. Behav. Organ. 24 (1994), 369-377.
[6] L.C. Wang and X.Q. Wu, Bifurcation analysis of a Kaldor-Kalecki model of business cycle with delay, Elec. J. Qual. Theory Diff. Equs. Spec. Ed. I, (2009) 1-20.
[7] J.Z. Cao and H.Y. Sun, Bifurcation analysis for the Kaldor-Kalecki model with two delays, Adv. Diff. Equs. (2019) 2019:107.
[8] J. Yu and M. Peng, Stability and bifurcation analysis for the Kaldor-Kalecki model with a discrete delay and a distributed delay, Phys. A, Stat. Mech. Appl. 460, (2016), 66-75.
[9] X.P. Wu, Zero-Hopf bifurcation analysis of a Kaldor-Kalecki model of business cycle with delay, Nonlinear Anal. RWA, 13 (2012), 736-754.
[10] A. Kaddar and H.T. Alaoui, Local Hopf bifurcation and stability of limit cycle in a delayed Kaldor-Kalecki model, Nonlinear Anal. Model Control, 14 (2009), 333-343.
[11] C. Zhang and J. Wei, Stability and bifurcation analysis in a kind of business cycle model with delay, Chaos Solitons and Fractals, 22 (2004), 883-896.
[12] X.P. Wu, Codimension-2 bifurcations of the Kaldor model of business cycle, Chaos Solitons Fractals, 44 (2011), 28-42.
[13] J. Grasman and J.J. Wentzel, Co-existence of a limit cycle and an equilibrium in Kaldor's business cycle model and its consequences, J. Econ. Behav. Organ. 24 (1994), 369-377.
[14] M.E. Fatini, A. Kaddar and A. Laaribi, Direction of Hopf bifurcation in a delayed Kaldor-Kalecki model of business cycle, Int. J. Stati. Econ. 14 (2014), 13-24.
[15] D.M. Dubois, Extension of the Kaldor-Kalecki model of business cycle with a computational anticipated capital stock, J. Org. Trans. Soc. Change, 1 (2004), 63-80.
[16] W.J. Hu, H. Zhao and T. Dong, Dynamic analysis for a Kaldor-Kalecki model of business cycle with time delay and diffusion effect, Complexity, 2018 (2018), 1263602.
[17] H.Y. Al  , Semi-Analytical solutions for the diffusive Kaldor-Kalecki business cycle model with a time delay for gross product and capital stock, Complexity, 2021 (2021), 9998756.
[18] T. Caraballo and A.P. Silva, Stability analysis of a delay differential Kaldor's model with government policies, Mathematica Scandinavica, 126 (2020), 116243.
[19] B.J. Zduniak, U. Grzybowska and A. Orowski, Numerical analysis of two coupled Kaldor-Kalecki models with delay, Acta Physica Polonica A, 127 (2015), 70-74.
[20] C.A. Desoer and M. Vidyasagar, Feedback System: Input-Output properties, Academic Press, New York, 1977.
[21] J.K. Hale, Theory of functional differential equations, Springer Verlag, Berlin, 1997.
[22] N. Chafee, A bifurcation problem for a functional differential equation of  nitely retarded type, J. Math. Anal. Appl. 35 (1971), 312-348.
[23] C. Feng and R. Plamondon, An oscillatory criterion for a time delayed neural ring network model, Neural Networks, 79 (2012), 70-79.