The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지

TRAVELING WAVE GLOBAL PRICE DYNAMICS OF LOCAL MARKETS WITH LOGISTIC SUPPLIES

  • Published : 2010.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

We employ the methods of Lattice Dynamical System to establish a global model extending the Walrasian evolutionary cobweb model in an independent single local market to the global market evolution over an infinite chain of many local markets with interaction of each other through a diffusion of prices between them. For brevity of the model, we assume linear decreasing demands and logistic supplies with naive predictors, and investigate the traveling wave behaviors of global price dynamics and show that their dynamics are conjugate to those of H$\acute{e}$non maps and hence can exhibit complicated behaviors such as period-doubling bifurcations, chaos, and homoclic orbits etc.

Keywords

References

  1. Afraimovich, V. S. & Bunimovich, L. A. : Simplest structures in coupled map lattices and their stability. preprint in GIT (1992).
  2. Arai, Z. & Mischaikow, K.: Rigorous Computations of Homoclinic Tangencies. Preprint. Kyoto University (2006).
  3. Aranson, I.S., Afraimovich, V.S. & Rabinovich, M.I.: Stability of spatially homogeneous chaotic regimes in unidirectional chains. Nonlinearity 3 (1990), 639-651. https://doi.org/10.1088/0951-7715/3/3/006
  4. Brock, W. A. & Hommes, C. H. : A Rational Route to Randomness. Econometrica 65 (1997), no. 5, 1059-1095. https://doi.org/10.2307/2171879
  5. Brock, W. A. & Hommes, C. H. : Heterogeneous Beliefs and Routes to Chaos in a Simple Asset Pricing Model. Journal of Economic Dynamics and Control 22 (1998), 1235-1274. https://doi.org/10.1016/S0165-1889(98)00011-6
  6. Brock, W. A., Hommes, C. H. & Wagner, F.O.O. : Evolutionary dynamics in markets with many trader types. Journal of Mathematical Economics 41 (2005), 7-42. https://doi.org/10.1016/j.jmateco.2004.02.002
  7. Bunimovich, L. A. et al : Trivial Maps. Chaos 2 (1992).
  8. Bunimovich, L. A. & Sinai, Y. G. : 1988. Spacetime Chaos in Coupled Map Lattices. Nonlinearity 1 (1998), 491-516.
  9. Choudhary, M. A. & Orszag, J. M. : A cobweb model with local externalities. Journal of Economic Dynamics & Control (2007).
  10. Curry, J.: On the structure of the Henon attracter. Preprint. National center for atmospheric research (1997).
  11. Devaney, R. L. : An introduction to Chaotic Dynamical Systems. Addison-Wesley, 1989.
  12. Devaney, R. L. & Nitecki, Z.: Shift Automorphisms in the Henon Mapping. Communications in Mathematical Physics 67, 137-146.
  13. Feit, S.: Characteristic exponents and strange attracters. Communications in Mathematical Physics 61 (1978), 249-260. https://doi.org/10.1007/BF01940767
  14. Henon, M.: A Two-dimensional Mapping with a Strange Attracter. Communications in Mathematical Physics 50 (1976), 69-77. https://doi.org/10.1007/BF01608556
  15. Hommes, C. H. : On the consistency of backward-looking expectations: The case of the cobweb. Journal of Economic Behavior & Organization 33 (1998), 333-362. https://doi.org/10.1016/S0167-2681(97)00062-0
  16. Kirchgraber, U. & Stoffer, D.: Transversal homoclinic points of the Henon map. Annali di Mathematica 185 (2006), 187-204. https://doi.org/10.1007/s10231-004-0142-4
  17. Mora, L. & Viana, M.: Abundance of Strange Attractors. Acta Mathematica 171 (1993), 1-71. https://doi.org/10.1007/BF02392766
  18. Robinson, C.: Bifurcation to Infinitely many sinks. Communications in Mathematical Physics 90 (1983), 433-459. https://doi.org/10.1007/BF01206892
  19. Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. CRC Press
  20. Sterling, D., Dullin, H.R. & Meiss, J.D.: Homoclinic bifurcations for the Henon map. Physica D 134 (1999), 153-184 https://doi.org/10.1016/S0167-2789(99)00125-6
  21. Wiggins S.: Global Bifurcations and Chaos. Springer-Verlag, New York, 1988.
  22. Wiggins S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York, 1990