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Discrete Dynamics in Nature and Society
Volume 2009, Article ID 407913, 22 pages
http://dx.doi.org/10.1155/2009/407913
Research Article

Impulsivity in Binary Choices and the Emergence of Periodicity

1Department of Economics and Quantitative Methods, Faculty of Economics and Business, University of Urbino, 61029 Urbino, Italy
2Department of of Statistics and Applied Mathematics "de Castro", Corso Unione Sovietica 218 bis, 10134 Torino, Italy

Received 17 March 2009; Accepted 15 June 2009

Academic Editor: Xue-Zhong He

Copyright © 2009 Gian Italo Bischi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. T. C. Schelling, “Hockey helmets, concealed weapons and daylight saving,” Journal of Conflict Resolution, vol. 17, no. 3, pp. 381–428, 1973. View at Google Scholar
  2. D. Challet and Y. C. Zhang, “Emergence of cooperation and organization in an evolutionary game,” Physica A, pp. 246–407, 1997. View at Google Scholar
  3. M. Granovetter, “Threshold models of collective behavior,” The American Journal of Sociology, vol. 83, no. 6, pp. 1420–1443, 1978. View at Google Scholar
  4. G. I. Bischi and U. Merlone, “Global dynamics in binary choice models with social influence,” Journal of Mathematical Sociology, vol. 33, pp. 1–26, 2009. View at Google Scholar
  5. J. B. Rosser, “On the complexities of complex economic dynamics,” Journal of Economic Perspectives, vol. 13, pp. 169–172, 1999. View at Google Scholar
  6. F. G. Moeller, E. S. Barratt, D. M. Dougherty, J. M. Schmitz, and A. C. Swann, “Psychiatric aspects of impulsivity,” American Journal of Psychiatry, vol. 158, pp. 1783–1793, 2001. View at Google Scholar
  7. J. H. Patton, M. S. Stanford, and E. S. Barratt, “Factor structure of the Barratt impulsiveness scale,” Journal of Clinical Psychology, vol. 51, pp. 768–774, 1995. View at Google Scholar
  8. R. A. Eve, S. Horsfall, and M. E. Lee, Eds., Chaos, Complexity, and Sociology: Myths, Models, and Theories, Sage Publications, London, UK, 1997.
  9. R. L. Devaney, Chaotic Dynamical Systems, Perseus Books, Reading, Mass, USA, 2nd edition, 1989.
  10. C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 2nd edition, 1999. View at MathSciNet
  11. V. Avrutin and M. Schanz, “On multi-parametric bifurcations in a scalar piecewise-linear map,” Nonlinearity, vol. 19, no. 3, pp. 531–552, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. V. Avrutin and M. Schanz, “On the fully developed bandcount adding scenario,” Nonlinearity, vol. 21, no. 5, pp. 1077–1103, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. V. Avrutin, M. Schanz, and S. Banerjee, “Multi-parametric bifurcations in a piecewise-linear discontinuous map,” Nonlinearity, vol. 19, no. 8, pp. 1875–1906, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. di Bernardo, M. I. Feigin, S. J. Hogan, and M. E. Homer, “Local analysis of C-bifurcations in n-dimensional piecewise-smooth dynamical systems,” Chaos, Solitons & Fractals, vol. 10, no. 11, pp. 1881–1908, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  15. I. Sushko, L. Gardini, and T. Puu, “Tongues of periodicity in a family of two-dimensional discontinuous maps of areal Möbius type,” Chaos, Solitons & Fractals, vol. 21, no. 2, pp. 403–412, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  16. I. Sushko, A. Agliari, and L. Gardini, “Bifurcation structure of parameter plane for a family of unimodal piecewise smooth maps: border-collision bifurcation curves,” Chaos, Solitons & Fractals, vol. 29, no. 3, pp. 756–770, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Yu. L. Maĭstrenko, V. L. Maĭstrenko, and L. O. Chua, “Cycles of chaotic intervals in a time-delayed Chua's circuit,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 3, no. 6, pp. 1557–1572, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Yu. L. Maĭstrenko, V. L. Maĭstrenko, S. I. Vikul, and L. O. Chua, “Bifurcations of attracting cycles from time-delayed Chua's circuit,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 5, no. 3, pp. 653–671, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. Yu. L. Maĭstrenko, V. L. Maĭstrenko, and S. I. Vikul, “On period-adding sequences of attracting cycles in piecewise linear maps,” Chaos, Solitons & Fractals, vol. 9, no. 1-2, pp. 67–75, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S. Banerjee and C. Grebogi, “Border-collision bifurcations in two-dimensional piecewise smooth maps,” Physical Review E, vol. 59, no. 4, pp. 4052–4061, 1999. View at Google Scholar
  21. S. Banerjee, M. S. Karthik, G. Yuan, and J. A. Yorke, “Bifurcations in one-dimensional piecewise smooth maps—theory and applications in switching circuits,” IEEE Transactions on Circuits and Systems I, vol. 47, no. 3, pp. 389–394, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. Banerjee, P. Ranjan, and C. Grebogi, “Bifurcations in two-dimensional piecewise smooth maps—theory and applications in switching circuits,” IEEE Transactions on Circuits and Systems I, vol. 47, no. 5, pp. 633–643, 2000. View at Google Scholar
  23. O. Feely, D. Fournier-Prunaret, I. Taralova-Roux, and D. Fitzgerald, “Nonlinear dynamics of bandpass sigma-delta modulation. An investigation by means of the critical lines tool,” International Journal of Bifurcation and Chaos, vol. 10, no. 2, pp. 303–327, 2000. View at Google Scholar
  24. Z. T. Zhusubaliyev and E. Mosekilde, Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems, vol. 44 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 2003. View at MathSciNet
  25. Z. T. Zhusubaliyev, E. Mosekilde, S. Maity, S. Mohanan, and S. Banerjee, “Border collision route to quasiperiodicity: numerical investigation and experimental confirmation,” Chaos, vol. 16, no. 2, Article ID 023122, 11 pages, 2006. View at Publisher · View at Google Scholar · View at PubMed · View at Zentralblatt MATH · View at MathSciNet
  26. Z. T. Zhusubaliyev, E. Soukhoterin, and E. Mosekilde, “Quasiperiodicity and torus breakdown in a power electronic dc/dc converter,” Mathematics and Computers in Simulation, vol. 73, no. 6, pp. 364–377, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. L. Gardini, T. Puu, and I. Sushko, “A goodwin-type model with a piecewise linear investment function,” in Business Cycles Dynamics. Models and Tools, T. Puu and I. Sushko, Eds., Springe, New York, NY, USA, 2006. View at Google Scholar
  28. L. Gardini, I. Sushko, and A. Naimzada, “Growing through chaotic intervals,” Journal of Economic Theory, vol. 143, pp. 541–557, 2008. View at Google Scholar
  29. T. Puu, L. Gardini, and I. Sushko, “A Hicksian multiplier-accelerator model with floor determined by capital stock,” Journal of Economic Behavior and Organization, vol. 56, pp. 331–348, 2005. View at Google Scholar
  30. T. Puu, “The Hicksian trade cycle with floor and ceiling dependent on capital stock,” Journal of Economic Dynamics & Control, vol. 31, no. 2, pp. 575–592, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. F. Tramontana, L. Gardini, and T. Puu, “Duopoly games with alternative technologies,” Journal of Economic Dynamic and Control, vol. 33, pp. 250–265, 2008. View at Google Scholar
  32. H. E. Nusse and J. A. Yorke, “Border-collision bifurcations including “period two to period three” for piecewise smooth systems,” Physica D, vol. 57, no. 1-2, pp. 39–57, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. H. E. Nusse and J. A. Yorke, “Border-collision bifurcations for piecewise smooth one-dimensional maps,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 5, no. 1, pp. 189–207, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. N. N. Leonov, “Map of the line onto itself,” Radiofisica, vol. 3, no. 3, pp. 942–956, 1959. View at Google Scholar
  35. N. N. Leonov, “On a discontinuous point transformation of the line into the line,” Doklady Akademii Nauk SSSR, vol. 143, no. 5, pp. 1038–1041, 1962. View at Google Scholar · View at MathSciNet
  36. C. Mira, “Sur la structure des bifurcations des difféomorphismes du cercle,” Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Series A, vol. 287, no. 13, pp. 883–886, 1978. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. C. Mira, Chaotic Dynamics: From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism, World Scientific, Singapore, 1987. View at MathSciNet
  38. B. L. Hao, Elementary Symbolic Dynamics and Chaos in Dissipative Systems, World Scientific, Teaneck, NJ, USA, 1989. View at MathSciNet
  39. G. I. Bischi, L. Gardini, and U. Merlone, “Periodic cycles and bifurcation curves for one-dimensional maps with two discontinuities,” Journal of Dynamical Systems and Geometric Theory. In press.