Coexisting Attractors and Complex Basins in Discrete-time Economic Models

  • Gian Italo Bischi
  • Fabio Lamantia
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 476)


In this lesson we consider discrete time dynamical systems with coexisting attractors, and we analyze the problem of the structure of the boundaries that separate their basins of attraction. This problem may become particularly challenging when the discrete dynamical system is represented by the iteration of a noninvertible map, because in this case nonconnected basins can be obtained, formed by several (even infinitely many) disjoint portions. Measure theoretic attractors, known as Milnor attractors, are also described, together with riddled basins, an extreme form of complex basin’s structure that can be observed in the presence of such attractors. Some tools for the study of global bifurcations that lead to the creation of complex structures of the basins are described, as well as some applications in discrete time models taken from economic dynamics.


Nash Equilibrium Chaotic Attractor Global Bifurcation Discrete Dynamical System Basin Boundary 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. R. Abraham, L. Gardini and C. Mira. Chaos in discrete dynamical systems (a visual introduction in two dimension). Springer-Verlag, 1997.Google Scholar
  2. H.N. Agiza, G.I. Bischi, and M. Kopel. Multistability in a Dynamic Cournot Game with Three Oligopolists. Mathematics and Computers in Simulation, 51, 63–90, 1999.CrossRefMathSciNetGoogle Scholar
  3. A. Agliari, G.I. Bischi, and L. Gardini. Some methods for thr Global Analysis of Dynamic Games represented by Noninvertible Maps. In T. Puu and I. Sushko, editors, Oligopoly Dynamics: Models and Tools, Springer Verlag, pp. 31–83, 2002.Google Scholar
  4. J.C. Alexander, J.A. Yorke, Z. You, and I. Kan. Riddled basins. Int. Jou. of Bif. & Chaos, 2, 795–813, 1992.MATHCrossRefMathSciNetGoogle Scholar
  5. M. Aoki, New Approaches to Macroeconomic Modelling. Cambridge University Press, New York, 1996.CrossRefGoogle Scholar
  6. P. Ashwin, J. Buescu, and I. Stewart. From attractor to chaotic saddle: a tale of transverse instability. Nonlinearity, 9, 703–737, 1996.MATHCrossRefADSMathSciNetGoogle Scholar
  7. E. Barucci, G.I. Bischi, and L. Gardini. Endogenous fluctuations in a bounded rationality economy: learning non perfect foresight equilibria. Journal of Economic Theory, 87, 243–253, 1999.MATHCrossRefMathSciNetGoogle Scholar
  8. K. Binmore. Fun and Games. D.C. Heath, 1992.Google Scholar
  9. G.I. Bischi, R. Carini, L. Gardini, and P. Tenti. Sulle Orme del Caos. Comportamenti complessi in modelli matematici semplici. Bruno Mondadori Editore, 2004.Google Scholar
  10. G.I. Bischi, H. Dawid, and M. Kopel. Gaining the Competitive Edge Using Internal and External Spillovers: A Dynamic Analysis. Journal of Economic Dynamics and Control, vol. 27, 2171–2193, 2003a.CrossRefMathSciNetGoogle Scholar
  11. G.I. Bischi, H. Dawid, and M. Kopel. Spillover Effects and the Evolution of Firm Clusters. Journal of Economic Behavior and Organization, vol. 50, 47–75, 2003b.CrossRefGoogle Scholar
  12. G.I. Bischi, M. Gallegati, and A. Naimzada. Symmetry-Breaking bifurcations and representative firm in dynamic duopoly games. Annals of Operations Research, 89, 253–272, 1999a.MATHCrossRefMathSciNetGoogle Scholar
  13. G.I. Bischi and L. Gardini. Role of invariant and minimal absorbing areas in chaos synchronization. Physical Review E, 58, 5710–5719, 1998.CrossRefADSMathSciNetGoogle Scholar
  14. G.I. Bischi and L. Gardini. Global Properties of Symmetric Competition Models with Riddling and Blowout Phenomena. Discrete Dynamics in Nature and Society, vol. 5, 149–160, 2000.MATHCrossRefGoogle Scholar
  15. G.I. Bischi, L. Gardini, and M. Kopel. Analysis of Global Bifurcations in a Market Share Attraction Model. Journal of Economic Dynamics and Control, 24, 855–879, 2000a.MATHCrossRefMathSciNetGoogle Scholar
  16. G.I. Bischi, L. Gardini, and C. Mira. Plane maps with denominator. Part I: some generic properties. International Journal of Bifurcation and Chaos, 9(1), 119–153, 1999b.MATHCrossRefMathSciNetGoogle Scholar
  17. G.I. Bischi, L. Gardini, and C. Mira. Plane Maps with Denominator. Part II: Nonin-vertible maps with simple focal points. International Journal of Bifurcation and Chaos, vol. 13, No. 8, 2253–2277, 2003c.MATHCrossRefMathSciNetGoogle Scholar
  18. G.I. Bischi and M. Kopel. Equilibrium Selection in a Nonlinear Duopoly Game with Adaptive Expectations. Journal of Economic Behavior and Organization, 46(1), 73–100, 2001.CrossRefGoogle Scholar
  19. G.I. Bischi and F. Lamantia. Nonlinear Duopoly Games with Positive Cost Externalities due to Spillover Effects. Chaos, Solitons & Fractals, vol. 13, 805–822, 2002a.CrossRefMathSciNetGoogle Scholar
  20. G.I. Bischi and F. Lamantia. Chaos Synchronization and Intermittency in a Duopoly Game with Spillover Effects. Chapter 8 in T. Puu and I. Sushko, editors, Oligopoly Dynamics: Models and Tools, Springer Verlag, 195–217, 2002b.Google Scholar
  21. G.I. Bischi, C. Mammana, and L. Gardini. Multistability and cyclic attractors in duopoly games. Chaos, Solitons & Fractals, 11, 543–564, 2000b.MATHCrossRefMathSciNetGoogle Scholar
  22. G.I. Bischi, L. Mroz, and H. Hauser. Studying basin bifurcations in nonlinear triopoly games by using 3D visualization. Nonlinear Analysis, Theory, Methods & Applications, 47(8), 5325–5341, 2001.MATHCrossRefMathSciNetGoogle Scholar
  23. G.I. Bischi and A. Naimzada. Global Analysis of a Duopoly game with Bounded Rationality. Advances in Dynamic Games and applications, vol.5, 361–385, Birkhauser, 2000.MathSciNetGoogle Scholar
  24. W.A. Brock, C. Hommes. A Rational Route to Randomness. Econometrica, 65,1059–1095, 1997.MATHCrossRefMathSciNetGoogle Scholar
  25. J. Buescu. Exotic Attractors, Birkhäuser, Boston, 1997.MATHGoogle Scholar
  26. C. Chiarella, R. Dieci, and L. Gardini. Asset price dynamics in a financial market with fundamentalists and chartists. Discrete Dynamics in Nature and Society, Vol 6, 69–99, 2001.MATHCrossRefGoogle Scholar
  27. C. Chiarella. R. Dieci, and L. Gardini. Speculative Behaviour and Complex Asset Price Dynamics: A Global Analysis. Journal of Economic Behavior and Organization, 49(1) 173–192, 2002.CrossRefGoogle Scholar
  28. P. Collet and J.P. Eckmann. Iterated maps on the interval as dynamical systems, Birkhäuser, Boston, 1980.MATHGoogle Scholar
  29. R.L. Devaney. An Introduction to Chaotic Dynamical Systems. The Benjamin / Cummings Publishing Co., Menlo Park, California, 1987.Google Scholar
  30. R. Dieci, G.I. Bischi, and L. Gardini. Multistability and role of noninvertibility in a discrete-time business cycle model. CEJOR, vol. 9, 71–96, 2001.MATHMathSciNetGoogle Scholar
  31. A. Ferretti and N.K. Rahman. A study of coupled logistic maps and its applications in chemical physics. Chemical Physics, 119, 275–288, 1988.CrossRefMathSciNetGoogle Scholar
  32. H. Fujisaka and T. Yamada. Stability theory of synchronized motion in coupled-oscillator systems. Progress of Theoretical Physics, 69(1), 32–47, 1983.MATHCrossRefADSMathSciNetGoogle Scholar
  33. C. Grebogi, E. Ott, and J.A. Yorke. Crises, sudden changes in chaotic attractors and transient chaos. Physica 7D, 81–200, 1983.MathSciNetGoogle Scholar
  34. J. Guckenheimer and T. Holmes. Non-linear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, 1983.Google Scholar
  35. R. Guesnerie and M. Woodford. Endogeneous fluctuations. In J. Laffont, editor, Advances in Economic Theory, 288–412, Cambridge University Press, 1993.Google Scholar
  36. I. Gumowski and C. Mira. Dynamique Chaotique, Cepadues Editions, Toulose, 1980.Google Scholar
  37. M. Hasler and Y. Maistrenko. An introduction to the synchronization of chaotic systems: coupled skew tent maps. IEEE Trans. Circuits Syst., 44(10), 856–866, 1997.MathSciNetGoogle Scholar
  38. A. P. Kirman. Whom or What Does the Representative Individual Represent? Jou. of Econ. Perspectives, 6, 117–136, 1992.Google Scholar
  39. M. Kopel. Simple and Complex Adjustment Dynamics in Cournot Duopoly Models. Chaos, Solitons & Fractals, 7(12), 2031–2048, 1996.MATHCrossRefMathSciNetGoogle Scholar
  40. M. Kopel, G.I. Bischi, and L. Gardini. On new phenomena in dynamic promotional competition models with homogeneous and quasi-homo-geneous firms. In D. Delli Gatti, M. Gallegati, and A.P. Kirman, editors, Interaction and Market Structure. Essays on Heterogeneity in Economics. Springer-Verlag, pp. 57–87, 2000.Google Scholar
  41. Y.C. Lai, C. Grebogi, and J.A. Yorke. Riddling bifurcation in chaotic dynamical systems. Physical Review Letters, 77, 55–58, 1996.PubMedCrossRefADSGoogle Scholar
  42. Y.C. Lai and C. Grebogi. Noise-induced riddling in chaotic systems. Physical Review Letters 77, 5047–5050, 1996.PubMedCrossRefADSGoogle Scholar
  43. H.W. Lorenz. Multiple attractors, Complex Basin Boundaries, and Transient Motion in Deterministic Economic Systems. In G. Feichtinger, editor, Dynamic Economic Models and Optimal Control, 411–430, North-Holland, Amsterdam, 1992.Google Scholar
  44. Y. Maistrenko, T. Kapitaniak, and P. Szuminski. Locally and globally riddled basins in two coupled piecewise-linear maps. Physical Review E, 57(3), 6393–6399, 1997.CrossRefADSMathSciNetGoogle Scholar
  45. Y. Maistrenko, V.L. Maistrenko, A. Popovich, and E. Mosekilde. Role of the Absorbing Area in Chaotic Synchronization. Physical Review Letters, 80(8), 1638–1641, 1998a.CrossRefADSGoogle Scholar
  46. Y. Maistrenko, V.L. Maistrenko, A. Popovich, and E. Mosekilde. Transverse instability and riddled basins in a system of two coupled logistic maps. Physical Review E, 57(3), 2713–2724, 1998b.CrossRefADSMathSciNetGoogle Scholar
  47. J. Milnor. On the concept of attractor. Commun. Math Phys, 99, 177–195, 1985.MATHCrossRefADSMathSciNetGoogle Scholar
  48. C. Mira. Chaotic Dynamics. Word Scientific, Singapore, 1987.MATHGoogle Scholar
  49. C. Mira, J.P. Carcasses, G. Millerioux, and L. Gardini. Plane foliation of two-dimensional noninvertible maps. International Journal of Bifurcation & Chaos, vol.6(8), 1439–1462, 1996.MATHCrossRefMathSciNetGoogle Scholar
  50. C. Mira, D. Fournier-Prunaret, L. Gardini, H. Kawakami, and J.C. Cathala. Basin bifurcations of two-dimensional noninvertible maps: fractalization of basins. International Journal of Bifurcation and Chaos, 4, 343–381, 1994.MATHCrossRefMathSciNetGoogle Scholar
  51. C. Mira, L. Gardini, A. Barugola, and J.C. Cathala. Chaotic Dynamics in Two-Dimensional Noninvertible Maps, World Scientific, Singapore, 1996.MATHCrossRefGoogle Scholar
  52. C. Mira and C. Rauzy. Fractal aggregation of basin islands in two-dimensional quadratic noninvertible maps. International Journal of Bifurcations and Chaos, 5(4), 991–1019, 1995.MATHCrossRefMathSciNetGoogle Scholar
  53. Y. Nagai and Y.C. Lai. Periodic-orbit theory of the blowout bifurcation. Physical Review E, 56(4), 4031–4041, 1997.CrossRefADSMathSciNetGoogle Scholar
  54. E. Ott and J.C. Sommerer. Blowout bifurcations: the occurrence of riddled basins and on-off intermittency. Phys. Lett A, 188, 39–47, 1994.CrossRefADSGoogle Scholar
  55. L.M. Pecora and T.L. Carrol. Synchronization in chaotic systems. Physical Review Letters, 64(8) pp. 821–824, 1990.PubMedCrossRefADSMathSciNetGoogle Scholar
  56. T. Puu. Attractors, Bifurcations and Chaos. Springer Verlag, Berlin, 2000.MATHGoogle Scholar
  57. A.S. Soliman. An analysis of the effectiveness of policy based on basins of attraction. Journal of Macroeconomics, 21, 165–178, 1999.CrossRefGoogle Scholar
  58. S.C. Venkataramani, B.R. Hunt, and E. Ott. Bubbling transition. Physical Review E, 54, 1346–1360, 1996.CrossRefADSGoogle Scholar
  59. J.W. Weibull, Evolutionary Game Theory. The MIT Press, 1995.Google Scholar

Copyright information

© CISM, Udine 2005

Authors and Affiliations

  • Gian Italo Bischi
    • 1
  • Fabio Lamantia
    • 2
  1. 1.Istituto di Scienze EconomicheUniversity of UrbinoUrbinoItaly
  2. 2.S.A. DepartmentUniversity of CalabriaArcavacata di Rende (CS)Italy

Personalised recommendations