Global Bifurcations in a Three-Dimensional Financial Model of Bull and Bear Interactions

  • Fabio TramontanaEmail author
  • Laura Gardini
  • Roberto Dieci
  • Frank Westerhoff


In a previous paper Tramontana et al. (2009), we developed a three-dimensional discrete-time dynamic model in which two stock markets of two countries, say H(ome) and A(broad), are linked via and with the foreign exchange market. The latter is modelled in the sense of Day and Huang (1990), i.e. it is characterized by a nonlinear interplay between technical traders (or chartists) and fundamental traders (or fundamentalists). In the absence of connections, the foreign exchange market is driven by the iteration of a one-dimensional cubic map, which has the potential to produce a regime of alternating and unpredictable bubbles and crashes for sufficiently large values of a key parameter, which captures the speculative behavior of chartists. Such a dynamic feature, first observed and explained by Day and Huang (1990) in their stylized model of financial market dynamics, can be understood with the help of bifurcation analysis: an initial situation of bi-stability (two coexisting, attracting non-fundamental steady states around an unstable fundamental equilibrium) evolves into coexistence of cycles or chaotic intervals within two disjoint bull and bear regions, which eventually merge via a homoclinic bifurcation. By introducing connections between markets (i.e. by allowing stock market traders to be active abroad), the endogenous fluctuations originating in one of the markets spread throughout the whole three-dimensional system. It therefore becomes interesting to investigate how the coupling of the markets affects the bull and bear dynamics of the model. With regard to this, in Tramontana et al. (2009) we already performed a thorough analytical and numerical study of two simplified lower-dimensional cases, where connections are either totally absent (each market evolves according to an independent one-dimensional map) or occur in one direction (a two-dimensional system evolves independently of the third dynamic equation). Also a short analysis of the stability of the equilibria of the three-dimensional model was there started, arguing that the global (homoclinic) bifurcations may still be a characteristic of the dynamics. This investigation is precisely the object of the present paper. We shall analyze the dynamic behavior of the complete three-dimensional model, following the approach adopted in Tramontana et al. (2009), based mainly on the numerical and graphical detection of the relevant global bifurcations. Although analytical conditions for such global bifurcation, mainly homoclinic bifurcations, are difficult to be formalized, their existence and occurrence can be numerically detected. As it is standard in the qualitative study of dynamic behaviors, the transverse crossing between stable and unstable sets of unstable cycles, leading to homoclinic trajectories, give numerical tools which may be considered as proofs in a given numerical example.


Chaotic Attractor Foreign Exchange Market Global Bifurcation Homoclinic Bifurcation Divergent Trajectory 
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. Day, R. H., & Huang, W. (1990). Bulls, bears and market sheep. Journal of Economic Behavior and Organization, 14, 299–329.CrossRefGoogle Scholar
  2. Elaydi, S. N. (1970). An introduction to difference equations. New York: Springer.Google Scholar
  3. Farebrother, R. W. (1973). Simplified Samuelson conditions for cubit and quartic equations. The Manchester School of Economic and Social Studies, 41, 396–400.CrossRefGoogle Scholar
  4. Gandolfo, G. (1980). Economic dynamics: Methods and models. Amsterdam: North Holland.Google Scholar
  5. Grebogi, C., Ott, E., & Yorke, J. A. (1983). Crises, sudden changes in chaotic attractors, and transient chaos. Physica D, 7, 231.CrossRefGoogle Scholar
  6. Mira, C., Gardini, L., Barugola, A., & Chatala, J. C. (1996). Chaotic dynamics in two-dimensional noninvertible maps. Singapore: World Scientific.Google Scholar
  7. Okuguchi, K., & Irie, K. (1990). The Shur and Samuelson conditions for a cubic equation. The Manchester School of Economic and Social Studies, 58, 414–418.CrossRefGoogle Scholar
  8. Tramontana, F., Gardini, L., Dieci, R., & Westerhoff, F. (2009). The emergence of “bull and bear” dynamics in a nonlinear model of interacting markets. Discrete Dynamics in Nature and Society, 2009, 310471.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Fabio Tramontana
    • 1
    Email author
  • Laura Gardini
  • Roberto Dieci
  • Frank Westerhoff
  1. 1.Dipartimento di EconomiaUniversità di AnconaAnconaItaly

Personalised recommendations