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Genericity analysis of split bifurcations

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  • Fan-chin Kung

    (Academia Sinica)

Abstract

This paper analyzes the genericity of bifurcations of one-parameter families of smooth (C1) vector fields that are embedded in an underlying multi-dimensional parameter space. Bifurcations with crossing equilibrium loci are called 'split bifurcations.' They include, for example, the pitchfork bifurcation and the transcritical bifurcation. In a regular parameter space where the system's Jacobian matrix with respect to endogenous variables and parameters has full rank at every equilibrium for all parameter values, there is a generic (open and dense) set of one-parameter C1 families of vector fields without split bifurcations. It is not difficult to obtain a regular parameter space when there are enough parameters. A regional migration model (a la Fujita, Krugman and Venables 1999) featuring the pitchfork bifurcation is presented as an example.

Suggested Citation

  • Fan-chin Kung, 2004. "Genericity analysis of split bifurcations," GE, Growth, Math methods 0410008, EconWPA, revised 24 Nov 2004.
  • Handle: RePEc:wpa:wuwpge:0410008
    Note: Type of Document - pdf; pages: 24
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    File URL: https://econwpa.ub.uni-muenchen.de/econ-wp/ge/papers/0410/0410008.pdf
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    References listed on IDEAS

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    1. Marcus Berliant & Yves Zenou, 2014. "Labor Differentiation and Agglomeration in General Equilibrium," International Regional Science Review, , vol. 37(1), pages 36-65, January.
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    3. Debreu, Gerard, 1970. "Economies with a Finite Set of Equilibria," Econometrica, Econometric Society, vol. 38(3), pages 387-392, May.
    4. Masahisa Fujita & Paul Krugman & Anthony J. Venables, 2001. "The Spatial Economy: Cities, Regions, and International Trade," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262561476, March.
    5. Fujita, Masahisa & Mori, Tomoya, 1997. "Structural stability and evolution of urban systems," Regional Science and Urban Economics, Elsevier, vol. 27(4-5), pages 399-442, August.
    6. Jorgen W. Weibull, 1997. "Evolutionary Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262731215, March.
    7. Berliant, Marcus & Kung, Fan-chin, 2006. "The indeterminacy of equilibrium city formation under monopolistic competition and increasing returns," Journal of Economic Theory, Elsevier, vol. 131(1), pages 101-133, November.
    8. Krugman, Paul, 1991. "Increasing Returns and Economic Geography," Journal of Political Economy, University of Chicago Press, vol. 99(3), pages 483-499, June.
    9. Antinolfi, Gaetano & Keister, Todd & Shell, Karl, 2001. "Growth Dynamics and Returns to Scale: Bifurcation Analysis," Journal of Economic Theory, Elsevier, vol. 96(1-2), pages 70-96, January.
    10. Timothy J. Kehoe, 1985. "Multiplicity of Equilibria and Comparative Statics," The Quarterly Journal of Economics, Oxford University Press, vol. 100(1), pages 119-147.
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    Cited by:

    1. Marcus Berliant, 2005. "Well Isn't That Spatial?! Handbook of Regional and Urban Economics: A View From Economic Theory," Urban/Regional 0503001, EconWPA, revised 08 Apr 2005.

    More about this item

    Keywords

    Bifurcation; Genericity analysis; Regular parameterization; Migration dynamics;

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • R23 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - Household Analysis - - - Regional Migration; Regional Labor Markets; Population
    • F12 - International Economics - - Trade - - - Models of Trade with Imperfect Competition and Scale Economies; Fragmentation

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