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Genericity analysis of split bifurcations

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  • Fan-chin Kung

    (Academia Sinica)

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    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.

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    File URL: http://128.118.178.162/eps/ge/papers/0410/0410008.pdf
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    Bibliographic Info

    Paper provided by EconWPA in its series GE, Growth, Math methods with number 0410008.

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    Length: 24 pages
    Date of creation: 13 Oct 2004
    Date of revision: 13 Oct 2004
    Handle: RePEc:wpa:wuwpge:0410008

    Note: Type of Document - pdf; pages: 24
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    Web page: http://128.118.178.162

    Related research

    Keywords: Bifurcation; Genericity analysis; Regular parameterization; Migration dynamics;

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    1. Berliant, Marcus & Zenou, Yves, 2012. "Labor Differentiation and Agglomeration in General Equilibrium," CEPR Discussion Papers 8840, C.E.P.R. Discussion Papers.
    2. Debreu, Gerard, 1970. "Economies with a Finite Set of Equilibria," Econometrica, Econometric Society, vol. 38(3), pages 387-92, May.
    3. Jorgen W. Weibull, 1997. "Evolutionary Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262731215, January.
    4. 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.
    5. 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, January.
    6. Krugman, Paul, 1991. "Increasing Returns and Economic Geography," Journal of Political Economy, University of Chicago Press, vol. 99(3), pages 483-99, June.
    7. Marcus Berliant & Fan- chin Kung, 2004. "The Indeterminacy of Equilibrium City Formation under Monopolistic Competition and Increasing Returns," Urban/Regional 0407011, EconWPA, revised 29 Apr 2005.
    8. Kehoe, Timothy J, 1985. "Multiplicity of Equilibria and Comparative Statics," The Quarterly Journal of Economics, MIT Press, vol. 100(1), pages 119-47, February.
    9. 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.
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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.

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