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Nonlinear difference equations bifurcations and chaos : an introduction

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  • Grandmont Jean-michel

Abstract

The aim of these lecture notes is to present a few mathematical factsabout the bifurcations of nonlinear difference equations, in a concise andsimple form that might be useable by economic theorists.
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Suggested Citation

  • Grandmont Jean-michel, 1988. "Nonlinear difference equations bifurcations and chaos : an introduction," CEPREMAP Working Papers (Couverture Orange) 8811, CEPREMAP.
  • Handle: RePEc:cpm:cepmap:8811
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    1. Gaunersdorfer, Andrea & Hommes, Cars H. & Wagener, Florian O.O., 2008. "Bifurcation routes to volatility clustering under evolutionary learning," Journal of Economic Behavior & Organization, Elsevier, vol. 67(1), pages 27-47, July.
    2. Grandmont, Jean-Michel, 1985. "On Endogenous Competitive Business Cycles," Econometrica, Econometric Society, vol. 53(5), pages 995-1045, September.
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    5. Grandmont, Jean-Michel & Pintus, Patrick & de Vilder, Robin, 1998. "Capital-Labor Substitution and Competitive Nonlinear Endogenous Business Cycles," Journal of Economic Theory, Elsevier, vol. 80(1), pages 14-59, May.
    6. Medio, Alfredo & Raines, Brian, 2007. "Backward dynamics in economics. The inverse limit approach," Journal of Economic Dynamics and Control, Elsevier, vol. 31(5), pages 1633-1671, May.
    7. Woodford, Michael, 1986. "Stationary sunspot equilibria in a finance constrained economy," Journal of Economic Theory, Elsevier, vol. 40(1), pages 128-137, October.
    8. Kennedy, Judy A. & Stockman, David R., 2008. "Chaotic equilibria in models with backward dynamics," Journal of Economic Dynamics and Control, Elsevier, vol. 32(3), pages 939-955, March.
    9. Cass, David & Shell, Karl, 1983. "Do Sunspots Matter?," Journal of Political Economy, University of Chicago Press, vol. 91(2), pages 193-227, April.
    10. Azariadis, Costas, 1981. "Self-fulfilling prophecies," Journal of Economic Theory, Elsevier, vol. 25(3), pages 380-396, December.
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    1. Gardini, Laura & Hommes, Cars & Tramontana, Fabio & de Vilder, Robin, 2009. "Forward and backward dynamics in implicitly defined overlapping generations models," Journal of Economic Behavior & Organization, Elsevier, vol. 71(2), pages 110-129, August.
    2. Ajevskis, Viktors, 2019. "Nonlocal Solutions To Dynamic Equilibrium Models: The Approximate Stable Manifolds Approach," Macroeconomic Dynamics, Cambridge University Press, vol. 23(6), pages 2544-2571, September.
    3. Dufourt, Frédéric & Nishimura, Kazuo & Venditti, Alain, 2015. "Indeterminacy and sunspots in two-sector RBC models with generalized no-income-effect preferences," Journal of Economic Theory, Elsevier, vol. 157(C), pages 1056-1080.
    4. Antoine Le Riche & Francesco Magris, 2016. "Decreasing Transaction Costs and Endogenous Fluctuations in a Monetary Model," Economics Bulletin, AccessEcon, vol. 36(4), pages 2381-2393.
    5. Theodore Palivos & Dimitrios Varvarigos, 2010. "Education and growth: A simple model with complicated dynamics," International Journal of Economic Theory, The International Society for Economic Theory, vol. 6(4), pages 367-384, December.
    6. Zhang, Guang & Zhang, Ruixuan & Yan, Yubin, 2020. "The diffusion-driven instability and complexity for a single-handed discrete Fisher equation," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    7. Magris, Francesco, 2012. "Indeterminacy and multiple steady states with sector-specific externalities," Economic Modelling, Elsevier, vol. 29(6), pages 2664-2672.
    8. Frédéric Dufourt & Kazuo Nishimura & Carine Nourry & Alain Venditti, 2017. "Sunspot Fluctuations in Two-Sector Models with Variable Income Effects," Studies in Economic Theory, in: Kazuo Nishimura & Alain Venditti & Nicholas C. Yannelis (ed.), Sunspots and Non-Linear Dynamics, chapter 0, pages 71-96, Springer.
    9. Paolo Mattana & Kazuo Nishimura & Tadashi Shigoka, 2012. "Homoclinic Bifurcation and Global Indeterminacy of Equilibrium in a Two-Sector Endogenous Growth Model," Springer Books, in: John Stachurski & Alain Venditti & Makoto Yano (ed.), Nonlinear Dynamics in Equilibrium Models, edition 127, chapter 0, pages 427-451, Springer.
    10. Stefano Bosi & David Desmarchelier, 2017. "A simple method to study local bifurcations of three and four-dimensional systems: characterizations and economic applications," Working Papers of BETA 2017-07, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
    11. Alain Raybaut, 2021. "A first French episode in the renewal of nonlinear theory of economic cycles," Post-Print hal-03375804, HAL.
    12. Bosi, Stefano & Desmarchelier, David, 2019. "Local bifurcations of three and four-dimensional systems: A tractable characterization with economic applications," Mathematical Social Sciences, Elsevier, vol. 97(C), pages 38-50.
    13. Amable, Bruno, 1995. "Endogenous growth and cycles through radical and incremental innovation," CEPREMAP Working Papers (Couverture Orange) 9504, CEPREMAP.
    14. Ferraris, Leo & Watanabe, Makoto, 2011. "Collateral fluctuations in a monetary economy," Journal of Economic Theory, Elsevier, vol. 146(5), pages 1915-1940, September.
    15. Deng, Liuchun & Khan, M. Ali, 2018. "On Mitra’s sufficient condition for topological chaos: Seventeen years later," Economics Letters, Elsevier, vol. 164(C), pages 70-74.
    16. Al-Hdaibat, Bashir & Govaerts, Willy & Neirynck, Niels, 2015. "On periodic and chaotic behavior in a two-dimensional monopoly model," Chaos, Solitons & Fractals, Elsevier, vol. 70(C), pages 27-37.
    17. Orlando Gomes, 2010. "Ordinary Least Squares Learning And Nonlinearities In Macroeconomics," Journal of Economic Surveys, Wiley Blackwell, vol. 24(1), pages 52-84, February.
    18. Ingrid Kubin & Laura Gardini, 2022. "On the significance of borders: the emergence of endogenous dynamics," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 17(1), pages 41-62, January.
    19. Kenji Sato & Makoto Yano, 2013. "Optimal ergodic chaos under slow capital depreciation," International Journal of Economic Theory, The International Society for Economic Theory, vol. 9(1), pages 57-67, March.
    20. Francesco MAGRIS & Daria ONORI, 2020. "The (de)Stabilizing Role of Fiscal and Monetary Policy," LEO Working Papers / DR LEO 2746, Orleans Economics Laboratory / Laboratoire d'Economie d'Orleans (LEO), University of Orleans.
    21. Antoine Riche & Francesco Magris & Daria Onori, 2020. "Monetary rules in a two-sector endogenous growth model," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 69(4), pages 1049-1100, June.

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    More about this item

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • E32 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Business Fluctuations; Cycles

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