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Growing through chaos in the Matsuyama map via subcritical flip and bistability


  • Laura Gardini

    () (Department of Economics, Society & Politics, Universit? di Urbino "Carlo Bo")

  • Iryna Sushko

    () (Institute of Mathematics, NASU, and Kyiv School of Economics, Ukraine)


Recent publications reconsider the growth model proposed by Matsuyama ("Growing through cycles"), also called M-map, presenting new interpretation of the model as well as new results on its dynamic behaviors. The goal of the present paper is to give the rigorous proof of some results which were remaining open, related to the dynamics of this model. We prove that in the whole parameter range of interest an attracting 2-cycle appears via border collision bifurcation, we give the explicit flip bifurcation value of the 2-cycle proving that it is always of subcritical type. This leads to bistability related to coexistence of an attracting 2-cycle with attracting 4-cyclic chaotic intervals. We give the conditions related to the sharp transition to chaos, proving that the cascade of stable cycles of even periods cannot occur. The parameter range in which repelling cycles of odd period exist is further investigated, giving an explicit boundary, as well as its relation to the non existence of cycles of period three. Length: 25 pages

Suggested Citation

  • Laura Gardini & Iryna Sushko, 2018. "Growing through chaos in the Matsuyama map via subcritical flip and bistability," Working Papers 1801, University of Urbino Carlo Bo, Department of Economics, Society & Politics - Scientific Committee - L. Stefanini & G. Travaglini, revised 2018.
  • Handle: RePEc:urb:wpaper:18_01

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    References listed on IDEAS

    1. Matsuyama, Kiminori, 2001. "Growing through Cycles in an Infinitely Lived Agent Economy," Journal of Economic Theory, Elsevier, vol. 100(2), pages 220-234, October.
    2. Sunaga, Miho, 2017. "Endogenous growth cycles with financial intermediaries and entrepreneurial innovation," Journal of Macroeconomics, Elsevier, vol. 53(C), pages 191-206.
    3. Anjan Mukherji, 2005. "Robust cyclical growth," International Journal of Economic Theory, The International Society for Economic Theory, vol. 1(3), pages 233-246, September.
    4. Mitra, Tapan, 2001. "A Sufficient Condition for Topological Chaos with an Application to a Model of Endogenous Growth," Journal of Economic Theory, Elsevier, vol. 96(1-2), pages 133-152, January.
    5. Kiminori Matsuyama, 1999. "Growing Through Cycles," Econometrica, Econometric Society, vol. 67(2), pages 335-348, March.
    6. Gardini, Laura & Sushko, Iryna & Naimzada, Ahmad K., 2008. "Growing through chaotic intervals," Journal of Economic Theory, Elsevier, vol. 143(1), pages 541-557, November.
    Full references (including those not matched with items on IDEAS)

    More about this item


    Endogenous growth models; Matsuyama map; Piecewise smooth map; Subcritical flip bifurcation; Border collision bifurcation; Skew tent map as a normal form.;

    JEL classification:

    • O41 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - One, Two, and Multisector Growth Models
    • E32 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Business Fluctuations; Cycles
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D90 - Microeconomics - - Micro-Based Behavioral Economics - - - General

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