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Continuous unimodal maps in economic dynamics: On easily verifiable conditions for topological chaos

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  • Deng, Liuchun
  • Khan, M. Ali
  • Mitra, Tapan

Abstract

In this paper, we offer necessary and sufficient conditions for the presence of odd-period cycles and turbulence in a continuous unimodal interval map. The characterizations we present are original both to the economic and the mathematical literature, and go beyond existential assertions to easy verifiability. We apply our two theorems to six different canonical models in the literature on economic dynamics, all being grounded in the fact that their policy functions are given by continuous unimodal maps. An unintended outcome of the work presented here is to alert the economics profession to a richer conception of erratic and chaotic dynamics, one that goes considerably beyond the 1964-1975 Sharkovsky-Li-Yorke emphasis on three-period cycles and on uncountable scrambled sets.

Suggested Citation

  • Deng, Liuchun & Khan, M. Ali & Mitra, Tapan, 2022. "Continuous unimodal maps in economic dynamics: On easily verifiable conditions for topological chaos," Journal of Economic Theory, Elsevier, vol. 201(C).
    • Handle: RePEc:eee:jetheo:v:201:y:2022:i:c:s0022053122000369
    DOI: 10.1016/j.jet.2022.105446
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    Cited by:

    1. Tomohiro Uchiyama, 2024. "Characterising the existence of an odd period cycle in price dynamics for an exchange economy," Economics Bulletin, AccessEcon, vol. 44(3), pages 983-989.
    2. Tomohiro Uchiyama, 2023. "A necessary and sufficient condition for the existence of chaotic dynamics in an overlapping generations model," Papers 2310.15755, arXiv.org.
    3. Tomohiro Uchiyama, 2023. "A necessary and sufficient condition for the existence of chaotic dynamics in a neoclassical growth model with a pollution effect," Papers 2311.03594, arXiv.org.
    4. Tomohiro Uchiyama, 2024. "A Necessary and Sufficient Condition for the Existence of Chaotic Dynamics in an Overlapping Generations Model," SN Operations Research Forum, Springer, vol. 5(2), pages 1-12, June.
    5. Tomohiro Uchiyama, 2023. "Odd period cycles and ergodic properties in price dynamics for an exchange economy," Papers 2309.09176, arXiv.org, revised Apr 2024.
    6. Tomohiro Uchiyama, 2024. "A necessary and sufficient condition for the existence of chaotic dynamics in a neoclassical growth model with a pollution effect," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 12(1), pages 79-86, June.

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    More about this item

    Keywords

    Odd-period cycles; Turbulence; Topological chaos; Economic dynamics;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • O11 - Economic Development, Innovation, Technological Change, and Growth - - Economic Development - - - Macroeconomic Analyses of Economic Development
    • O40 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - General

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