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Parrondo’s paradox or chaos control in discrete two-dimensional dynamic systems

Author

Listed:
  • Mendoza, Steve A.
  • Matt, Eliza W.
  • Guimarães-Blandón, Diego R.
  • Peacock-López, Enrique

Abstract

In ecological modeling, seasonality can be represented as an alternation between environmental conditions. This concept of alternation holds common ground between ecologists and chemists, who design time-dependent settings for chemical reactors to influence the yield of a desired product. In this study and for a variety of maps, we consider a switching strategy that alternates between two undesirable dynamics that yields a stable desirable dynamic behavior. By comparing bifurcation diagrams of a map and its alternate version, we can easily find parameter values, which, on their own, yield chaotic orbits. When alternated, however, the parameter values yield a stable periodic orbit. Our analysis of the two-dimensional (2-D) maps is an extension of our previous work with one-dimensional (1-D) maps. In the case of 2-D maps, we consider the Beddington, Free, and Lawton and Udwadia and Raju maps. For these 2-D maps, we not only show that we can find “chaotic” parameters for the so-called “chaos” + “chaos” = “periodic” case, but we find two new “desirable” dynamic situations: “quasiperiodic” + “quasiperiodic” = “periodic” and “chaos” + “chaos” = “periodic coexistence.” In the former case, the alternation of chaotic dynamics yield two different periodic stable orbits implying the coexistence of attractors.

Suggested Citation

  • Mendoza, Steve A. & Matt, Eliza W. & Guimarães-Blandón, Diego R. & Peacock-López, Enrique, 2018. "Parrondo’s paradox or chaos control in discrete two-dimensional dynamic systems," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 86-93.
  • Handle: RePEc:eee:chsofr:v:106:y:2018:i:c:p:86-93
    DOI: 10.1016/j.chaos.2017.11.011
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    References listed on IDEAS

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    1. Silva, Emily & Peacock-Lopez, Enrique, 2017. "Seasonality and the logistic map," Chaos, Solitons & Fractals, Elsevier, vol. 95(C), pages 152-156.
    2. Levinsohn, Erik A. & Mendoza, Steve A. & Peacock-López, Enrique, 2012. "Switching induced complex dynamics in an extended logistic map," Chaos, Solitons & Fractals, Elsevier, vol. 45(4), pages 426-432.
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    Cited by:

    1. Mendoza, Steve A. & Peacock-López, Enrique, 2018. "Switching induced oscillations in discrete one-dimensional systems," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 35-44.
    2. Lai, Joel Weijia & Cheong, Kang Hao, 2022. "Risk-taking in social Parrondo’s games can lead to Simpson’s paradox," Chaos, Solitons & Fractals, Elsevier, vol. 158(C).

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