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<p>In this paper, the emergence of Parrondo’s paradox for fractional-order (FO) dynamical systems driven by the Parameter Switching (PS) algorithm is investigated. The PS algorithm runs by switching the control parameter within a predetermined set of values during the numerical integration of the initial value problem, so that the resulting trajectory approximates the existing attractor corresponding to the averaged parameter. By extending the operation and application of PS algorithm to FO systems, it is demonstrated analytically, numerically, and experimentally verified that Parrondo’s paradox occurs in FO systems. The analysis relies on the convergence properties of the PS algorithm applied via the fractional Adams-Bashforth-Moulton (ABM) scheme, showing that the switched solution approximates the averaged solution accurately. Numerical simulations on a dark-matter–dark-energy system, on a highly nonlinear system, and also on an electronic circuit implementation of a FO Chen system all confirm the presence of Parrondo-like behavior. The findings quantify the convergence of the PS algorithm in FO systems and confirm that the switching scheme consistently yields the paradoxical behavior predicted by the averaged parameter model.</p>

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Parrondo’s paradox in fractional order systems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月8日

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  • Marius-F. Danca,
  • Guanrong Chen

摘要

In this paper, the emergence of Parrondo’s paradox for fractional-order (FO) dynamical systems driven by the Parameter Switching (PS) algorithm is investigated. The PS algorithm runs by switching the control parameter within a predetermined set of values during the numerical integration of the initial value problem, so that the resulting trajectory approximates the existing attractor corresponding to the averaged parameter. By extending the operation and application of PS algorithm to FO systems, it is demonstrated analytically, numerically, and experimentally verified that Parrondo’s paradox occurs in FO systems. The analysis relies on the convergence properties of the PS algorithm applied via the fractional Adams-Bashforth-Moulton (ABM) scheme, showing that the switched solution approximates the averaged solution accurately. Numerical simulations on a dark-matter–dark-energy system, on a highly nonlinear system, and also on an electronic circuit implementation of a FO Chen system all confirm the presence of Parrondo-like behavior. The findings quantify the convergence of the PS algorithm in FO systems and confirm that the switching scheme consistently yields the paradoxical behavior predicted by the averaged parameter model.