Chaos and Parrondo's paradox: An overview

Abstract

Parrondo's paradox (PP) is a fundamental principle in nonlinear science where the alternation of individually losing strategies leads to a winning outcome. In this topical review, we provide the first systematic panorama of the synergy between PP and chaos. We observe a bidirectional connection between the two areas. The first direction is the translation of PP into the interplay between Order and Chaos through either Chaos + Chaos $\to$ Order (CCO) or Order + Order $\to$ Chaos (OOC). In this vein, many quantifiers, such as Lyapunov Exponents, $\lambda$, and entropic measures, are used. Second, we note that chaos can be used to engineer switching protocols that can lead to nontrivial effects in diverse PP cases. Our review clarifies the universality of PP and highlights its robust theoretical and practical applications across several areas of science and technology. Finally, we delineate key open questions, emphasizing the unresolved theoretical limits, the role of high-dimensional maps and continuous flows, and the critical need for more experimental verification of the dynamic PP in chaotic systems. For completeness, we also provide a full Python code that allows the reader to observe the many facets of the PP.

24 pages and 7 figures. Python code complements this review: https://github.com/PiresMA/chaos_parrondo_paradox

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