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Historic Behavior in Rock–Paper–Scissor Dynamics

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Abstract

In the evolutionary dynamics of the Rock–Paper–Scissor game, it is common to see the emergence of heteroclinic cycles. The dynamics in the vicinity of a stable heteroclinic cycle is marked by intermittency, where an orbit remains close to the heteroclinic cycle, repeatedly approaching and lingering at the saddles for increasing periods of time, and quickly transitioning from one saddle to the next. This causes the time spent near each saddle to increase at an exponential rate. This highly erratic behavior causes the time averages of the orbit to diverge, a phenomenon known as historic behavior. The problem of describing persistent families of systems exhibiting historic behavior, known as Takens’ Last Problem, has been widely studied in the literature. In this paper, we propose a persistent and broad class of replicator equations generated by increasing functions which exhibit historic behavior wherein the slow oscillation of time averages of the orbit ultimately causes the divergence of higher-order repeated time averages.

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Notes

  1. The phenomenological scenario responsible for the formation of a persistent strange attractor in 3D polymatrix replicators was discussed in the literature [35]. This strange attractor is an attractive whirlpool that is generated by a repelling fixed point with complex eigenvalues.

  2. It is possible to choose mutually disjoint convex compact sets \({\mathbb {U}}_1, {\mathbb {U}}_2, {\mathbb {U}}_3\) while maintaining the validity of the results and their proofs. This is due to the fact that the only fixed points of the replicator equation are the vertices of the simplex and the interior point \(\textbf{p}\). As a result, the function \(\Vert \textbf{x}-{\mathcal {R}}(\textbf{x})\Vert \) is positive over the compact subset \(\Delta \) of the simplex which is composed of the complement of sufficiently small neighborhoods around the fixed points. By defining \(\delta :=\min \nolimits _{\textbf{x}\in \Delta }\Vert \textbf{x}-{\mathcal {R}}(\textbf{x})\Vert >0\), it is possible to choose convex compact sets \({\mathbb {U}}_1\subset (G_1\cup G_2){\setminus } {\mathbb {U}}_0, \ {\mathbb {U}}_2\subset (G_3\cup G_4){\setminus } {\mathbb {U}}_0, \ {\mathbb {U}}_3\subset (G_5\cup G_6){\setminus } {\mathbb {U}}_0\) such that the gap between any two of them is less than \(\delta \) in diameter along the boundary of the simplex. It can be demonstrated that as the orbit of the replicator equation approaches the boundary of the simplex, the transition from one convex compact set \({\mathbb {U}}_{i}\) to \({\mathbb {U}}_{i+1}\) results in at most one term of the orbit remaining within the gap between the sets. This deviation does not negatively impact the validity of the proof.

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Acknowledgements

The author is greatly indebted to two anonymous referees for carefully reading the manuscript and for providing such constructive comments and suggestions which substantially contributed to improving the quality and presentation of the paper.

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  1. College of Engineering and Technology, American University of the Middle East, Egaila, 54200, Kuwait

    Mansoor Saburov

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Saburov, M. Historic Behavior in Rock–Paper–Scissor Dynamics. Qual. Theory Dyn. Syst. 22, 121 (2023). https://doi.org/10.1007/s12346-023-00820-0

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