IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v199y2025ip2s0960077925007258.html

Complex dynamics of discrete-time replicators in repeated Snowdrift Games with four strategies

Author

Listed:
  • Zhang, Yafei
  • Tian, Haiyan
  • Zhang, Gang

Abstract

This paper investigates the dynamics of discrete-time replicators in repeated Snowdrift Games with four strategies. A three-dimensional discrete-time dynamical system is proposed to model these repeated Snowdrift Games. The existence of equilibrium points within the system is classified, and their local stabilities are thoroughly studied. Utilizing the center manifold theorem and bifurcation theory, it is demonstrated that the system undergoes flip bifurcations. The chaos control is studied by feedback control strategies for the understudied discrete system. Numerical simulations are conducted to validate the theoretical results, revealing that the system exhibits complex dynamical behaviors, including multiple periodic orbits and chaotic behavior. The maximum Lyapunov exponent, time series graphs, and bifurcation diagrams confirm the chaotic dynamical behaviors of the system.

Suggested Citation

  • Zhang, Yafei & Tian, Haiyan & Zhang, Gang, 2025. "Complex dynamics of discrete-time replicators in repeated Snowdrift Games with four strategies," Chaos, Solitons & Fractals, Elsevier, vol. 199(P2).
    • Handle: RePEc:eee:chsofr:v:199:y:2025:i:p2:s0960077925007258
    DOI: 10.1016/j.chaos.2025.116712
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077925007258
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2025.116712?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Jorgen W. Weibull, 1997. "Evolutionary Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262731215, December.
    2. Stephen Lynch, 2007. "Dynamical Systems with Applications using Mathematica®," Springer Books, Springer, number 978-0-8176-4586-1, August.
    3. repec:fth:iniesr:487 is not listed on IDEAS
    4. Ross Cressman, 2003. "Evolutionary Dynamics and Extensive Form Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262033054, December.
    5. Xu, Yan & Feng, Meiling & Zhu, Yuying & Xia, Chengyi, 2022. "Multi-player snowdrift game on scale-free simplicial complexes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 604(C).
    6. Wang, Qiushuang & Chen, Xiaojie & Szolnoki, Attila, 2025. "Coevolutionary dynamics of feedback-evolving games in structured populations," Chaos, Solitons & Fractals, Elsevier, vol. 193(C).
    7. Christoph Hauert & Michael Doebeli, 2004. "Spatial structure often inhibits the evolution of cooperation in the snowdrift game," Nature, Nature, vol. 428(6983), pages 643-646, April.
    8. Cui, Qianqian & Zhang, Qiang & Qiu, Zhipeng & Hu, Zengyun, 2016. "Complex dynamics of a discrete-time predator-prey system with Holling IV functional response," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 158-171.
    9. Ernst Fehr & Simon Gächter, 2002. "Altruistic punishment in humans," Nature, Nature, vol. 415(6868), pages 137-140, January.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Yue, Hongyu & Xiong, Xiaojin & Feng, Minyu & Szolnoki, Attila, 2025. "Dynamic evolution of cooperation based on adaptive reputation threshold and game transition," Chaos, Solitons & Fractals, Elsevier, vol. 199(P1).
    2. Yanlong Zhang, 2015. "Partially and Wholly Overlapping Networks: The Evolutionary Dynamics of Social Dilemmas on Social Networks," Computational Economics, Springer;Society for Computational Economics, vol. 46(1), pages 1-14, June.
    3. Xiang Wei & Peng Xu & Shuiting Du & Guanghui Yan & Huayan Pei, 2021. "Reputational preference-based payoff punishment promotes cooperation in spatial social dilemmas," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 94(10), pages 1-7, October.
    4. Zhao, Zhengwu & Zhang, Chunyan, 2023. "The mechanisms of labor division from the perspective of task urgency and game theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 630(C).
    5. Bin Wu & Julián García & Christoph Hauert & Arne Traulsen, 2013. "Extrapolating Weak Selection in Evolutionary Games," PLOS Computational Biology, Public Library of Science, vol. 9(12), pages 1-7, December.
    6. Rong-Hua Li & Jeffrey Xu Yu & Jiyuan Lin, 2013. "Evolution of Cooperation in Spatial Traveler's Dilemma Game," PLOS ONE, Public Library of Science, vol. 8(3), pages 1-11, March.
    7. Wang, Zhen & Chen, Tong & Wang, Yongjie, 2017. "Leadership by example promotes the emergence of cooperation in public goods game," Chaos, Solitons & Fractals, Elsevier, vol. 101(C), pages 100-105.
    8. Cárdenas, Juan-Camilo & Gómez, Santiago & Mantilla, César, 2019. "Between-group competition enhances cooperation in resource appropriation games," Ecological Economics, Elsevier, vol. 157(C), pages 17-26.
    9. Chen, Qiao & Chen, Tong & Wang, Yongjie, 2017. "Publishing the donation list incompletely promotes the emergence of cooperation in public goods game," Applied Mathematics and Computation, Elsevier, vol. 310(C), pages 48-56.
    10. Silveira, Douglas & Vasconcelos, Silvinha, 2020. "Essays on duopoly competition with asymmetric firms: Is profit maximization always an evolutionary stable strategy?," International Journal of Production Economics, Elsevier, vol. 225(C).
    11. Quan, Ji & Yang, Haoliang & Wang, Xianjia, 2025. "Cooperation dynamics in a two-state spatial prisoner's dilemma game with reputation-driven transitions," Chaos, Solitons & Fractals, Elsevier, vol. 197(C).
    12. Balkenborg, Dieter & Hofbauer, Josef & Kuzmics, Christoph, 2016. "Refined best reply correspondence and dynamics," Center for Mathematical Economics Working Papers 451, Center for Mathematical Economics, Bielefeld University.
    13. Jean Rabanal & Daniel Friedman, 2014. "Incomplete Information, Dynamic Stability and the Evolution of Preferences: Two Examples," Dynamic Games and Applications, Springer, vol. 4(4), pages 448-467, December.
    14. Egas, Martijn & Riedl, Arno, 2005. "The Economics of Altruistic Punishment and the Demise of Cooperation," IZA Discussion Papers 1646, IZA Network @ LISER.
    15. Dirk Helbing & Anders Johansson, 2010. "Cooperation, Norms, and Revolutions: A Unified Game-Theoretical Approach," PLOS ONE, Public Library of Science, vol. 5(10), pages 1-15, October.
    16. Zhu, Wenqiang & Pan, Qiuhui & Song, Sha & He, Mingfeng, 2023. "Effects of exposure-based reward and punishment on the evolution of cooperation in prisoner’s dilemma game," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    17. Marc Harper & Dashiell Fryer, 2015. "Lyapunov Functions for Time-Scale Dynamics on Riemannian Geometries of the Simplex," Dynamic Games and Applications, Springer, vol. 5(3), pages 318-333, September.
    18. Yang, Yujin & Zhao, Dawei & Wang, Juan, 2025. "Evolution of cooperation in spatial public goods games driven by reinforcement learning and environmental feedback," Chaos, Solitons & Fractals, Elsevier, vol. 199(P1).
    19. Rocio Botta & Gerardo Blanco & Christian E. Schaerer, 2021. "Fractional Punishment of Free Riders to Improve Cooperation in Optional Public Good Games," Games, MDPI, vol. 12(1), pages 1-21, February.
    20. Cheng, Jiangjiang & Mei, Wenjun & Su, Wei & Chen, Ge, 2023. "Evolutionary games on networks: Phase transition, quasi-equilibrium, and mathematical principles," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 611(C).

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:199:y:2025:i:p2:s0960077925007258. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.