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Stability of heteroclinic cycles: a new approach

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  • Telmo Peixe
  • Alexandre A. Rodrigues

Abstract

This paper analyses the stability of cycles within a heteroclinic network lying in a three-dimensional manifold formed by six cycles, for a one-parameter model developed in the context of game theory. We show the asymptotic stability of the network for a range of parameter values compatible with the existence of an interior equilibrium and we describe an asymptotic technique to decide which cycle (within the network) is visible in numerics. The technique consists of reducing the relevant dynamics to a suitable one-dimensional map, the so called \emph{projective map}. Stability of the fixed points of the projective map determines the stability of the associated cycles. The description of this new asymptotic approach is applicable to more general types of networks and is potentially useful in computational dynamics.

Suggested Citation

  • Telmo Peixe & Alexandre A. Rodrigues, 2022. "Stability of heteroclinic cycles: a new approach," Papers 2204.00848, arXiv.org.
    • Handle: RePEc:arx:papers:2204.00848
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    References listed on IDEAS

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    1. Hassan Najafi Alishah & Pedro Duarte & Telmo Peixe, 2019. "Asymptotic Poincaré Maps along the Edges of Polytopes," Working Papers REM 2019/70, ISEG - Lisbon School of Economics and Management, REM, Universidade de Lisboa.
    2. Telmo Peixe, 2019. "Permanence in Polymatrix Replicators," Working Papers REM 2019/69, ISEG - Lisbon School of Economics and Management, REM, Universidade de Lisboa.
    3. Rodrigues, Alexandre A.P., 2013. "Persistent switching near a heteroclinic model for the geodynamo problem," Chaos, Solitons & Fractals, Elsevier, vol. 47(C), pages 73-86.
    4. Gaunersdorfer Andrea & Hofbauer Josef, 1995. "Fictitious Play, Shapley Polygons, and the Replicator Equation," Games and Economic Behavior, Elsevier, vol. 11(2), pages 279-303, November.
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