Stability of Heteroclinic Cycles: A New Approach Based on a Replicator Equation
Tóm tắt
This paper analyses the stability of cycles within a heteroclinic network formed by six cycles lying in a three-dimensional manifold, for a one-parameter model developed in the context of polymatrix replicator equations. 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 projective map. The 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.
Tài liệu tham khảo
Afraimovich, V.S., Moses, G., Young, T.: Two-dimensional heteroclinic attractor in the generalized Lotka–Volterra system. Nonlinearity 29(5), 1645 (2016)
Aguiar, M.A.D.: Is there switching for replicator dynamics and bimatrix games? Physica D 240(18), 1475–1488 (2011)
Aguiar, M.A.D., Labouriau, I.S., Rodrigues, A.A.P.: Switching near a network of rotating nodes. Dyn. Syst. 25(1), 75–95 (2010)
Alishah, H.N., Duarte, P.: Hamiltonian evolutionary games. J. Dyn. Games 2(1), 33 (2015)
Alishah, H.N., Duarte, P., Peixe, T.: Conservative and dissipative polymatrix replicators. J. Dyn. Games 2(2), 157 (2015)
Alishah, H.N., Duarte, P., Peixe, T.: Asymptotic Poincaré maps along the edges of polytopes. Nonlinearity 33(1), 469 (2019)
Alishah, H.N., Duarte, P., Peixe, T.: Asymptotic dynamics of hamiltonian polymatrix replicators. Nonlinearity 36(6), 3182 (2023)
Ashwin, P., Postlethwaite, C.: On designing heteroclinic networks from graphs. Physica D 265, 26–39 (2013)
Barendregt, N.W., Thomas, P.J.: Heteroclinic cycling and extinction in May–Leonard models with demographic stochasticity. J. Math. Biol. 86(2), 30 (2023)
Bunimovich, L.A., Webb, B.Z.: Isospectral compression and other useful isospectral transformations of dynamical networks. Chaos Interdiscip. J. Nonlinear Sci. 22, 3 (2012)
Castro, S.B.S.D., Garrido-da Silva, L: Finite switching near heteroclinic networks. arXiv preprint arXiv:2211.04202 (2022)
Castro, S.B.S.D., Labouriau, I.S., Podvigina, O.: A heteroclinic network in mode interaction with symmetry. Dyn. Syst. 25(3), 359–396 (2010)
Castro, S.B., Ferreira, A., Garrido-da-Silva, L., Labouriau, I.S.: Stability of cycles in a game of Rock-Scissors-Paper-Lizard-Spock. SIAM J. Appl. Dyn. Syst. 21(4), 2393–2431 (2022)
Field, M.J.: Lectures on Bifurcations, Dynamics and Symmetry. CRC Press (2020)
Field, M., Swift, J.W.: Stationary bifurcation to limit cycles and heteroclinic cycles. Nonlinearity 4(4), 1001 (1991)
Garrido-da Silva, L., Castro, S.B.S.D.: Stability of quasi-simple heteroclinic cycles. Dyn. Syst. 34(1), 14–39 (2019)
Gaunersdorfer, A., Hofbauer, J.: Fictitious play, Shapley polygons, and the replicator equation. Games Econ. Behav. 11(2), 279–303 (1995)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42. Springer (2013)
Hofbauer, J., Sigmund, K.: Permanence for replicator equations. In: Dynamical Systems, pp. 70–91. Springer (1987)
Hofbauer, J., Sigmund, K., et al.: Evolutionary Games and Population Dynamics. Cambridge University Press (1998)
Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems, vol. 54. Cambridge University Press (1997)
Krupa, M., Melbourne, I.: Asymptotic stability of heteroclinic cycles in systems with symmetry. Ergodic Theory Dyn. Syst. 15(1), 121–147 (1995)
Labouriau, I.S., Rodrigues, A.A.P.: On Takens’ last problem: tangencies and time averages near heteroclinic networks. Nonlinearity 30(5), 1876 (2017)
Lohse, A.: Unstable attractors: existence and stability indices. Dyn. Syst. 30(3), 324–332 (2015)
Melbourne, I.: An example of a nonasymptotically stable attractor. Nonlinearity 4(3), 835 (1991)
Milnor, J.: On the concept of attractor. In: The Theory of Chaotic Attractors, pp. 243–264. Springer (1985)
Palis, J., de Melo, W.: Local stability. In: Geometric Theory of Dynamical Systems, pp. 39–90. Springer (1982)
Peixe, T.: Lotka–Volterra Systems and Polymatrix Replicators. ProQuest LLC, Ann Arbor. Thesis (Ph.D.)–Universidade de Lisboa (Portugal) (2015)
Peixe, T.: Permanence in polymatrix replicators. J. Dyn. Games (2019)
Peixe, T., Rodrigues, A.: Persistent strange attractors in 3d polymatrix replicators. Physica D 438, 133346 (2022)
Podvigina, O.: Stability and bifurcations of heteroclinic cycles of type z. Nonlinearity 25(6), 1887 (2012)
Podvigina, O., Ashwin, P.: On local attraction properties and a stability index for heteroclinic connections. Nonlinearity 24(3), 887 (2011)
Podvigina, O., Chossat, P.: Simple heteroclinic cycles. Nonlinearity 28(4), 901 (2015)
Podvigina, O., Chossat, P.: Asymptotic stability of pseudo-simple heteroclinic cycles in \(\mathbb{R} ^{4}\). J. Nonlinear Sci. 27(1), 343–375 (2017)
Podvigina, O., Castro, S.B.S.D., Labouriau, I.S.: Stability of a heteroclinic network and its cycles: a case study from Boussinesq convection. Dyn. Syst. 34(1), 157–193 (2019)
Podvigina, O., Castro, S.B.S.D., Labouriau, I.S.: Asymptotic stability of robust heteroclinic networks. Nonlinearity 33(4), 1757 (2020)
Postlethwaite, C.M., Rucklidge, A.M.: Stability of cycling behaviour near a heteroclinic network model of Rock-Paper-Scissors-Lizard-Spock. Nonlinearity 35, 1702 (2021)
Rodrigues, A.A.P.: Persistent switching near a heteroclinic model for the geodynamo problem. Chaos Solitons Fractals 47, 73–86 (2013)
Rodrigues, A.A.P.: Attractors in complex networks. Chaos Interdiscip. J. Nonlinear Sci. 27(10), 103105 (2017)
Ruelle, D.: Elements of Differentiable Dynamics and Bifurcation Theory. Elsevier (2014)
Smith, J.M., Price, G.R.: The logic of animal conflict. Nature 246(5427), 15–18 (1973)