Abstract:
Robust heteroclinic cycles and networks are a common occurrence in game theory where the interaction of agents, or traits in a population, are described by continuous-time dynamics in the positive orthant of finite dimensional Euclidean space or in the simplex. The heteroclinic connections on the boundary of state space determine which is the winning agent or stronger trait indicating the evolution of the respective variables with time. Nevertheless, the dynamics of initial conditions near the cycle or network, but not part of a heteroclinic connection, can be very diverse.

The dynamics near a heteroclinic network can be better understood if the network is asymptotically stable and, in this case, additional information can be obtained from the stability of the cycles in the network. I shall present some tools to establish the stability of the cycles (and connections) in the heteroclinic network [2], and illustrate how the stability relates to the outcome of a game. Examples include the 2-person Rock-Scissors-Paper game [3], the Rock-Scissors-Paper-Lizard-Spock game [1], and the Jungle game [5].

The question of whether chaotic dynamics appears in these dynamics will be addressed briefly [4].

This is joint work with Ana Ferreira, Liliana Garrido-da-Silva, and Isabel Labouriau, all from the University of Porto, and may be found in

[1]  S.B.S.D. Castro, A. Ferreira, L. Garrido-da-Silva and I.S. Labouriau, Stability of cycles in a game of Rock-Scissors-Paper-Lizard-Spock, SIAM Journal on Applied Dynamical Systems (SIADS) (2022)

[2]  L. Garrido-da-Silva and S.B.S.D. Castro, Stability of quasi-simple heteroclinic cycles, Dynamical Systems: an International Journal, Vol. 34 (1), 14–39 (2019) 

[3]  L. Garrido-da-Silva and S.B.S.D. Castro, Cyclic dominance in a two-person rock-scissors-paper game, International Journal of Game Theory, Vol. 49, 885– 912 (2020)
[4]  L. Garrido-da-Silva and S.B.S.D. Castro, Finite switching near heteroclinic networks, Nonlinearity, Vol. 36, 6239–6259 (2023) 

[5]  S.B.S.D. Castro, A. Ferreira and I.S. Labouriau, Stability of cycles and survival in a Jungle Game with four species, Dynamical Systems: an international journal, to appear.