Is there switching for replicator dynamics and bimatrix games?

We consider heteroclinic networks for replicator dynamics and bimatrix games, that is, in a simplex or product of simplices, with equilibria at the vertices and connections at the edges–edge networks. Switching dynamics near a heteroclinic network occurs whenever every (infinite) sequence of connect...

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Bibliographic Details
Published in:Physica. D 2011-09, Vol.240 (18), p.1475-1488
Main Author: Aguiar, Manuela A.D.
Format: Article
Language:English
Subjects:
Divergence-free vector field
Dynamics
Evolutionary dynamics
Exact sciences and technology
Games
Heteroclinic network
Joints
Networks
Nonlinear dynamics
Physics
Switching
Switching dynamics
Switching theory
Trajectories
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Summary:We consider heteroclinic networks for replicator dynamics and bimatrix games, that is, in a simplex or product of simplices, with equilibria at the vertices and connections at the edges–edge networks. Switching dynamics near a heteroclinic network occurs whenever every (infinite) sequence of connections in the network is shadowed by at least one trajectory in its neighborhood. Aguiar and Castro [M.A.D. Aguiar, S.B.S.D. Castro Chaotic switching in a two-person game, Physica D 239 (16), 1598–1609] prove switching near an edge network for the dynamics of the rock–scissors–paper game. Here we give conditions for switching dynamics in general bimatrix games and show that switching near an edge network can never occur for replicator dynamics. ► We consider edge heteroclinic networks for replicator dynamics and bimatrix games. ► We study the existence of switching near such networks. ► We prove that there is no switching in replicator dynamics. ► We give conditions for switching dynamics in bimatrix games. ► The mechanism for switching is conservative dynamics and no Kirk & Silber subnetwork.
ISSN:0167-2789
1872-8022
DOI:10.1016/j.physd.2011.06.016