Cyclic R 2 (m) Lotka–Volterra Operators on Simplices: Classification on S 4 and Global Dynamics for Odd m
We study discrete Lotka–Volterra operators V : S m−1 → S m−1 whose interaction
graph is the 2-regular cycle C m , referred to as R 2 (m) configurations. For m = 5 (the simplex S 4 )
we classify all orientation patterns under the dihedral group D 5 via Burnside’s lemma, obtaining
exactly four isomorphism classes (I)–(IV). For each class we determine the P → Q repeller–attractor
structure, describe invariant polytopes via coordinate signatures, and prove global convergence of
trajectories. For general odd m ≥ 3 we prove that the directed-cycle operator possesses a unique
interior fixed point M, construct an explicit Lyapunov function ϕ M , and show that every non-fixed
interior trajectory converges to the boundary in forward time while its α-limit set is {M}.
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