Public Goods with Reward and Reputation. Dynamics on the four faces of the simplex *S*_{4} and on the invariant manifold *W*_{K} with *K* = 1. Introducing reputation destabilizes the asocial segment **G**_{3}**Q** (see Fig. 4), leading to complex dynamical behavior. As before, the line of fixed points **G**_{2}**G**_{3} is replaced by the transversal line *L* running through *S*_{4} and intersecting *W*_{K} in **m**. For *K* = 1, periodic orbits appear with a center in **m**. Depending on the parameter values, Δ determines the dynamics on *W*_{K} for *K* ≠ 1. If Δ < 0, then for *K* < 1, **m** turns into a source, and the state spirals towards the heteroclinic cycle **G**_{1}**G**_{2}**G**_{3}**G**_{4} and for *K* > 1, **m** becomes a sink, and all states spiral inwards and converge to **m**. If Δ > 0, the converse holds, and for Δ = 0 all orbits are periodic. Small random shocks send the state from one manifold to another and hence change the value of *K*. Therefore, the system never converges. Parameters: *c* = 1, *g* = 3, β = 2, γ = 1, μ = 0.1, ν = 0.1.

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