Stability of Cycles in a Game of Rock-Scissors-Paper-Lizard-Spock

Finite switching near heteroclinic networks

“…Such a result rectifies lemma 3.7 in [10] and is consistent with the correction in [11] that was first made to the intersection of the sets F i12 with C 12k in [10]. The heteroclinic networks (a) from [13] and (b) from [9]. All nodes are distribution nodes.…”

“…For simplicity, we ignore them because they do not enter into significant terms of shadow trajectories. 9 The image of ϕ ijk is the set of (v out , z out ) ∈ H out,k j whose pre-image under ϕ ijk is in C ijk . 10 Comparing with figure 11(b), the sets F i12 in figure 6 in [10] are not well-defined.…”

“…This provides yet another proof of the absence of switching in the Kirk and Silber network of [14], this time along the connection shared by the two heteroclinic cycles. Other examples may be found in the heteroclinic network of the two-person RPS game studied in [13] and in the heteroclinic network of the Rock-Scissors-Paper-Lizard-spock (RSPLS) game of [9], see figure 13. Proposition 3.8.…”

“…Several forms of finite switching exist, that is, a wide variety of finite heteroclinic paths can be followed by trajectories near the RSPLS network for a specific set of parameter values such that the heteroclinic cycles in the network are unstable. See [9,23].…”

“…We focus on the wide class of quasi-simple heteroclinic networks, defined in section 2, but our results apply to any heteroclinic network provided the geometric constrains we identify are satisfied. For the reader concerned with applications we stress that replicator dynamics in game theory and population dynamics naturally exhibits quasi-simple heteroclinic sub-networks 4 satisfying the additional assumption on the global maps, as pointed out in Castro et al [9]. Our results apply also to sub-networks of the ac-heteroclinic networks studied in Podvigina et al [20], to the heteroclinic networks considered by Afraimovich et al [1], as well as to all simple heteroclinic networks of types B, C, and Z that first appear in the work of Krupa and Melbourne [15] for the first two types, and Podvigina [19], for the latter.…”