Conjecture on limit cycles in regions 1, 2, 4–8, and 12

Let $(\rho,\phi)$ be the parameters of the collective Parrondo game, and let the parameter plane be divided into the regions shown in Figure. For a finite word $w$ in the game symbols, write $\Delta_w$ for the set of initial states whose greedy trajectories begin with $w$, and let $[1,2]$ denote the corresponding limit cycle. Limit-cycle conjecture. If $(\rho,\phi)$ belongs to region 1, region 2, or one of regions 4, 5, 6, 7, and 8, then there is a unique asymptotically stable limit cycle of the form $[1,2]$, together with an unstable equilibrium. If $(\rho,\phi)$ belongs to region 12, then there is a unique globally asymptotically stable limit cycle of the form $[1,2]$. The basin descriptions are respectively given by the decompositions of $\Delta_A$ displayed in the conjecture, and trajectories from the indicated portions of $\Delta_B$ eventually enter $\Delta_A$. This conjecture describes the dynamics in regions not covered by the preceding theorems; the displayed decompositions provide the proposed mechanism for eventual entry into the attracting cycle, but the claims are not established in the supplied text.