Persistence of aligned or connected singular-cycle families

Let $\varepsilon$ and $\delta$ be positive perturbation parameters, and suppose that the assumptions concerning the parameters and reduced flow hold. Assume that the singularities of the system are aligned or connected, and let $mu_{SH}^-$ and $mu_{SH}^+$ denote the relevant parameter thresholds. Persistence conjecture. There exist positive sufficiently small $\varepsilon_0$ and $\delta_0$ such that, for $(\varepsilon,\delta)\in(0,\varepsilon_0)\times(0,\delta_0)$ and $mu$ in an appropriate subinterval of $(mu_{SH}^-,mu_{SH}^+)$, the system exhibits either MMOs with double epochs of SAOs and a Farey segment of type $1^s1_k$, with $s,k>0$; MMOs with one epoch of SAOs and one non-SAO epoch of slow dynamics and a Farey segment of type $1^s1_0$ or $1_s1^0$, with $s>0$; or three-timescale relaxation oscillations and a Farey segment of type $1^01_0$. These predicted MMO and relaxation-oscillation families are expected to persist under sufficiently small perturbations.