Persistence of remote-fold singular cycles in three-timescale systems

Let the assumptions concerning the parameters, reduced flow, and slow drift hold, and suppose that the singularities of the system are remote. Let $\theta$ denote the system parameter, and let $\theta_{SH}^{\pm}$ and $\theta_r^{\pm}$ be the corresponding saddle-homoclinic and return thresholds. For sufficiently small positive $\varepsilon$ and $\delta$, consider $mu$ in the parameter intervals specified below. Persistence conjecture. There exist positive sufficiently small $\varepsilon_0$ and $\delta_0$ such that, for $(\varepsilon,\delta)\in(0,\varepsilon_0)\times(0,\delta_0)$, the system exhibits either MMOs with single epochs of SAOs and Farey segments of type $L^s$ or $L_s$, with $s>0$, for $mu$ in appropriate subintervals of $(mu_{SH}^-,mu_r^-+\mathcal{O}(\varepsilon+\delta))\cup(mu_r^++\mathcal{O}(\varepsilon+\delta),mu_{SH}^+)$, or two-timescale relaxation oscillations for $mu$ in an appropriate subinterval of $(mu_r^-,mu_r^+)+\mathcal{O}(\varepsilon+\delta)$. This conjecture asserts that the corresponding families of singular cycles persist in a full neighborhood of both $\delta=0$ and $\varepsilon=0$, uniformly in $\varepsilon$ and $\delta$.