Global isochrons of a planar system near a phaseless set with saddle equilibria

Given an attracting periodic orbit of a system of ordinary differential equations, one can assign an asymptotic phase to any initial condition that approaches such a periodic orbit. All initial conditions with the same asymptotic phase lie on what is known as an isochron. Isochrons foliate the basin of attraction, and may have intriguing geometric properties. We present here two cases of a planar vector field for which the basin boundary — also referred to as the phaseless set — contains saddle equilibria and their stable manifolds. A continuation-based approach, in combination with Poincare compactification when the basin is unbounded, allows us to compute isochrons accurately and visualise them as smooth curves to clarify their overall geometry.