Abstract We explore two low-dimensional dynamical systems modeling transition to turbulence in shear flows to try to understand the nature of the boundary ∂B of the basin of attraction B of the stable, laminar point at the origin of coordinates. Components of ∂B are found to exist in two types: one (the ‘strong’ type) separating B from a complementary set where orbits never relaminarize, and a second (the ‘weak’ type) separating B into two parts locally but not globally. For a boundary of weak type, orbits on each side relaminarize but may be distinguished from one another by features such as orbital complexity and time to relaminarize. The basin boundary may be of a single type, or may be a union of components of different types. The models are parametrized and may transform from one type to another at a critical parameter value. In the models studied here the change from purely strong type to a union of the two types occurs via the collapse of two sheets of a strong boundary into a single sheet. This is accompanied, at the critical value of the parameter, by the appearance of a homoclinic orbit and the subsequent occurrence of a periodic orbit on the strong part of the boundary.