Suppose I have an autonomous system of ordinary differential equations and I want to show that I have a trapping region: a region of phase space which trajectories can enter but can never leave. One way to do this is to show that the vector field points "inward" on the boundary of the region, which is how trapping regions are presented in many books.
Is it possible to weaken this condition and just require that the vector field not point "outward"? In other words, if the vector field is inward pointing or tangent to the boundary everywhere on the boundary, is that enough to have a trapping region?
I am assuming that the system has unique solutions, otherwise there is the following counter-example: Let $\frac{dy}{dt} = -3 y^{2/3}$. Consider the region to be the positive real line. Then at the boundary of the region the vector field is zero, but there is still the solution $y=-t^3$ that goes from inside the region to outside.