Asymptotically autonomous differential equations in the plane II. Stricter Poincaré/Bendixson type results

We consider planar asymptotically autonomous ordinary differential equations and their limit equations. We show that thew-limit sets of an asymptotically autonomous equation get very close to having the same properties (of Poincare/Bendixson type) as the Γ-limit sets of the corresponding limit equation provided that the non-autonomous perturbation decreases to 0 fast enough as time tends to infinity.