Stability of planar nonlinear switched systems

Let X and Y be two smooth vector fields on R-2, globally asymptotically stable at the origin, and consider the time-dependent nonlinear system q(t) = u(t) X (q(t)), where u : (0, infinity) -> {0, 1} is an arbitrary measurable function. Analyzing the topology of the set where X and Y are parallel, we give some sufficient and some necessary conditions foe global asymptotic stability, uniform with respect to u(.). Such conditions can be verified without any intergration or construction of a Lyapunov function, and they do not change under small perturbations of the vector fields.