The effect of inversely unstable solutions on the attractor of the forced pendulum equation with friction

Consider the attractor of a periodically forced equation of pendulum type with linear friction, in the cylinder. Levi and independently Min, Xian and Jinyan show that if the friction coefficient is larger than a certain bound then is homeomorphic to the circle. We shall give a topological version of the definition of inversely unstable solution of N. Levinson and show that the appearance of such solutions imply that is not homeomorphic to the circle. As an application we shall show that the bounds on the friction coefficient obtained before are optimal.