The attractor of an equation of Tricomi's type

Consider the attractor A of an equation of pendulum type with friction driven by a constant torque. The results of M. Levi and also obtained independently by Q. Min, S. Xian, and Z. Jinyan show that if the friction coefficient is larger than a certain bound then A is homeomorphic to the circle. We shall study the bifurcation diagram of a particular class of equations of pendulum type and show that the bounds obtained before are optimal.