I found this problem in strogatz nonlinear dynamics . The theorem says, A closed phase space trahectory must enclose a fixed point .
The question is asked as, is this true for phase surfaces other that plane , e.g sphere, torus ?
Now, the question seems very unclear to me . For cylindrical or spherical phase surfaces, if the exception is permissible, it must be owing to the return of phi to same point after rotating by 2pi . But in the question, strogatz asked to check this . Is the way of checking completely trial and error method or there is some intuitive way of constructing equations ?