On the existence and stability of periodic orbits in non ideal problems: General results

In this work, motivated by non-ideal mechanical systems, we investigate the following O.D.E. x = f (x) + epsilon g (x, t) + epsilon(2)(g) over cap (x, t, epsilon), where x is an element of Omega R-n, g,(g) over cap are T periodic functions of t and there is a(0) is an element of Omega such that f (a(0)) = 0 and f'(a(0)) is a nilpotent matrix. When n = 3 and f(x) = (0, q (x(3)), 0) we get results on existence and stability of periodic orbits. We apply these results in a non ideal mechanical system: the Centrifugal Vibrator. We make a stability analysis of this dynamical system and get a characterization of the Sommerfeld Effect as a bifurcation of periodic orbits.