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

Abstract.

In this work, motivated by non-ideal mechanical systems, we investigate the following O.D.E.

${\dot{x} = f (x) + \varepsilon g (x, t) + \varepsilon^{2}\widehat{g} (x, t, \varepsilon)}$ , where ${x \in \Omega \subset \mathbb{R}^n}$ , ${g,\widehat{g}}$ are T periodic functions of t and there is a ₀ ∈ Ω 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.