The stationary Instability in quasi-Reversible Systems and the Lorenz Pendulum

We study the resonance at zero frequency in presence of a neutral mode in quasi-reversible systems. The asymptotic normal form is derived and it is shown that in the presence of a reflection symmetry it is equivalent to the set of real Lorenz equations. Near the critical point an analytical condition for the persistence of an homoclinic curve is calculated and chaotic behavior is then predicted and its existence verified by direct numerical simulation. A simple mechanical pendulum is shown to be an example of the instability, and preliminary experimental results agree with the theoretical predictions.