Non-Local Methods for Pendulum-Like Feedback Systems

0 are already deecribed by I. Newton (116]. However it was 250 years later that F. Tricorni (147] carried out the first non-local qualitative investigation of equation (0.1) with arbitrary o ~ 0 and "'{ ~ 0. It was proved by F. Tricorni that any solution of (0.1) with o > 0 corresponds either to a rotatory motion or to a damped oscillatory motion. Moreover, he showed that in the non-trivial case "'! :::; 1 there exists a bifurcation value ocr("'!) corresponding to a separatrix-loop, i.e. to a double-asymptotic to a saddle-point trajectory. For o < ocr("'!) equation (0.1) admits damped oscillations as weil as rotatory motions. For o > ocr("'') global asymptotic stability takes place, i.e. every motion is a damped oscillation. The papers of F. Tricorni became familiar immediately.