Stability of nonlinear phase motion from the study of oscillator mappings

A fundamental approach to study the intrinsic stability of phase motion at resonances of oscillator differential equations is reported. An averaging technique was developed to replace the continuous problem by a discrete phase mapping. A criterium of structural stability of resonances was found to numerically plot the diagram of frequency locked states. Typical diagrams have several layers corresponding to seperate power laws in the frequency fluctuation. By studying Hoedler continuity of frequency in parameter space (Omega, c) the phenomenon of Hoedler exponent resonance associated to fractal sets was discovered. In the case epsilon = 0, previous power laws associated to mode lock-in or irrational windings are recovered. Scaling laws associated to the decay in time of frequency fluctuations are studied. In the case of a small epsilon not equal to 0, a wide region of 1/f frequency noise is observed at the bottom of Arnold tongues.