Algorithm for the Determination of the Resonances of Anharmonic Damped Oscillators

In addition to the passive observation of a non­ linear oscillator and the description of the measured data with Lyapounow exponents [1, 2], fractal dimensions [3-5], entropies [6], etc. it is possible to charcterize a non-linear oscillator by an active method, namely by determining its response to specific perturbations. The resonances of harmonic systems which are brought about by a sinusoidal perturbation and a systematic variation of the fre­ quency are of major significance. Resonance spec­ troscopy has proved very successfull in many fields of physics. Huberman and Crutchfield [7] and many other groups (see, for example, [8 10]) have shown that damped oscillators with marked nonlinearity respond to a purely periodic perturbation with complex, in many cases chaotic dynamics. This chaotic response is difficult to characterize, com­ paratively small, nonresonant, because the driving force and velocity of the oscillator are out of phase [11]. The following approach shall attempt to derive resonant driving mechanisms for damped non-linear oscillators.