Autoparametric resonance of relaxation oscillations

Abstract

Stable normal mode vibrations in engineering can be undesirable and one of the possibilities for quenching these vibrations is by embedding the oscillator in an autoparametric system by coupling to a damped oscillator. We have the possibility of destabilising the undesirable vibrations by a suitable tuning and choice of coupling parameters. In the case of normal mode vibration derived from a relaxation oscillations we need low-frequency tuning of the attached oscillator.An additional feature is that to make the quenching effective we also have to deform the slow manifold by choosing appropriate coupling; this is illustrated for van der Pol relaxation.A number of numerical experiments have been done and show some interesting phenomena, such as a chaotic attractor and effective quenching.