Bifurcation Sequence in a Physical Model of Trumpet-like Instruments: From a Fixed Point to Chaos

cote interne IRCAM: Vergez98a/National audienceWe have built a numerical model of trumpet-like instruments. Since theunderstanding of the model's behavior is desirable for a musicalusage, we have studied the model in the framework of the theory of thenonlinear dynamical systems. The blowing pressure has been chosen asthe bifurcation parameter. We have been able to predict, according tothe frequential version of the Hopf theorem, the critical threshold atwhich a stable fixed point looses its stability and gives birth to aunique stable limit cycle. Moreover, amplitude and frequency of thelimit cycle have been forecasted to an excellent approximation. Bystill increasing the blowing pressure, a secondary supercritical Hopfbifurcation has been obtained, leading to a quasi-periodic motion on atwo-torus. Finally, with a further increase in blowing pressure, theprogressive destruction of the two-torus has been observed, leading toa chaotic motion