Basins of attraction in a dynamical system with a time-varying control parameter: The case of attack transients in a simple model of reed musical instrument

Multistability, a common phenomenon in dynamical systems in general and in musical instruments in particular, corresponds to the coexistence of several stable regimes for a given set of parameters. When a system is multistable, which regime is observed depends on the initial conditions on the state variables, and therefore relates to the basins of attraction. Here we consider a simple model of reed musical instrument. A bifurcation analysis shows that the system is bistable on a range of the blowing pressure, which is one of the main control parameters for the musician. In the bistability region, the boundary between basins of attraction is computed for the classical case of constant control parameters in the one hand, and for saturating ramps of the blowing pressure on the other hand. These profiles of the blowing pressure are chosen as prototypical representations of the so-called attack transients of the blowing pressure, during which the musician starts to blow in the instrument to create a musical note from silence. Attack transients are known to be of crucial importance in a musical context, yet mostly overlooked in physical models of the musician–instrument system. The separatrix between basins of attraction is computed numerically by combining time-domain simulations with a machine learning technique relying on support vector machine. Our results demonstrate that attack transients of the blowing pressure significantly affect the practical observability of the different stable regimes, suggesting that they might be used by musicians to navigate between basins. From a more general point of view, these results relate to rate-induced tipping phenomena and might therefore be of interest beyond the particular case of musical instruments.