Basins of attraction of periodic solutions in a multistable model of self-oscillating musical instrument

Abstract

Self-sustained musical instruments, modeled as nonlinear dynamical systems, can exhibit a wide range of dynamical regimes. This includes non-oscillating regimes, periodic oscillations corresponding to musical notes and non-periodic behaviors such as quasiperiodicity. Several regimes may coexist stably for identical parameter values, a phenomenon known as multistability. In this case, which regime is observed depends only on the initial conditions. We consider a simple model of single-reed instrument written as a system of four ordinary differential equations. A bifurcation analysis with the blowing pressure as bifurcation parameter shows that several stable periodic regimes – corresponding to distinct musical notes – coexist on a range of the blowing pressure. The implications of this multistable dynamics are explored by calculating the boundaries of the basins of attraction associated with each stable regime, i.e. the set of initial conditions leading to a particular regime. Since the system has a four-dimensional phase space, the direct visualization of the basins boundaries is not straightforward. We introduce a method inspired by the construction of the Poincaré section to visualize basins boundaries (referred to as separatrices) in a three-dimensional subspace of the phase space, by computing intersections of separatrices with cross-sections of the phase space defined as hyperplanes orthogonal to a particular stable periodic orbit. This yields a parametrized description of the basins boundaries, which can be visualized as a movie. Finally, we argue that the geometry of the basins of attraction provides insight into the sensitivity of periodic regimes to perturbations and, as such, on the instrument’s playability.