Analysis with slow-fast decomposition approach in a birdsong production model

Abstract

In this thesis, we investigate a dynamical system that models sound production in birds. This model was developed by Amador at the University of Buenos Aires and involves quantitative details of the physiological process. Our focus is on a full investigation of the amplitude and frequency content that the model can generate. Mathematically, the birdsong production model can be studied using a slow-fast decomposition approach. This is a common method to characterise complex oscillating behaviour in systems with two different time scales. It is a four-dimensional model with two fast and two slow variables. The slow variables for this model are scaled pressure pr and a non-linear restitution force k1. These two components are responsible for the oscillatory motion of the membrane that generates different spectral content, producing sound of different frequencies. We compute the bifurcation diagram in the (pr; k1)-plane, that is, we calculate all equilibrium points, their stability, and observed types of local and global bifurcations like saddle-node, cusp, Hopf, and homoclinic bifurcations. We characterise all possible dynamics in the physically realistic model and find that it is different from what is reported by Amador. In particular, we find an additional fourth equilibrium that is involved in a third saddle-node bifurcation. As a consequence, there are parameter regimes where no equilibria are present, which means that any oscillatory motion is also ruled out in these regimes. Furthermore, the presence of a fourth equilibrium increases the spectral richness displayed by the model, which improves the similarity with that found in bird songs. Periodic motion arises from a Hopf bifurcation and there exist periodic orbits in the lower and upper regions of the (pr; k1)-parameter plane. We focus on the lower region, which has not been studied before. The two key ingredients needed to create sound for birds are amplitude and frequency. Hence, we compute curves along which the periodic orbits have a specific, fixed period. We also calculate the amplitudes along these curves, which provides information about the locus of curves with fixed amplitudes. Our overall findings are that the scaled pressure pr is primarily responsible for selecting the frequency of the produced sound, while the restitution force k1, along with pr, are used to determine its amplitude. We further conclude that this lower parameter region not only corresponds to physically relevant values, but the spectral richness present in this regime also suggest that the birds may well produce their songs from this range of pressures and restitution forces.