Source-tract coupling in birdsong production.

Birdsong is a complex phenomenon, generated by a nonlinear vocal device capable of displaying complex solutions even under simple physiological motor commands. Among the peripheral physical mechanisms responsible for the generation of complex sounds in songbirds, the understanding of the dynamics emerging from the interaction between the sound source and the upper vocal tract remains most elusive. In this work we study a highly dissipative limit of a simple sound source model interacting with a tract, mathematically described in terms of a delay differential equation. We explore the system numerically and, by means of reducing the problem to a phase equation, we are capable of studying its periodic solutions. Close in parameter space to the point where the resonances of the tract match the frequencies of the uncoupled source solutions, we find coexistence of periodic limit cycles. This hysteresis phenomenon allows us to interpret recently reported features found in the vocalization of some songbirds, in particular, "frequency jumps."