Figure 1

A syllable from the song of the magpie tanager (Cissopis leveriana) [18], an oscine songbird (bilateral syrinx with two sound sources). (a) Sonogram. In the interval labeled B, the two sound sources act independently. Something different apparently happens in intervals A and C, however, when the fundamental frequencies are close to each other and sidebands appear as parallel, additional strokes. The two sound sources seem to interact in some way, since the detailed time evolution of their fundamental frequencies now closely resemble each other. (b) Sound envelope. The occurrence of subharmonic frequencies and sidebands coincides with intervals of large sound amplitude and close fundamental frequencies (A and C). At the end of interval C the amplitude is small, but the frequencies are even closer. On the other hand, at the beginning of interval B the frequencies are very close but the amplitude is very small. Different renditions of the same sound by the bird share the same features described here.Reuse & Permissions

Figure 2

Unfoldings for the system of Eqs. (15, 16). (a) Case Ia. (b) Case II. (c) Case III. (d) Case IVa. Every unfolding has a nontrivial stable fixed point in the shaded regions. The dashed line [Eq. (17), with slope $\sim 1$ and perturbative offset from the origin] is the direction of increasing air sac pressure $p_s$. On increasing air sac pressure $p_s$, the system enters into the region of the nontrivial fixed point always very quickly, because its boundaries are always either very steep or very flat (slopes $\left|1/b\right|>40$ and $\left|c\right|<0.1$, respectively, for the parameter values listed in Table ).Reuse & Permissions

Figure 3

Map of the spectral content of the model Eqs. (1, 2, 3, 4, 5) as a function of air sac pressure $p_s$ and labial tension difference $\Delta k=k_1-k_2$ ($k_2$ is fixed at $8\times {10}^8{\text{s}}^{-2}$). Region 1 (open circles): constant solutions, no oscillation takes place. Region 2 (crosses): periodic solutions with two independent frequencies $F_1$ and $F_2$ (the fundamental frequencies of the two modes, both active) and their corresponding harmonic overtones at integer multiples of them, $n{F}_1$ and $m{F}_2$ (“harmonic frequencies”), and no other frequencies. Region 3 (black squares): oscillatory solutions where heterodyne frequencies are found (“subharmonic frequencies”). Heterodyne frequencies are not integer multiples of the fundamental frequencies, but are sums and differences of integer multiples of them: $n{F}_1+m{F}_2$, with $n,m∊\mathbb{Z}$. The Hopf-Hopf bifurcation takes place at the boundary between regions 1 and 2. Notice that the region 3 of subharmonic frequencies can be reached by either increasing air sac pressure $p_s$ (and thus increasing sound amplitude) or decreasing the difference in labial tension between the sides $\Delta k$ (and thus decreasing the difference between the fundamental frequencies). Parameter values other than $p_s$ and $k_1$ are listed in Table . The weak coupling approximation is not valid for the first few rows at the bottom $\left(\Delta k\sim 0.0\right)$.Reuse & Permissions

Figure 4

Contour plot of the radius vector $\mu$ in the unfolding parameter space, Eq. (18), as a function of air sac pressure $p_s$ and labial tension difference $\Delta k$. Larger values of $\mu$ are represented by darker shading. $\mu$ is increased by either increasing $p_s$ or decreasing $\Delta k$, moving the system farther from the bifurcation point. The shapes of these contours are in qualitative agreement with the shape of the boundary between regions 2 and 3 shown in Fig. 3.Reuse & Permissions

Figure 5

Simulation of two different vocal maneuvers that move the system into region 3. In both cases the spectrogram of the resulting acoustic pressure $p_i$ is plotted. (a) Air sac pressure $p_s$ changes abruptly around 0.07 s, from $3\times {10}^6$ to $1.5\times {10}^7\text{cm}{\text{s}}^{-2}$, and sidebands appear. Labial tensions are held constant at $k_1=1.1\times {10}^9{\text{s}}^{-2}$ and $k_2=8\times {10}^8{\text{s}}^{-2}$ $\left(\Delta k=3\times {10}^8{\text{s}}^{-2}\right)$. (b) Labial tension $k_1$ changes abruptly around 0.07 s, from $1.4\times {10}^9$ to $9\times {10}^8{\text{s}}^{-2}$, and sidebands appear. Parameters $k_2=8\times {10}^8{\text{s}}^{-2}$ and $p_s=7.5\times {10}^6\text{cm}{\text{s}}^{-2}$ are held constant. (c) The two vocal maneuvers plotted as paths in parameter space of Fig. 3. Both cases show harmonic spectral content before the change in the control parameter (every frequency is an integer multiple of one of the two lowest frequencies). After the change, frequencies appear that are not integer multiples of the fundamental frequencies before. Transients not shown.Reuse & Permissions

Figure 6

Numerical integration of our model’s Eqs. (7, 8). $R_1$ and $R_2$ are the amplitudes of the two interacting modes defined by Eq. (25). A simulation with parameter values near the Hopf-Hopf bifurcation shows that both modes are turned on with constant amplitudes (the dot near the bottom left corner, $p_s=1\times {10}^6\text{cm}{\text{s}}^{-2}$). However, when the system is set far from the bifurcation $\left(p_s=1.5\times {10}^7\text{cm}{\text{s}}^{-2}\right)$, the amplitudes of the modes $R_1$ and $R_2$ are not constant, but oscillatory. The spectrogram corresponding to this simulation is shown in Fig. 5a (second half). $\Delta k=3\times {10}^8{\text{s}}^{-2}$.Reuse & Permissions