Naturally occurring bifurcation in the amplitude parameter. (a)–(c) Three examples of syllables sang by a canary that transition naturally between a period 2 syllable (grey shaded) and a period 1 (white). Top experimental panels show the sonogram, top simulation panels the model average activity of the forcing (a.u.) and lower panels show the pressure patterns (a.u.). Acoustical almost identical output occurs twice in each period 2 respiratory pattern. Modeled syllables go through a period 2 bifurcation when the forcing amplitude at in a 2:1 locking regime is increased linearly ( ) until reaching a 1:1 regime at . (d) Period of the syllables against time aligned to the first period 1 syllable. Linear fits before and after transition show a constant syllabic frequency ( ) before and after transition with 0 slope and a y-intercept halved. Full symbols and line are for experimental data and open symbols and dashed line are for the modeled data. Linear y-intercepts before transition are and for experiment and model, respectively, and after transition for both.
Naturally occurring bifurcation in the frequency parameter is manipulated with temperature. (a) Three examples of experimental song renditions of pulsatile syllables (grey shaded) followed by period 2 syllables (white) at different HVC temperature variations: normal, and . As temperature is decreased, the time of regime change is shifted to the left. (b) Syllabic period against time aligned to the first p2 syllable. Period rises with temperature decrease, showing the effect of syllabic frequency reduction of the cooling. There is a naturally occurring period increase before transition. (c) Total duration of pulsatile syllables is decreased with temperature variation in HVC, showing a second effect of cooling: the earlier reach of syllable transition. Tendency line has a slope of . Total songs analyzed are 71 across all temperatures.
Locking regimes and bifurcations in canary respiratory gestures. (a) Diagram shows colored regions representing the obtained locking regimes between the forcing frequency and the output pressure pattern. Full vertical and horizontal arrows correspond to the postulated path in parameter space used for the generation of syllables of the canaries of Figs. 1 and 2 , respectively. Cooling induces an increase in pattern length, giving slower frequencies and shifting to the left the original values. Dashed arrow shows this effect on the horizontal path: Pulsatile syllable length is shorter, since green region is reached earlier. Bifurcations occur when following an arrow one changes to a different locking region. Red is 1:1 locking, green 2:1, orange 1:1 above constant expiration, cyan 2:1 above constant expiration, grey 4:1. (b) Modeled pressure pattern along the horizontal full arrow path in (a). Starting at and 2.25 in amplitude, a linear decrease of is applied in the forcing activity (shown in the upper panel in a.u.), finishing at , same as experimental gesture of Fig. 2 . (c) Modified diagram using forcing deltas of constant duration (see Eq. (3) ). Here, we find also locking regimes 3:1 in blue and 6:1 in white. Rearrangement of regions retains main bifurcations of the map.
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