When we reconstruct a bird song as a time-delay attractor, we start to see beautiful patterns: spirals, twists, petals, and loops. But where do these shapes come from?
The answer lies in the biomechanics of how birds produce sound — and how we mathematically capture it.
A bird's song is shaped by several interacting physical components:
The syrinx — the bird’s vocal organ, with vibrating membranes (labia)
Air pressure — controlled by the lungs and air sacs
Muscle control — modulating membrane tension and frequency
Amplitude envelope — how loudness changes over time
Frequency modulation — how pitch varies moment by moment
These components create a sound wave whose amplitude and frequency are constantly varying.
To understand the attractor patterns, we can model a bird song as:
A sinusoidal wave with both:
A slowly changing amplitude envelope
A rapidly changing frequency (FM signal)
Mathematically, something like:
y(t) = A(t) · sin(2π · f(t) · t)
Where:
A(t) = slowly varying envelope
f(t) = frequency modulation
t = time
This interaction between fast and slow changes is crucial.
When you reconstruct the attractor using time-delay embedding, you’re capturing both:
The fast oscillations (like pitch cycles)
The slow modulations (envelope, breathing, syllable structure)
Slow changes in A(t) cause the attractor to spiral inward or outward
Sudden jumps or frequency changes cause twists and bends
If there are two interacting periodicities (e.g. vibrato + envelope), we get torus-like structures
The sampling frequency used during recording can also alias patterns into petal shapes — a digital artifact of a continuous system
Birdsong often shows attractors that wrap around like a donut or torus. Why?
Because:
One frequency represents the carrier (e.g. pitch oscillation)
The other represents the modulation (e.g. breathing or phrase rhythm)
This dual-periodicity causes the attractor to “wrap around” in phase space — a behavior seen in many natural systems like heartbeats, speech, and even planetary motion.
By just looking at these attractors, we’re seeing a geometric fingerprint of the bird's physical control system.
The attractor doesn’t just “look pretty” — it encodes the vocal behavior of the bird.
Differences in shape reflect different muscle patterns, breathing rhythms, or control strategies.
Birdsong emerges from two intertwined rhythms:
Fast oscillations of the syrinx membranes (carrier frequency, tens–hundreds of Hz)
Slow modulations of airflow pressure and membrane tension (amplitude envelope and pitch slides, over tens–hundreds of ms)
Together, these create a 2‑torus: a mathematical shape that combines one circle for the fast oscillation and one for the slow envelope.
We only ever record one number at a time—sound pressure y(t)y(t). To recover the hidden 2‑torus, we use Takens’ time‑delay embedding. In plain text:
X(t) = [ y(t), y(t + τ), y(t + 2τ) ]
y(t)y(t) is the audio amplitude at time tt.
τ (tau) is a time delay chosen so that y(t)y(t) and y(t+τ)y(t+τ) carry new information (e.g. via mutual‑information).
Embedding dimension m≥2m ≥ 2 unfolds the torus so its loops and twists become visible.
When you plot those points in 2D (y(t)y(t) vs. y(t+τ)y(t+τ)) or 3D (adding y(t+2τ)y(t+2τ)), you often see the flower‑ or spiral‑like attractor shapes.
To explain the shapes, consider a toy signal with both amplitude modulation (AM) and frequency modulation (FM):
y(t) = E(t) · sin( φ(t) )
where
E(t) = 1 + α · sin(ω_slow · t) (slow amplitude envelope)
φ(t) = ω_fast · t + β · sin(ν · t) (fast carrier + slow frequency chirp)
E(t) makes the loudness rise and fall (loop radius).
ω_fast sets one loop per oscillation.
β · sin(ν · t) adds a gradual pitch slide or trill (twist of the loops).
When embedded, the slow envelope makes loops grow and shrink; the fast carrier makes each loop; and the FM term rotates each loop relative to the last—producing spirals and twists.
Loop count = number of carrier cycles in your time window.
Radial expansion/contraction = amplitude envelope E(t)E(t).
Angular twist = frequency modulation term eta·sin(ν·t).
Together these effects trace out a toroidal shape in state-space that looks like petals or a spiral when projected into 2D or 3D.
Resolution: you need enough samples per carrier period (fs/ωfastf_s/ω_fast) to see smooth loops.
Petal count: the number of petals equals how many fast cycles occur in your recording. Under‑sampling can blur or distort petals.
Because two frequencies interact—fast and slow—the reconstructed attractor is topologically a torus (S¹ × S¹). Embedding projects that torus into 2D/3D so we see its donut‑flower shape.
Interpretation: loops ↔ syringeal oscillations; radius ↔ volume envelope; twist ↔ pitch modulation.
Quantification: measure torus radii, twist angles, sampling effects to compare species or individual variability.
Modeling: fit the toy AM+FM equation (or use Sparse ID / Neural ODE) to infer physiological parameters (α, β, ω_fast, ω_slow).
Experiments: build physical rigs that emulate the envelope and chirp to validate the mapping from code to craft.