A dynamical system is one whose state evolves over time according to a set of rules — often described by differential equations. The complete set of all possible system states forms a phase space (or state space), where each point represents a unique configuration of the system.
As the system evolves from a starting point, it traces out a trajectory in phase space. The collection of all such trajectories forms a phase portrait, which gives insight into the system’s long-term behavior.
In 1963, Edward Lorenz developed a simplified model of atmospheric convection. His system describes the motion of a 2D fluid layer heated from below and cooled from above, using three coupled nonlinear differential equations:
x: rate of convection
y: temperature difference between rising/falling air
z: vertical temperature difference
The system depends on three parameters:
σ (sigma): Prandtl number
ρ (rho): Rayleigh number
β (beta): geometric factor
Though deterministic, the Lorenz system exhibits a range of dynamic behaviors depending on these parameters.
As we vary the parameters of the Lorenz system, we observe different attractors — regions in phase space toward which trajectories tend to evolve:
Fixed Point Attractor (ρ = 13.0):
Trajectories converge to a stationary state.
Limit Cycle Attractor (ρ = 350.0):
Trajectories settle into a periodic orbit.
Strange Attractor (ρ = 28.0):
Trajectories form a chaotic, butterfly-shaped path. They never repeat but remain confined, and are sensitive to initial conditions — the hallmark of chaos.
These attractors reveal the long-term structure of the system: where it goes, whether it repeats, and how it reacts to tiny changes.
In practical cases — like analyzing birdsong — we don’t have full access to the underlying system variables (e.g., brain signals or syrinx muscle dynamics). We often have just a single scalar measurement: the sound pressure, recorded as a time series.
So, the question arises:
Can we reconstruct the hidden dynamics of the system from a single observed variable?
The answer is yes, using a method called delay coordinate embedding.
Takens' Theorem (1981) proves that, under certain conditions, the dynamics of a system can be faithfully reconstructed using just one observable — by plotting time-delayed copies of the signal.
Given a scalar time series y(t), we form vectors like:
X(t)=[y(t), y(t+τ), y(t+2τ),…, y(t+(m−1)τ)]
τ is the time delay
m is the embedding dimension
Each vector becomes a point in an embedding space. The collection of such points reconstructs the attractor — a proxy for the original phase space.
Even though we don't observe the full system, we recover its topological structure, preserving key properties like attractor dimension and Lyapunov exponents.
To demonstrate how embedding works, we can take the x-variable from the Lorenz system and reconstruct its dynamics using delay coordinates.
Using:
m=3m = 3m=3 (3D embedding)
different values of τ\tauτ
We observe:
For small τ (e.g., τ = 3), the embedded points are tightly clustered — highly correlated.
For large τ (e.g., τ = 60), the structure becomes complex and overfolded.
An intermediate τ (e.g., τ = 16) gives a reconstructed attractor that closely resembles the true Lorenz attractor.
This shows how critical the choice of τ and m is for effective reconstruction.
Birdsong production involves complex dynamics in the bird's brain and vocal system. But we only measure the sound pressure — a 1D signal.
By applying Takens’ method to the audio waveform, we reconstruct a multi-dimensional attractor that reveals:
The structure of song patterns
Potential markers of species, individuals, or states
New ways to visualize and classify vocalizations based on their dynamics
This is what our project does:
It turns birdsong into geometry — letting us “see” the hidden system behind the sound.