Interactively explore the reconstruction of dynamical attractors from birdsong recordings using Takens' embedding theorem.
This demo allows you to:
Apply time-delay embedding to real birdsong data.
Adjust the embedding dimension and time lag parameters.
Visualize the reconstructed attractor in 3D.
Observe how changes in parameters influence the topological structure of the attractor.
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:
Ο (sigma): Prandtl number
Ο (rho): Rayleigh number
Ξ² (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 β regions in phase space toward which trajectories tend to evolve:
Try teh following demonstrator for that
Takens' Theorem (1981) proves that, under certain conditions, the dynamics of a system can be faithfully reconstructed using just one observable β by plotting time-delayed copies of the signal.Β Each vector becomes a point in an embedding space. The collection of such points reconstructs the attractor β 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.Β
To demonstrate how embedding works, we can take the x-variable from the Lorenz system and reconstruct its dynamics using delay coordinates.