make_lorenz63

kooplearn.datasets.make_lorenz63(X0, n_steps=100, dt=0.01, sigma=10.0, mu=28.0, beta=2.667)[source]

Generate a trajectory from the Lorenz-63 system.

The Lorenz-63 system is a simplified mathematical model of atmospheric convection that exhibits chaotic behavior. It is one of the most famous examples of deterministic chaos.

The system is governed by:

\[\begin{split}\begin{cases} \frac{dx}{dt} = \sigma(y - x) \\ \frac{dy}{dt} = x(\mu - z) - y \\ \frac{dz}{dt} = xy - \beta z \end{cases}\end{split}\]
Parameters:

  • X0 (array-like, shape (3,)) – Initial conditions [x, y, z].

  • n_steps (int, default 100) – Number of time steps to simulate.

  • dt (float, default 0.01) – Time step size for numerical integration.

  • sigma (float, default 10.0) – The \(\sigma\) parameter, controlling the ratio of the rate of heat conduction to the rate of convection.

  • mu (float, default 28.0) – The \(\mu\) parameter (also called \(\rho\)), representing the Rayleigh number.

  • beta (float, default 8/3) – The \(\beta\) parameter, related to the physical dimensions of the convection layer.

Returns:

df – Trajectory of the Lorenz-63 system with columns ['x', 'y', 'z'] and n_steps + 1 samples. Has a MultiIndex with levels ['step', 'time']. Metadata stored in df.attrs includes:

  • 'generator': 'make_lorenz63';

  • 'X0': initial conditions;

  • 'params': dict of all parameters.

Return type:

pandas.DataFrame

Examples

>>> import numpy as np
>>> X0 = np.array([1.0, 0.0, 0.0])
>>> df = make_lorenz63(X0, n_steps=1000, dt=0.01)
>>> df.shape
(1001, 3)
>>> df.columns.tolist()
['x', 'y', 'z']
>>> df.attrs['generator']
'make_lorenz63'

Access time values from index:

>>> times = df.index.get_level_values('time')

Generate classic chaotic trajectory from near the attractor:

>>> df_chaotic = make_lorenz63(
...     X0=[0.0, 1.0, 1.05],
...     n_steps=10000,
...     dt=0.01,
...     sigma=10.0,
...     mu=28.0,
...     beta=8.0/3.0
... )