Lorenz84

Systems / ODEs / Chaotic attractors

Lorenz's low-order atmospheric circulation model — regime behavior in the mid-latitude jet.

continuous · ODE3 dimensionschaoticclimate

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= a f - a x - y^{2} - z^{2} \\ \dot{y} &= - b x z + g + x y - y \\ \dot{z} &= b x y + x z - z \end{aligned} \]

Parameters

Symbol	Default	Role
a	1.32	damping of the wind current
b	7.91	damping of the eddies
f	4.83	thermal (symmetric) forcing amplitude
g	4.194	asymmetric forcing amplitude

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - a + 2 x - 2$
state-dependent

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.Lorenz84()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference

Lorenz (1984), Tellus 36A, 98-110
doi:10.3402/tellusa.v36i2.11473

BibTeX

@misc{lorenz84,
  title = {Lorenz84 system},
  note = {Lorenz (1984), Tellus 36A, 98-110},
  doi = {10.3402/tellusa.v36i2.11473}
}