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

\[ \begin{aligned} \dot{y_{0}} &= a f - a y_{0} - y_{1}^{2} - y_{2}^{2} \\ \dot{y_{1}} &= - b y_{0} y_{2} + g + y_{0} y_{1} - y_{1} \\ \dot{y_{2}} &= b y_{0} y_{1} + y_{0} y_{2} - y_{2} \end{aligned} \]

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

Properties

Lyapunov spectrum

$+0.4477,\; +0.02054,\; -2.709$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.173$

Divergence ∇·f

$\nabla\!\cdot f = - a + 2 y_{0} - 2$
state-dependent

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.Lorenz84()
traj = sys.integrate(final_time=100.0, dt=0.01)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

Reference

Lorenz (1984), Tellus 36A, 98-110

BibTeX

@misc{lorenz84,
  title = {Lorenz84 system},
  note = {Lorenz (1984), Tellus 36A, 98-110}
}