Lorenz

Systems / ODEs / Chaotic attractors

The 1963 convection model — three coupled quadratic ODEs whose butterfly attractor launched modern chaos theory.

continuous · ODE3 dimensionschaoticdissipative

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Definition

\[ \begin{aligned} \dot{x} &= \sigma \left(- x + y\right) \\ \dot{y} &= \rho x - x z - y \\ \dot{z} &= - \beta z + x y \end{aligned} \]

Parameters

Symbol	Default	Role
sigma	10	Prandtl number
rho	28	Rayleigh ratio (drives the bifurcation)
beta	2.66667	geometric aspect ratio

State variables: x, y, z

Properties

Lyapunov spectrum

$+0.906,\; 0,\; -14.57$
Sprott (2003), Chaos and Time-Series Analysis

Kaplan–Yorke dimension

$D_{KY} = 2.062$

Divergence ∇·f

$\nabla\!\cdot f = - \beta - \sigma - 1$
constant

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Lorenz (1963), J. Atmos. Sci. 20, 130-141

BibTeX

@misc{lorenz,
  title = {Lorenz system},
  note = {Lorenz (1963), J. Atmos. Sci. 20, 130-141}
}