ArnoldBeltramiChildress

Systems / ODEs / Climate & geophysics

An exact steady Beltrami solution of Euler's equations whose streamlines are generically chaotic — the classic example of Lagrangian chaos and fast-dynamo theory.

continuous · ODE3 dimensionschaotic advection

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= a \sin{\left(z \right)} + c \cos{\left(y \right)} \\ \dot{y} &= a \cos{\left(z \right)} + b \sin{\left(x \right)} \\ \dot{z} &= b \cos{\left(x \right)} + c \sin{\left(y \right)} \end{aligned} \]

Parameters

Symbol	Default	Role
a	1.73205	amplitude of the first Beltrami component
b	1.41421	amplitude of the second Beltrami component
c	1	amplitude of the third Beltrami component

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = 0$
constant

Equilibria

none found (no equilibria)

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.ArnoldBeltramiChildress()
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

Arnold (1966), J. Appl. Math. Mech. 30, 223-226
doi:10.1016/0021-8928(66)90070-0

BibTeX

@misc{arnoldbeltramichildress,
  title = {ArnoldBeltramiChildress system},
  note = {Arnold (1966), J. Appl. Math. Mech. 30, 223-226},
  doi = {10.1016/0021-8928(66)90070-0}
}