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

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Definition

\[ \begin{aligned} \dot{y_{0}} &= a \sin{\left(y_{2} \right)} + c \cos{\left(y_{1} \right)} \\ \dot{y_{1}} &= a \cos{\left(y_{2} \right)} + b \sin{\left(y_{0} \right)} \\ \dot{y_{2}} &= b \cos{\left(y_{0} \right)} + c \sin{\left(y_{1} \right)} \end{aligned} \]

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

Properties

Lyapunov spectrum

$+0.007897,\; +0.001071,\; -0.008968$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 3$

Divergence ∇·f

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

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Arnold (1966), J. Appl. Math. Mech. 30, 223-226

BibTeX

@misc{arnoldbeltramichildress,
  title = {ArnoldBeltramiChildress system},
  note = {Arnold (1966), J. Appl. Math. Mech. 30, 223-226}
}