Halvorsen

Systems / ODEs / Chaotic attractors

A cyclically symmetric quadratic flow with a tightly wound multi-scroll attractor.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

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

Parameters

Symbol	Default	Role
a	1.4	linear damping coefficient
b	4	coupling coefficient between the three variables

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - 3 a$
constant

Equilibria

2 equilibria
0 stable · 2 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.Halvorsen()
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

Sprott (2010), Elegant Chaos, World Scientific
doi:10.1142/9789812838827

BibTeX

@misc{halvorsen,
  title = {Halvorsen system},
  note = {Sprott (2010), Elegant Chaos, World Scientific},
  doi = {10.1142/9789812838827}
}