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¶

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

Parameters¶

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

Properties¶

Lyapunov spectrum

$+0.614,\; +0.01294,\; -4.827$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.13$

Divergence ∇·f

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

Equilibria

2 equilibria
0 stable · 2 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Sprott (2010), Elegant Chaos, World Scientific

BibTeX

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