Skip to content

Arneodo

Systems / ODEs / Chaotic attractors

A third-order jerk system with a cubic nonlinearity whose spiral Shilnikov attractor arises from a homoclinic bifurcation.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= y \\ \dot{y} &= z \\ \dot{z} &= - a x - b y - c z + d x^{3} \end{aligned} \]

Parameters

Symbol Default Role
a -5.5 linear feedback coefficient
b 4.5 linear feedback coefficient
c 1 linear feedback coefficient
d -1 cubic nonlinearity coefficient

State variables: x, y, z

Properties

Lyapunov spectrum
TODO — lyapunov_spectrum failed: AttributeError
Kaplan–Yorke dimension
TODO — requires a numeric Lyapunov spectrum
Divergence ∇·f
$\nabla\!\cdot f = - c$
constant
Equilibria
3 equilibria
0 stable · 3 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.Arneodo()
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

Arneodo, Coullet & Tresser (1980), Phys. Lett. A 79, 259-263
doi:10.1016/0375-9601(80)90342-4

BibTeX 
@misc{arneodo,
  title = {Arneodo system},
  note = {Arneodo, Coullet & Tresser (1980), Phys. Lett. A 79, 259-263},
  doi = {10.1016/0375-9601(80)90342-4}
}