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¶

\[ \begin{aligned} \dot{y_{0}} &= y_{1} \\ \dot{y_{1}} &= y_{2} \\ \dot{y_{2}} &= - a y_{0} - b y_{1} - c y_{2} + d y_{0}^{3} \end{aligned} \]

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

Properties¶

Lyapunov spectrum

$+0.204,\; +0.001341,\; -1.205$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.17$

Divergence ∇·f

$\nabla\!\cdot f = - c$
constant

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Arneodo, Coullet & Tresser (1980), Phys. Lett. A 79, 259-263

BibTeX

@misc{arneodo,
  title = {Arneodo system},
  note = {Arneodo, Coullet & Tresser (1980), Phys. Lett. A 79, 259-263}
}