Coullet¶

Systems / ODEs / Chaotic attractors

The Arneodo–Coullet–Tresser jerk flow in its Volterra regime, whose homoclinic (Shilnikov) connection spawns a spiral strange attractor.

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`	`-0.8`	linear position feedback
`b`	`1.1`	linear velocity feedback
`c`	`0.45`	linear acceleration feedback
`d`	`-1`	cubic nonlinearity coefficient

State variables: x, y, z

Properties¶

Lyapunov spectrum

$+0.1546,\; -0.003396,\; -0.6012$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.251$

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