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

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