ArnoldWeb

Systems / ODEs / Exotic systems

A perturbed three-degree-of-freedom Hamiltonian whose overlapping resonance lines form the Arnold web governing slow chaotic diffusion in nearly integrable systems.

continuous · ODE5 dimensionsconservative

spatiotemporal field

Definition

\[ \begin{aligned} \dot{p1} &= - \frac{\mu \sin{\left(x_{1} \right)}}{\left(\cos{\left(x_{1} \right)} + \cos{\left(x_{2} \right)} + \cos{\left(z \right)} + 4\right)^{2}} \\ \dot{p2} &= - \frac{\mu \sin{\left(x_{2} \right)}}{\left(\cos{\left(x_{1} \right)} + \cos{\left(x_{2} \right)} + \cos{\left(z \right)} + 4\right)^{2}} \\ \dot{x1} &= p_{1} \\ \dot{x2} &= p_{2} \\ \dot{z} &= w \end{aligned} \]

Parameters

Symbol	Default	Role
mu	0.01	perturbation strength coupling the three actions through the web
w	1	frequency of the driving angle

State variables: p1, p2, x1, x2, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = 0$
constant

Equilibria

none found (no equilibria)

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.ArnoldWeb()
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

Froeschlé, Guzzo & Lega (2000), Science 289, 2108
doi:10.1126/science.289.5487.2108

BibTeX

@misc{arnoldweb,
  title = {ArnoldWeb system},
  note = {Froeschlé, Guzzo & Lega (2000), Science 289, 2108},
  doi = {10.1126/science.289.5487.2108}
}