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

\[ \begin{aligned} \dot{y_{0}} &= - \frac{\mu \sin{\left(y_{2} \right)}}{\left(\cos{\left(y_{2} \right)} + \cos{\left(y_{3} \right)} + \cos{\left(y_{4} \right)} + 4\right)^{2}} \\ \dot{y_{1}} &= - \frac{\mu \sin{\left(y_{3} \right)}}{\left(\cos{\left(y_{2} \right)} + \cos{\left(y_{3} \right)} + \cos{\left(y_{4} \right)} + 4\right)^{2}} \\ \dot{y_{2}} &= y_{0} \\ \dot{y_{3}} &= y_{1} \\ \dot{y_{4}} &= w \end{aligned} \]

Parameters

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

Properties

Lyapunov spectrum

$+0.01226,\; +0.01147,\; -0.01224,\; -0.01149,\; 0$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 5$

Divergence ∇·f

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

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Froeschlé, Guzzo & Lega (2000), Science 289, 2108

BibTeX

@misc{arnoldweb,
  title = {ArnoldWeb system},
  note = {Froeschlé, Guzzo & Lega (2000), Science 289, 2108}
}