Duffing

Systems / ODEs / Chaotic attractors

A driven double-well oscillator — the textbook forced-chaos system with a fractal basin boundary.

continuous · ODE3 dimensionsforcedchaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= y_{1} \\ \dot{y_{1}} &= - \alpha y_{0} - \beta y_{0}^{3} - \delta y_{1} + \gamma \cos{\left(y_{2} \right)} \\ \dot{y_{2}} &= \omega \end{aligned} \]

Parameters

Symbol	Default	Role
alpha	1	linear stiffness coefficient
beta	-1	cubic stiffness coefficient (negative = double well)
delta	0.1	linear damping coefficient
gamma	0.35	amplitude of the periodic forcing
omega	1.4	angular frequency of the periodic forcing

Properties

Lyapunov spectrum

TODO — Lyapunov compute exceeded 20s (killed)

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

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

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Duffing (1918), Erzwungene Schwingungen bei veränderlicher Eigenfrequenz, Vieweg, Braunschweig

BibTeX

@misc{duffing,
  title = {Duffing system},
  note = {Duffing (1918), Erzwungene Schwingungen bei veränderlicher Eigenfrequenz, Vieweg, Braunschweig}
}