SwingingAtwood

Systems / ODEs / Physical systems

The swinging Atwood machine — a mass on a pulley that swings into chaotic motion.

continuous · ODE4 dimensionsconservativechaotic

projection (x, y, z)

projection (x, y, w)

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= \frac{y_{2}}{m_{1} + m_{2}} \\ \dot{y_{1}} &= \frac{y_{3}}{m_{1} y_{0}^{2}} \\ \dot{y_{2}} &= 9.82 m_{1} \cos{\left(y_{1} \right)} - 9.82 m_{2} + \frac{y_{3}^{2}}{m_{1} y_{0}^{3}} \\ \dot{y_{3}} &= - 9.82 m_{1} y_{0} \sin{\left(y_{1} \right)} \end{aligned} \]

Parameters

Symbol	Default	Role
m1	1	swinging pendulum mass
m2	4.5	non-swinging counterweight mass (ratio m2/m1 controls the dynamics)

Properties

Lyapunov spectrum

$+0.005442,\; +0.0177,\; -0.01728,\; -0.00586$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 4$

Divergence ∇·f

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

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Tufillaro, Abbott & Griffiths (1984), Am. J. Phys. 52, 895-903

BibTeX

@misc{swingingatwood,
  title = {SwingingAtwood system},
  note = {Tufillaro, Abbott & Griffiths (1984), Am. J. Phys. 52, 895-903}
}