DoublePendulum

Systems / ODEs / Physical systems

Two coupled pendula — the iconic mechanically chaotic system, exquisitely sensitive to initial conditions.

continuous · ODE4 dimensionschaoticconservative

projection (th1, th2, p1)

projection (th1, th2, p2)

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{th1} &= \frac{6 \left(2 p_{1} - 3 p_{2} \cos{\left(th_{1} - th_{2} \right)}\right)}{d^{2} m \left(16 - 9 \cos^{2}{\left(th_{1} - th_{2} \right)}\right)} \\ \dot{th2} &= \frac{6 \left(- 3 p_{1} \cos{\left(th_{1} - th_{2} \right)} + 8 p_{2}\right)}{d^{2} m \left(16 - 9 \cos^{2}{\left(th_{1} - th_{2} \right)}\right)} \\ \dot{p1} &= - 0.5 d^{2} m \left(\frac{29.46 \sin{\left(th_{1} \right)}}{d} + \frac{36 \left(2 p_{1} - 3 p_{2} \cos{\left(th_{1} - th_{2} \right)}\right) \left(- 3 p_{1} \cos{\left(th_{1} - th_{2} \right)} + 8 p_{2}\right) \sin{\left(th_{1} - th_{2} \right)}}{d^{4} m^{2} \left(16 - 9 \cos^{2}{\left(th_{1} - th_{2} \right)}\right)^{2}}\right) \\ \dot{p2} &= - 0.5 d^{2} m \left(\frac{9.82 \sin{\left(th_{2} \right)}}{d} - \frac{36 \left(2 p_{1} - 3 p_{2} \cos{\left(th_{1} - th_{2} \right)}\right) \left(- 3 p_{1} \cos{\left(th_{1} - th_{2} \right)} + 8 p_{2}\right) \sin{\left(th_{1} - th_{2} \right)}}{d^{4} m^{2} \left(16 - 9 \cos^{2}{\left(th_{1} - th_{2} \right)}\right)^{2}}\right) \end{aligned} \]

Parameters

Symbol	Default	Role
d	1	arm length (sets the gravitational frequency scale)
m	1	arm mass (both arms equal)

State variables: th1, th2, p1, p2

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

n/a — too large to display — pathological rational divergence

Equilibria

108 equilibria
0 stable · 108 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.DoublePendulum()
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

Marion (1965), Classical Dynamics of Particles and Systems, Academic Press
doi:10.1016/c2013-0-12598-6

BibTeX

@misc{doublependulum,
  title = {DoublePendulum system},
  note = {Marion (1965), Classical Dynamics of Particles and Systems, Academic Press},
  doi = {10.1016/c2013-0-12598-6}
}