Double Pendulum Ensemble: Chaos Fan with Lyapunov Exponent | mysimulator.uk

What is modelled

A planar double pendulum: two point masses on rigid, massless rods, the upper rod hinged to a wall. Not one pendulum but a whole fan of them, all hinged at the same point and started almost identically: pendulum k begins k times the chosen gap further round. In the chaotic regime the fan opens up like a bouquet, which is the practical meaning of sensitive dependence on initial conditions.

Equations

• Lagrangian for angles measured from straight down: T = ½(m1+m2) l1² ω1² + ½ m2 l2² ω2² + m2 l1 l2 ω1 ω2 cos(θ1−θ2), U = −(m1+m2) g l1 cosθ1 − m2 g l2 cosθ2.
• The Euler-Lagrange equations give the angular accelerations θ1″ and θ2″ (written out in the page source) with the common denominator D = 2m1 + m2 − m2 cos(2θ1−2θ2). They are integrated with fourth-order Runge-Kutta at 960 steps per second.
• Chaos: the gap between two neighbours grows as δ(t) ≈ δ0 eλt. The page fits λ, the largest Lyapunov exponent, to the straight part of ln δ against time. A 0.1 rad error is reached after about t = ln(0.1/δ0) / λ; making the start a million times more precise adds only ln(106)/λ seconds.
• Small oscillations: M ω² = K with M = [[(m1+m2) l1², m2 l1 l2],[m2 l1 l2, m2 l2²]] and K = diag((m1+m2) g l1, m2 g l2); the two roots of det(K − ω² M) = 0 are the normal modes shown in the readout.

Checks run in the page

• Energy is conserved to better than 1e-6 of m g (l₁+l₂) over 20 s for a chaotic, a wild and a regular motion.
• For tiny amplitudes the simulated period of each normal mode matches 2π/ω from the eigenvalue equation, for two different mass and length sets.
• A completely separate formulation, Newton's laws in Cartesian coordinates with rod tensions and no angles at all, reproduces the Lagrangian trajectory over 4 s.
• Reversing the velocities after a chaotic run brings the pendulum back to its start.
• The chaotic start gives a clearly positive λ and a fan that fully disperses; the regular start gives λ close to zero and a fan that stays together.

Honest simplifications

• Point masses on massless, perfectly rigid rods with frictionless hinges and no air drag. A real pendulum has distributed mass, bearing friction and drag, and these damp the chaos away within a minute or so.
• The Lyapunov fit is only meaningful while the gap is small (a few times the start value up to about 0.05 rad); afterwards the gap saturates because the angles are bounded on the energy surface.
• The wall, protractor and table are drawn for the picture only; the pendulums are slightly spaced in depth so that they can be seen, which has no physical effect.
• Round-off error is real too: at gaps near 1e-9 rad the numerical noise of the integrator is not far below the signal, so very small gaps show the limit of double-precision arithmetic as well as the chaos of the pendulum.

What to try

• Press Gentle: stays regular and compare the gap graph with the chaotic fan.
• Set the gap to 1e-9 rad: the fan opens only about ten seconds later than with 1e-6, because the delay is logarithmic.
• Make the upper arm heavy (mass ratio 0.25) or the lower arm longer and see how the normal-mode periods change.