Chaos & Predictability Notation How to run

Chapter 4 · Regular motion and why it is predictable

Part II — From regular motion to chaos Knob: $\theta_0$, $\delta_0$ live notebook

Overview

Weather models are nonlinear, and nonlinearity usually takes the blame for the two-week forecast limit. That explanation is not good enough. The pendulum in this chapter is thoroughly nonlinear — its period depends on amplitude, its phase portrait is nothing like a circle, and no closed-form solution in elementary functions exists — and it is perfectly predictable forever. Give it an initial condition wrong by one part in 10810^8 and the forecast is wrong by one part in 10810^8 a thousand swings later.

Add a second rod and the same integrator, at the same tolerance, loses all skill within a few seconds.

So the ingredient that matters is not nonlinearity. It is phase-space dimension. A single pendulum has one degree of freedom, so its state (θ,ω)(\theta,\omega) lives in a plane, and conservation of energy confines the motion to a one-dimensional level curve. The Poincaré–Bendixson theorem then leaves no room for anything but fixed points and closed orbits: a trajectory confined to a curve cannot stretch and fold, so it cannot be chaotic no matter how nonlinear the restoring force. The double pendulum has two degrees of freedom, a four-dimensional phase space, and a three-dimensional energy surface — the minimum in which stretch-and-fold can operate.

That is the argument this chapter exists to make concrete, and it is the argument the rest of Part II builds on.

The model

The exact nonlinear pendulum,

θ¨=−gLsin⁡θ,\ddot\theta = -\frac{g}{L}\sin\theta,

whose period is not 2πL/g2\pi\sqrt{L/g} except in the small-angle limit, but

T=4L/g  K ⁣(sin⁡2(θ0/2))T = 4\sqrt{L/g}\;K\!\left(\sin^2(\theta_0/2)\right)

with KK the complete elliptic integral of the first kind. The notebook overlays the numerically integrated orbit on the exact phase-space curve and compares the measured period against this formula, so the reader can confirm the integrator is faithful before trusting anything it says about the double pendulum.

For two rods the full Euler–Lagrange equations apply, with the state (θ1,θ2,ω1,ω2)(\theta_1,\theta_2,\omega_1,\omega_2) and the characteristic sin⁡(θ1−θ2)\sin(\theta_1-\theta_2) coupling. Energy is conserved but is no longer enough to pin the motion to a curve.

Both systems come from chaoslib.systems — pendulum, double_pendulum, pendulum_energy, double_pendulum_energy and pendulum_period_exact — and are tested there against energy conservation over long integrations, the elliptic-integral period at four amplitudes, and the small-angle limit. Integration is adaptive RK45 at rtol=10−10r_{\mathrm{tol}} = 10^{-10}, tight enough that the divergence you see is the physics rather than the solver.

Prefer to run it yourself? Download as a Jupyter notebook — it installs its own dependencies in the first cell. The sliders are static in Jupyter; edit value= and re-run to change a parameter.

Exercises

Analytic. Show that a one-degree-of-freedom autonomous system whose energy is conserved cannot have a positive Lyapunov exponent. Where exactly does the argument use autonomy? Construct a driven pendulum that is chaotic, and identify which step of the argument it defeats.

Computational. Measure the single pendulum’s period as a function of θ0\theta_0 from 5∘5^\circ to 175∘175^\circ and plot the ratio to the small-angle result. At what amplitude does the small-angle approximation err by 1 %? By 10 %?

Exploratory. Set both double-pendulum angles small — say θ1=10∘\theta_1 = 10^\circ, θ2=5∘\theta_2 = 5^\circ — and run the twin-trajectory experiment. Is the separation growing exponentially, or linearly? Now raise the energy until it does grow exponentially. Chaos in this system is not a property of the equations alone but of the equations and the energy: locate that boundary as sharply as you can, and say what it means for a system whose “energy” varies from day to day.

Further reading