Deterministic chaos: the double pendulum

Two pendulums start almost identically, a millionth of a radian apart. The equations come from the Lagrangian and are integrated with fourth-order Runge–Kutta. Watch the two trajectories diverge exponentially, and measure the Lyapunov exponent live while total energy stays conserved.

Drag a bob to set the starting angles

Phase space (θ₂, ω₂)

Separation log₁₀|δ(t)|

Energy error |ΔE/E₀|

From the Lagrangian to the equations of motion

With point masses on massless rigid rods, $L = T - V$ where

$$ T = \tfrac12 (m_1+m_2) l_1^2\dot\theta_1^2 + \tfrac12 m_2 l_2^2\dot\theta_2^2 + m_2 l_1 l_2 \dot\theta_1\dot\theta_2\cos(\theta_1-\theta_2) $$ $$ V = -(m_1+m_2) g l_1\cos\theta_1 - m_2 g l_2 \cos\theta_2 .$$

The Euler–Lagrange equations $\frac{d}{dt}\frac{\partial L}{\partial\dot\theta_i} = \frac{\partial L}{\partial\theta_i}$ are solved for $\ddot\theta_1, \ddot\theta_2$ in closed form and integrated with classical RK4 (Δt = 0.5 ms). Optional joint friction enters as the generalised force $Q_i = -b\,\dot\theta_i$.

Checks you can see

With b = 0, total energy $E = T + V$ is conserved. The energy-error plot stays near 10⁻¹⁰ (RK4's local error is $O(\Delta t^5)$).
At small angles the motion is a superposition of two normal modes with $\omega^2 = \frac{g}{l}(2 \mp \sqrt2)$ (equal masses and lengths). The test suite checks the period to 1 part in 10⁴.

Measuring chaos: the Lyapunov exponent

Nearby trajectories separate as $|\delta(t)| \approx |\delta_0| e^{\lambda t}$. The slope of the separation plot is $\lambda/\ln 10$ until δ saturates at the size of the attractor.

The number shown as λ uses Benettin's method. A shadow trajectory is kept at a distance $d_0 = 10^{-8}$ in phase space and renormalised every τ = 0.05 s:

$$ \lambda = \lim_{t\to\infty}\frac{1}{t}\sum_i \ln\frac{d_i}{d_0} .$$

λ > 0 means chaos. For small swings, λ falls toward 0 and the motion is regular.

Things to try

Start at θ₁ = θ₂ = 10°. The twins stay together and the Poincaré section shows closed curves (invariant tori).
Start at θ₁ = 120°, θ₂ = −20°. The Poincaré section fills an area: a chaotic sea.
Make δθ₁ ten times smaller. Divergence is delayed by only $\ln 10/\lambda$ seconds, which is why long-range prediction fails.
Add friction: the energy now drops steadily and the motion settles into the stable equilibrium.