A chaotic double pendulum in MuJoCo

5th Jun 2026

Two identical double pendulums released side-by-side with their initial angles differing by 10⁻³ rad. They look the same for a second or two, then diverge.

The double pendulum is the textbook example of a deterministic system that is, in practice, unpredictable. Two arms hinged in series have four state variables ( $\theta_1, \theta_2, \dot\theta_1, \dot\theta_2$ ) and a Lagrangian with one nonlinear coupling term, and that is enough: trajectories that start arbitrarily close together separate exponentially in time.

The demo runs two of them at once, side by side in the same MuJoCo scene. Both pendulums have identical mass, length, and damping ( $0$ — no losses). The only thing that differs is the initial angle of the upper arm:

\theta_1^A(0) = 2.0,\quad \theta_1^B(0) = 2.0 + 10^{-3}.

For the first ~2 seconds the two pendulums trace what looks like the same motion. Around $t \approx 3\,\text{s}$ the tip-to-tip distance crosses 0.1 m, and from then on they are visibly doing completely different things. The Lyapunov-like blow-up is the whole story.

The MJCF model lives inline in src/clis/mujoco_cli/cli.py:

two <body> chains of two <joint type="hinge"> arms, anchored at $x = \pm 0.6\,\text{m}$
arms are $0.4\,\text{m}$ capsules; a small <site> marks each tip so we can read its world position
timestep $0.002\,\text{s}$ with the RK4 integrator — important for the chaotic regime, where forward Euler would let small numerical errors swamp the actual divergence
damping $0$ , so total energy is conserved (give or take RK4’s $O(\Delta t^4)$ drift)

The tip separation reported below is measured in each pendulum’s anchor-local frame — i.e. each tip position minus its own anchor — so the constant $1.2\,\text{m}$ horizontal offset between the two scenes doesn’t contaminate the metric.

Run parameters

Parameter	Value
theta1 (rad)	2.0
theta2 (rad)	0.0
epsilon (rad)	1.0e-3
duration (s)	8.0
fps	60
separation_threshold (m)	0.1
separation_time_s	2.968
max_separation_m	1.21
final_separation_m	0.565

The maximum separation hits $1.21\,\text{m}$ — basically the full extent of a single pendulum, $0.8\,\text{m}$ — meaning that at some point one pendulum’s tip is at the top of its swing while the other’s is at the bottom. The final separation of $0.56\,\text{m}$ is just where they happen to be at $t = 8\,\text{s}$ ; with a chaotic system, “where they end up” is meaningless past the divergence time.