A pendulum swinging from the end of another pendulum — and one of the simplest systems that behaves chaotically. Release it, then launch a "chaos twin" a millionth of a radian apart and watch the two paths track together for a heartbeat before diverging completely. Same laws, same start, wildly different fate. It all runs on your device.
Fully deterministic, yet impossible to predict. The double pendulum obeys exact, unchanging physical laws — no randomness anywhere. Run the exact same start twice and you get the exact same swing. But release a second pendulum just a millionth of a radian away and the two visibly diverge within seconds. That is deterministic chaos: sensitive dependence on initial conditions, in a system you can hold in your hand.
How It Works
Two coupled joints, the exact equations of motion, and a tiny nudge that changes everything.
1
The equations of motion
Two angles describe the state — the first rod's angle and the second rod's angle — along with how fast each is turning. From the masses, rod lengths and gravity, the coupled equations give each joint's angular acceleration at every instant. There is no closed-form solution; the motion has to be computed step by step.
2
RK4 integration
The simulator advances the state with fourth-order Runge–Kutta at a small fixed time step, taking several substeps per frame. RK4 samples the accelerations four times per step and blends them, which keeps the energy stable and the swing faithful far longer than a naive update would.
3
The chaos twin
Turn on the twin and a second pendulum starts from an angle offset by the tiny "twin offset" you set. Both obey identical rules, yet the gap between their tips — shown in the readout — grows exponentially. That runaway separation is sensitive dependence on initial conditions, the signature of chaos.
What is a double pendulum?
A double pendulum is one pendulum hung from the end of another: a rod swings from a fixed pivot, and a second rod swings freely from the tip of the first. With two coupled joints it has two degrees of freedom, and its motion quickly becomes intricate and unpredictable-looking even though it follows simple physical laws.
What is deterministic chaos?
Deterministic chaos means a system obeys exact, fixed rules with no randomness, yet is so sensitive to its starting conditions that its long-term behaviour is effectively impossible to predict. The double pendulum is a textbook example: the same starting state always produces the same motion, but the tiniest change in the start leads to a completely different path.
Why do two identical pendulums diverge?
Because the motion is chaotic, any difference between two pendulums grows exponentially over time. Two pendulums released from angles differing by a millionth of a radian track each other for a moment, then separate and swing completely differently within seconds — the "sensitive dependence on initial conditions" that defines chaos.
Is the double pendulum random?
No. Its motion is fully deterministic and governed by Newton's laws, and this simulator reproduces it by integrating the exact equations of motion. It only looks random because it is chaotic: predicting it far into the future would require knowing the starting angles with impossible precision.