A single double pendulum is the textbook chaotic system: two rods, two angles, and sensitive dependence that destroys predictability within a few swings. This simulation asks what happens when you build a whole material out of them — an $n \times n$ lattice, one double pendulum per site, each coupled to its four nearest neighbours by torsional springs, with periodic boundaries so the lattice is a torus with no edges.

Individually, every site wants to be chaotic. Collectively, they have to negotiate. The result is a competition between chaos and coupling that produces synchronized domains, travelling fronts, and phase textures that none of the individual pendulums knows anything about.

The left panel shows the pendulums themselves, coloured by the phase of the upper angle. The right panel is a live heatmap of the same lattice — choose the field: either phase angle, the site energy, or how well each site agrees with its neighbours. Click any pendulum to kick it (shift kicks the other way), and drag the Lattice size slider to resize the lattice mid-flight — the current pattern is resampled onto the new grid rather than discarded.

The model

Each site carries one double pendulum with unit masses and rod lengths, described by two angles from the vertical, $\theta_1$ (upper) and $\theta_2$ (lower). Writing $\Delta = \theta_1 - \theta_2$, the standard Lagrangian equations of motion take the mass-matrix form $M(q),\ddot q = F(q, \dot q)$ with

$$ M = \begin{pmatrix} 2 & \cos\Delta \\ \cos\Delta & 1 \end{pmatrix}, \qquad F = \begin{pmatrix} -\dot\theta_2^2 \sin\Delta - 2g\sin\theta_1 + \tau_1 \\[2pt] \;\;\,\dot\theta_1^2 \sin\Delta - g\sin\theta_2 + \tau_2 \end{pmatrix} $$

Neighbour interactions and damping enter as the generalized torques

$$ \tau_1 = K_1 \sum_{j \in \mathcal{N}(i)} \sin\!\big(\theta_1^{(j)} - \theta_1^{(i)}\big) - \gamma\,\dot\theta_1, \qquad \tau_2 = K_2 \sum_{j \in \mathcal{N}(i)} \sin\!\big(\theta_2^{(j)} - \theta_2^{(i)}\big) - \gamma\,\dot\theta_2 $$

— torsional springs acting on the corresponding joints of adjacent sites, exactly the coupling of the XY and Kuramoto models. Because the torques are applied through the inertia matrix rather than added to the accelerations by hand, the undamped lattice is genuinely Hamiltonian: the sum of site energies plus the bond potential $K\sum (1 - \cos\delta\theta)$ is a conserved quantity, which is also how the implementation is tested.

Two things about this coupling are worth being precise about. First, it acts on angles, not positions: the spacing between lattice sites is purely visual, the pendulums never collide, and no swing “reaches across” the gap — what travels from site to site is angular disagreement, not anything mechanical moving through the space between them. (A distance-dependent alternative — real springs connecting neighbouring bob tips, or pendulums sharing an elastic support the way Huygens’ clocks and tabletop metronomes synchronize — would be a different, heavier model; this one is the clean angular limit.) Second, the sign of $K$ chooses attraction or repulsion in phase. $K > 0$ is aligning — ferromagnetic in magnet language — and pulls neighbours toward the same angle. $K < 0$ is anti-aligning: each pendulum prefers to point opposite to its neighbours, and since a square lattice can satisfy that perfectly, the system orders into a checkerboard of alternating phases. Both signs are on the sliders.

The HUD tracks the Kuramoto order parameter for each joint,

$$ R_k = \frac{1}{n^2} \left| \sum_i e^{\,i\,\theta_k^{(i)}} \right|, $$

which reads 1 when every pendulum swings in unison and near 0 when the phases are incoherent.

What to look for

  • The synchronization transition. Set gravity to 0 (free rotors — this matters: with gravity on, damping alone drags every site to the hanging state and they end up aligned without ever talking to each other). Press Randomize with damping around 0.2 and sweep the coupling. At $K \approx 0$ the phases freeze wherever damping caught them and $R_1$ stays near zero; with coupling on, domains of agreement nucleate, coarsen, and swallow the lattice. The neighbour sync heatmap shows the domain walls as dark seams while they last.
  • Chaos spreading. Reset to the quiet lattice, then kick the centre and watch the site energy heatmap: the disturbance spreads outward as a front, and — because the boundaries are periodic — the fronts leaving opposite edges re-enter from the other side and collide.
  • Two-tier coupling. The upper and lower joints have separate coupling constants. Set $K_1$ high and $K_2 = 0$: upper angles lock while the lower rods stay chaotic — an ordered lattice wearing a disordered fringe, visible by flipping the heatmap between $\theta_1$ and $\theta_2$ phase.
  • Gravity off. With $g = 0$ the pendulums become free rotors and the model is essentially a two-layer XY dynamics; phase textures wind and anneal without a preferred direction to hang toward.
  • Repulsive coupling. Drag $K_1$ negative with $g = 0$ and some damping, then Randomize: instead of aligning, neighbours anti-align and the phase heatmap orders into a checkerboard — the antiferromagnet to the positive-$K$ ferromagnet. The neighbour sync field turns uniformly dark: perfect local disagreement is its own kind of order.
  • Size matters. Slide $n$ up mid-run. The pattern survives the resampling, but the balance shifts: the same kick that dominated an $8 \times 8$ lattice is a local ripple in a $32 \times 32$ one.

Copy link captures every slider, the heatmap field, and the seed in a shareable URL.

Implementation notes

Plain JavaScript, no dependencies — static/sims/pendulum-lattice/pendulum-lattice.js is the entire thing, with the DOM-free engine exported on window.PendulumLattice.

The full lattice — a $4n^2$-dimensional ODE system, up to 4096 variables at $n = 32$ — is integrated with classical RK4, evaluating the complete coupled derivative at all four stages. The trick that keeps this cheap: each stage computes $\sin$ and $\cos$ of both angles once per site, and every other trigonometric quantity — $\sin\Delta$, $\cos\Delta$, and all eight coupling terms — is reconstructed from that cache via angle-difference identities. The derivative costs exactly four trig calls per site regardless of coupling.

Rendering avoids per-site canvas state changes: pendulums are binned into 24 hue buckets by phase and drawn as one stroked path per bucket, and the heatmap is an $n \times n$ ImageData scaled up without smoothing, so each lattice site is one crisp pixel block. Changing $n$ mid-run does nearest-neighbour resampling of the state onto the new grid instead of resetting it.

RK4 is not symplectic, and a chaotic Hamiltonian lattice will eventually drift; at the default time step the drift is far below what the eye can see, and a guard resets the state cleanly if a large time step is pushed into a blow-up.

References

The double pendulum as a laboratory for chaos: T. Shinbrot, C. Grebogi, J. Wisdom, J. A. Yorke, Chaos in a double pendulum, Am. J. Phys. 60, 491 (1992). Synchronization of coupled oscillators: Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, 1984); S. H. Strogatz, From Kuramoto to Crawford, Physica D 143, 1 (2000); J. A. Acebrón et al., The Kuramoto model: a simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77, 137 (2005).