Think of the constellation as a field of clocks. Every satellite keeps its own time under one shared rule — each tugging its neighbours toward agreement — yet the way they fall into line can settle into three regimes. Knock them out of step and you get chaos: noise, no coordination, nothing you'd trust to hold a position. Pull them into perfect step and you get rigid sync: clean, uniform, and a single point of failure — when the one shared rhythm slips, every clock slips with it. The state you actually want sits between the two.
That held middle is a standing truce — never fully agreeing, never fully scattering. And it has a sound: a low thrum, the whole field beating in loose time. Hit Chaos · Balance · Sync (top-right) and listen. You can hear when it holds.
Chimera on the sphere
dθᵢ/dt = ωᵢ − (K/N) Σⱼ G(dᵢⱼ)·sin(θᵢ − θⱼ + αᵢ)
Each satelliteOne oscillator in the constellation — a clock orbiting the sphere, running at its own phase until coupling pulls it toward its neighbours. is one clock in the field, carrying a phasePhase θᵢ — where the oscillator sits in its cycle, an angle in [0, 2π). Here it is the clock's offset; coupling tries to align it with the rest of the constellation. θᵢ — where it sits in its cycle — and a natural frequencyNatural frequency ωᵢ — the rate the clock prefers with no coupling. Here each clock is given its own rate, drawn from a spread of width γ (this preset: γ = 0.13). That irreducible difference is what holds the truce open: with identical clocks (γ = 0) this field heals all the way to lockstep. The spread is deliberate — the difference the coupling isn't allowed to erase. ωᵢ, the rhythm it would keep alone. The coupling is the link between clocks: it pulls each toward the others, weighted by the kernel and lagged by αᵢ. Change that lag and the same clocks settle into a different regime.
Kernel & geodesic distance
G(d) = ½ + ½·A·cos(d)
cos(dᵢⱼ) = rᵢ · rⱼ
The coupling is non-localNon-local coupling — strength falls off with the geodesic distance dᵢⱼ between two satellites on the sphere, set by the kernel ½ + ½A·cos(d). Near beats far. This is the structural ingredient a chimera needs; uniform all-to-all coupling cannot produce one in a phase-only model like this.: nearby clocks pull on each other harder than distant ones — the field has structure, not one uniform pull. A sets how sharply near beats far. Because rᵢ and rⱼ are unit position vectors, their dot product is cos(dᵢⱼ) — no arccos in the loop. As the satellites orbit, the distances change, so the coupling field breathes.
Phase lag
αᵢ = π/2 − βᵢ
β is the phase lagPhase lag β — the offset inside the sine, α = π/2 − β. Partial coherence lives in a narrow band of β. Push β up and the field heals into full sync; pull it toward zero and coherence breaks down. Select a satellite and drop its β alone to stress one clock. — the one knob that carries the field between regimes. Large β → α near 0 → the clocks fall all the way into step (full sync). β → 0 → α near π/2 → the partial-coherence band: the standing truce, where the field settles between lock and scatter and holds there. Select a satellite and drop its β alone to stress a single clock and watch it drag on the rest.
Reading the colour
Each satellite is tinted by its local order parameterLocal order parameter Rᵢ — how synchronized a satellite is with its spatial neighbourhood. Rᵢ rides high (amber) where clocks fall toward step, low (blue) where they drift apart. At Balance the sphere doesn't split into locked and scattered patches — it settles into an even, intermediate glow: every region partly in step, none fully locked, none fully loose. That held middle is the standing truce. Rᵢ — amber where clocks fall into step, blue where they drift apart. At Balance the sphere holds an even, mid-band glow: no fully-locked amber, no fully-scattered blue — the truce, seen. The global bar lower-left tracks the whole field's R.
Reading the motion
Colour shows where the field is coherent; the heading arrows show the clocks moving together. Each satellite carries a short arrow whose direction is set by its phase, drawn in the constellation's mean-rotating frameMean-rotating frame (θᵢ − ψ) — headings are measured relative to the global mean phase ψ, not raw θ. A fully locked field would otherwise spin all arrows in unison and read as restlessness; in the mean frame, lock becomes stillness and alignment instead. (θᵢ − ψ), with a faint radial bob on the same phase. Where a region falls into step, its arrows settle into a steady, aligned flow and the satellites breathe in and out as one — coherence reading as stillness and alignment; where clocks scatter, the arrows churn and the motion turns to shimmer. Synchrony becomes stillness and alignment — the cue a Starlink train gives the eye for free. Toggle Headings to compare against colour alone.
Reading the sound
The LISTEN presets give the same coherence to the ear: each clock sounds as it comes round, so the beat falls out of the field's own sweep. At Balance you hear a low thrum — the whole field beating in loose time. Not one clean pulse (that's Sync), not clatter (that's Chaos), but a thick, breathing beat with its edges blurred: hundreds of clocks crossing close together and never quite together. The blur is the truce, made audible. Toggle Sound and hit a preset.
Live state
R = 0.000
t = 0.0
sel = 0
Ωorb = 0.00 rad/s
N = 300
At Balance, global R sits at an intermediate plateau — neither 0 nor 1. It lands there not because half the field locks and half runs free, but because the whole field settles at a partial coherence and holds it — everyone half in step, together. That steady in-between value is the truce holding: not locked, not scattered, the field keeping its standing compromise.
The re-run fingerprint
Identical clocks, identical rule — yet the field never settles the same way twice. This is partial synchronization holding its ground: a standing truceSymmetry breaking — nothing distinguishes one satellite from another, so nothing fixes which regions run warmer or cooler; the arrangement is set by the initial phases alone. It stays partial because the kernel's mean-field term puts a floor under local coherence — no region can go fully incoherent, so the split a true chimera needs can't form here. between falling into step and keeping your own time, and it stays there with no referee. Hit a preset or Reseed φ and the field re-forms its truce somewhere new every time — same character, never the same arrangement: proof the pattern is born from how the field starts, not from the recipe. (A true chimera goes one step further — one region locking completely while another stays fully incoherent, side by side, with a sharp boundary. That's the next card — on the sphere.)
PNT (Position, Navigation, Timing) systems coordinate distributed sensors and clocks. When one node's local oscillator drifts — vibration, thermal stress, GNSS denial — its coupling to the constellation degrades the same way β → 0 degrades coherence here. The visible failure is a gradual loss of position accuracy that traces back to a single clock no one was listening to.
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