A single oscillator's whole state is one phase, a point on a circle. Amplitude never enters; there is only "where in the cycle am I." Two oscillators, then, are a pair of angles (θ₁, θ₂), and the set of all such pairs, circle × circle, is a torus. Not a physical donut: the donut is the space of every joint state the two can be in.
Slit the donut open and unroll it and you get a square where the left edge is glued to the right and the top to the bottom, a Pac-Man screen. Walk off one side, reappear on the other. That wrap is what makes it a torus and not a sheet.
The winding flow
dθ₁/dt = ω₁ dθ₂/dt = ω₂
With no coupling each angle just turns at its own rate, so every tracer walks a straight line on the unrolled square. What matters is not position but slope: the ratio ω₂/ω₁The frequency ratio is the slope of the path, not a point on it. Two oscillators at a fixed ratio always trace the same line on the torus, wherever they start.. A rational ratio (2/1, 3/2) closes into a loop after finitely many trips around, a torus knot(p, q) torus knot. When ω₂/ω₁ = p/q the path wraps p times one way and q times the other, then joins its own tail: a closed curve living on the surface.. An irrational ratio never closes: a single thread that comes arbitrarily close to every point and fills the whole surfaceIrrational rotation, or dense winding line. An irrational slope never returns to its start, so one endless trajectory is dense on the torus. This is the Weyl equidistribution / Kronecker flow..
How many times a closed path wraps each way are its winding numbers: whole numbers that can't change smoothly. That is why, under coupling, a lock slips in a sudden ±1 jump rather than a gentle drift: an integer has to step.
Coupling & the locking state
dθd,i/dt = ωd,i + (Kd/N) Σⱼ Gij·sin(θd,j − θd,i − β)
Gij = 1 + A·cos( 2π(i−j)/N )
Turn on K and the tracers pull on each other. This is the Kuramoto–SakaguchiKuramoto–Sakaguchi model. Phase oscillators coupled through the sine of their phase difference, with a lag β inside the sine (Sakaguchi & Kuramoto, 1986). Past a critical coupling the population locks. rule, run separately on each angle with its own strength K₁, K₂. Past a critical coupling the surface-filling spray collapses into a single travelling bundle: synchronization.
A gradient of natural frequencies across the ring, plus a phase lag, open the middle. The kernel G is non-localNon-local coupling. G = 1 + A·cos(Δ) makes near neighbours (in the index ring) pull hardest and far ones weakest — but every term still pulls together (G stays ≥ 0, so nothing ever pulls against). This structure is what a chimera needs; uniform all-to-all coupling can't make one in a phase-only model. Here, though, the kernel isn't what splits the state: at the preset, dialling locality to zero still splits the ring — the frequency gradient and β carry it., so near neighbours pull harder than far ones — but here it's nearly flat, and not what carries the split. What does is that gradient of natural frequencies together with β, a phase lagPhase lag β. A frustration term inside the sine that keeps agreement from ever completing. Locking states and chimeras both live just below β = π/2 ≈ 1.57; the shipped preset sits at 1.43, right in that window. that keeps agreement from ever completing. Together they let one arc of the population lock while another stays incoherent: a locking state, coherence and incoherence holding side by side on the same donut.
Reading the picture
Each tracer is tinted by its local order RᵢLocal order parameter Rᵢ. How aligned an oscillator's θ₁ is with its near neighbours on the index ring. Amber = locked, cool blue = drifting. A warm arc beside a cool one is the locking state, seen directly on the surface.: amber where a neighbourhood is locked, cool where it drifts. In a locking state a warm arc winds beside a cool one. The locking-state strip (top-left, appears once coupling is on) lays the same thing out flat: θ₁ against oscillator index, so the coherent band and the scattered band sit next to each other. Show instruments adds the two order-parameter rings (loop θ₁, tube θ₂) with their mean-field arrows, and the sync bars.
Euler: why this is a torus (and the sphere isn't)
Tile any surface with V corners, E edges, F faces. The alternating count χ = V − E + F doesn't care how you tile: it is a property of the shape. For a sphere it is always 2. For a torus it is always 0.
And χ counts the hole: χ = 2 − 2g, where g is the genusGenus g. The number of handles/holes. Sphere g = 0 → χ = 2. Torus g = 1 → χ = 0. A two-holed pretzel g = 2 → χ = −2. Genus is a topological invariant: no bending, stretching, or denting changes it., the number of holes. Sphere: g = 0, χ = 2. Torus: g = 1, χ = 0.
A sphere and a torus are the textbook pair of surfaces that are genuinely not the same: 2 ≠ 0, and no amount of stretching, denting, or bending ever closes that gap. One hole versus none is a difference baked into the arithmetic, and nothing you do to the shape can smooth it away.
Live state
R₁ (loop) = 0.00
R₂ (tube) = 0.00
ratio = 1.62
N = 30
R is the order parameterOrder parameter R = |⟨e^{iθ}⟩|. 0 when phases are scattered, 1 when they all agree. Computed per angle: R₁ for the loop, R₂ for the tube. In a locking state the global R sits in the middle while the local field splits into locked and loose. for each angle: 0 scattered, 1 locked. A locking state keeps R off both ends.
Lineage
Every piece here is old and well-named: irrational rotation / winding lines on the torus (Kronecker, Weyl), Kuramoto–Sakaguchi coupled phase oscillators, and chimera states under non-local coupling (Kuramoto & Battogtokh, 2002; Abrams & Strogatz, 2004). The fresh edge is only the framing: two coupled angles, on the torus, interactive, walked from one circle up to a locking state on the surface. Cite the giants; claim the demonstration, not the discovery.
Kuramoto & Battogtokh, Nonlin. Phenom. Complex Syst. (2002) · Abrams & Strogatz, PRL (2004) · Sakaguchi & Kuramoto (1986) · Weyl equidistribution / Kronecker flow · Euler, Elementa doctrinae solidorum (1758).
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