Kuramoto

2D/3D WebGPU Visualization
FPS--
Oscillators--
Grid--
Local R̄(t)
χ(t)
View & Grid
View Mode
Manifold
S2 disables delay/harmonics/RC/LLE and uses RGB vectors.
Grid Size 256
📝 Valid: 64–1024 (multiples of 64)
Layers
Layer Count 1
Layers
Rebuilds simulation buffers.
Active Layer 0
Layer Z Offset 0.15
Coupling & Rules
mode: s1/grid/classic/static/layer0
frame:0.0ms | stats:0.0ms | gauge:0.0ms | probe:0.0ms
Coupling Mode
Classic: dθ/dt = ω + K·sin(θⱼ - θᵢ)
🌍 GLOBAL COUPLING ACTIVE
Coupling Strength (K₀) 1.00
Affects dynamics.
Neighborhood Range 2
Affects dynamics (grid topology only).
Affects S1 coupling argument as sin(Δθ - η).
Phase Lag η 0.00
Layer Coupling Up 0.00
Layer Coupling Down 0.00
Uses the active layer's kernel parameters.
Gauge Field (U(1))
Gauge Field (U(1), S1 only)
Coupling uses sin(θ_j - θ_i - qA_ij), invariant under local phase shifts compensated by link-field shifts.

Modes
• Static: fixed background links
• Dynamic: links co-evolve with oscillator currents (grid-only)

Parameters
• Enable Gauge Field: turn covariant coupling on/off
• Gauge Mode: static vs dynamic field evolution
• Charge q: scales phase shift by qA; larger q amplifies gauge frustration
• Matter Coupling g_J: current-to-field feedback strength in dynamic mode
• Magnetic Stiffness κ_B: smooths plaquette flux by penalizing sharp curvature
• Gauge Damping γ_A: relaxes links toward zero when undriven
• Gauge Noise: stochastic forcing on links
• Gauge dt Scale: relative integration speed of gauge dynamics
• Init Pattern: zero / uniform flux / random links / pure gauge
• Init Amplitude: magnitude for random-link and pure-gauge init
• Flux Bias B0: target uniform plaquette flux in uniform-flux init
• Gauge Graph Seed: deterministic graph-link initialization seed

Starter Settings
• Static: q=1, B0≈0.5–1.0, random θ
• Dynamic: g_J≈0.5–1.5, κ_B≈0.1–0.5, γ_A≈0.05–0.2, low noise
Gauge Mode
Charge q 1.00
Affects dynamics.
Matter Coupling g_J 1.00
Dynamic gauge only.
Magnetic Stiffness κ_B 0.20
Dynamic gauge only.
Gauge Damping γ_A 0.05
Dynamic gauge only.
Gauge Noise 0.00
Dynamic gauge only.
Gauge dt Scale 1.00
Dynamic gauge only.
Init Pattern
Init Amplitude 0.50
Flux Bias B0 0.00
Gauge Graph Seed 1
S1 only. Dynamic gauge requires grid topology.
Graph Topology
Topology avg deg 0
Seed 1
avg deg 0.00 | max 16
Integration & Noise
Time Step (dt) 0.03
Leak / Damping 0.00
0 = no leak; higher values damp dynamics to enforce fading memory
Time Scale 1.0×
Noise Strength 0.00
Layer Editor
Editing Layer 0
Active layer: 0
Pattern Presets
Pattern Presets
Each preset combines:
• Coupling Rules - Mathematical equations governing oscillation
• Initial Conditions - Starting phase/frequency patterns
• Parameters - Optimal values for each pattern

Presets are shortcuts for common behaviors like spirals, waves, and synchronization patterns.
Basic Patterns
Wave Patterns
Spiral Patterns
Special
Lenia / Artificial Life
Discovery Presets
Complex Patterns
Initial Conditions
Phase Pattern (θ)
Frequency Pattern (ω)
ω Amplitude 0.40
Experiment Seed 1
Controls randomness in init, presets, and RC. Interactive drawing is not logged.
State Snapshot —
Saves theta/omega/gauge + params with versioning. Large grids can produce big files.
Visualization
layer: phase | palette: rainbow
Data Layer Shift+C cycles
Affects rendering only.
Palette C cycles
Affects rendering only.
Flux Gain 1.00
Render-only amplification for Gauge Flux layer.
Covariant Gain 1.00
Render-only amplification for Covariant Gradient layer.
Render style only. Does not change equations.
Style Blend 1.00
Style Base Source
Trail Fade 0.15
Glow Scale 50
Core Threshold 0.40
Core Scale 0.30
Render-only local contrast normalization for layers 7/8.
Render-only signed mapping for clockwise/counterclockwise flux.
Render-only overlay. S1 grid recommended.
Render-only overlay. Shows local plaquette sign.
Shows local θ and covariant decomposition near cursor.
3D Surface Mode
Prismatic Aether
Prismatic Aether (S1 + Grid)
Style-first split: Prismatic visuals/audio can run without changing equations.

Prismatic Dynamics (optional)
• Inertial update: v'=(v+K(F+Fmouse))f
• Phase: θ' = θ + ω + v'
• Energy: E' = E·decay + |v'|·mix

Interaction
• Left-drag injects local entrainment force
• Cmd+left-drag pans in 3D
• In 3D, drag projects to active layer plane

Audio is reactive-only and follows the active visualization feature bus.
Advanced mode. Changes simulation equations (S1 + grid).
Left-drag injects force (S1 + grid). Cmd+left-drag pans in 3D.
Prismatic K 0.10
Friction 0.92
Energy Decay 0.88
Energy Mix 0.12
Drag Radius (px) 120
Drag Peak Force 4.00
Target Phase 0.00
Force Falloff 1.00
Audio uses shared phase-energy bins (output-only).
Audio Master 0.55
Audio Bells 0.45
Audio Bass 0.50
Reverb Mix 0.45
Reverb Time 8.00
Musical Mode
If on, Prismatic audio follows Prismatic Style view.
Audio source: active layer feature bus | profile: Phase
Scale Modulation
Scale Modulation
Spatially varies coupling strength (K₀) per oscillator:
• Base - Uniform scaling factor
• Radial - Stronger coupling at edges
• Random - Per-oscillator random variation
• Ring - Concentric ring pattern

Creates regions with different synchronization dynamics.
Flow Field
Flow Field
Adds directional bias to phase evolution (adds to ω):
• Radial - Inward/outward flow from center
• Rotate - System-wide rotation bias
• Swirl - Spiral flow pattern
• Bubble - Quadrupole expansion
• Ring - Circular flow
• Vortex - Vortex circulation
• Vertical - Up/down drift

Causes patterns to drift, rotate, or travel.
Center is zero; sliders snap to 0 (hold Ctrl to avoid snap).
Orientation Map
Orientation Map
Spatially modulates coupling effectiveness:
• Radial - Stronger coupling at left/right
• Circles - Stronger coupling at top/bottom
• Swirl - Diagonal modulation
• Bubble - Quadrupole pattern
• Linear - Vertical gradient

Creates spatially-varying synchronization behavior.
Positive linear = vertical, negative = horizontal.
Statistics
Local R̄: 0.000
Global R: 0.000
Susceptibility χ (temporal var): 0.0000
Std(Local R̄): 0.0000
min max
Est. Kc: —
Local R Histogram
Distribution of per-oscillator local order Rᵢ (bimodal suggests mixed regions).
Time Series
Local R̄(t)
χ(t) = N·Var(Local R̄)
Organisms
Count: 0
Born / Died: 0 / 0
Tracked: 0
Detection Threshold (R) 0.50
Min Area 4
Phase Space
🗺️ Phase Space
Subsampled θ plotted on unit circle.
Phase Diagram
Phase Diagram (Local R̄ vs K)
Parameter Sweep
Sweep Parameter
From
To
Steps
Settle Frames
idle
A/B Compare
Capture states with fixed seed for visual before/after checks.
A
A snapshot
empty
B
B snapshot
empty
ΔR: — | Δχ: —
Experiments
Warmup Steps
Measure Steps
Steps / Frame
Readback Every
Higher = faster; lower = more samples.
idle
Lyapunov Exponent
λ Lyapunov Exponent
LLE (λ): —
Status: idle
stable critical chaotic
Finite-Size Scaling
📐 Finite-Size Scaling
Run K-scan at multiple grid sizes to extrapolate true Kc
Kc (N→∞): —
Reservoir Computing
Task
Injection Mode
Feature Budget 512
Input / Output
Input Region
Output Region
Input Strength 2.0
Training
Status: idle
Samples: 0
Train NRMSE: —
Test NRMSE: —
Cond. Est: —
K Sweep: idle
K Sweep Best: —
RC Plots
Prediction vs Target
RC vs Criticality (Test NRMSE vs K)
Injection Mode Compare
idle
—
Keyboard Shortcuts
0-5 Switch rules
R Reset state
G Toggle global coupling
C Cycle palettes  |  Shift+C Cycle data layer
V Toggle 2D/3D view
Space Pause/Resume
[] Adjust speed
↑↓ Adjust coupling
←→ Adjust range
Shift+↑↓ Adjust grid size
+- Zoom in/out (2D)
Z Reset zoom (2D)
S Cycle smoothing modes
L Toggle log scale (χ plot)
K Start K-scan
Shift+K Go to Kc

Mouse
Left-drag: prismatic force (S1 + grid)
Cmd+Left-drag (3D): pan
Right-drag (3D): pan
Scroll to zoom (towards cursor)
Double-click to reset view