DYNAMICAL SYSTEMS / EDUCATIONAL MODEL
Emergent Systems Explorer
Explore how local oscillator interactions can produce collective behavior on a graph.
Click a node to inspect its phase and neighbors.
Local rules, collective behavior
This is a Kuramoto phase-oscillator model on a seeded random graph. Every node has an intrinsic frequency; connected nodes influence each other’s phase. The order parameter R ranges from 0 (phases dispersed) to 1 (phases aligned).
This is an educational dynamical-systems experiment. It does not reproduce or validate Vitaly Vanchurin’s proposal that fundamental physics might emerge from a neural network.
AUTHORED STUDY COMPANION / 3.4
Check synchronization against numerical resolution
Prerequisites. Ordinary differential equations, graphs, phases and numerical integration.
The model evolves phase θᵢ with intrinsic frequency ωᵢ and degree-normalized coupling KΣⱼAᵢⱼsin(θⱼ−θᵢ)/max(1,dᵢ). The order parameter R=|N⁻¹Σᵢexp(iθᵢ)| lies between zero and one and measures instantaneous phase concentration. It is not a measure of intelligence, a physical universe, or proof that every oscillator is frequency-locked.
The companion creates a fixed seeded random graph, frequencies and initial phases, then runs explicit Euler integration to the same final time with steps Δt and Δt/2. It compares phases using wrapped differences and reports both R values. Holding the graph and initial condition fixed isolates discretization effects; changing the random seed at the same time would confound that comparison.
One refinement comparison is a diagnostic rather than a convergence proof. Repeat another halving and then an ensemble of seeds before claiming a stable regime. The companion's NumPy random stream differs from the browser's seeded generator, so matching seed numbers alone does not imply identical graphs. The existing browser model and its animation remain unchanged.
| Element | Model | Interpretation |
|---|---|---|
| Dynamics | Intrinsic frequency + neighbor phase coupling | State the degree normalization |
| Observable | R from complex phase average | Instantaneous coherence |
| Numerical check | Same fixture, half the time step | Estimate discretization sensitivity |
Predict, run, compare
Use 12 nodes, edge probability 0.25, seed 7, K=1.2 and duration 4. Compare steps 0.02 and 0.01; every reported order parameter must remain in [0,1].
Download executable notebooks and source Inspect source ↗
Open kuramoto.ipynb in the package. Run cells in order, compare with the reference, then change the stated input. Restart the kernel to reset.
Computed reference result — local CPU fixture
{
"order_coarse": 0.4454460892751879,
"order_fine": 0.44600157746130203,
"phase_rms_difference": 0.0016291790813062115,
"edges": 16,
"seed": 7,
"dt": 0.02,
"duration": 4.0,
"scope": "Euler convergence comparison on one fixed seeded graph, not universal synchronization evidence"
}This is the recorded result of the included code under the package environment. It is not a live service or device measurement.
Extend the experiment
Run ten seeds at several coupling strengths. Report the seed distribution and a time-step refinement check instead of selecting the most coherent animation.
Check your reasoning: Which change isolates time-discretization error?
- Change seed and time step together — That mixes two sources of variation.
- Hold the fixture fixed and halve the time step — Correct. The comparison controls graph and initial conditions.
- Choose only the largest observed R — Selection by the desired outcome biases the report.
This is an explained self-check, not a certification of mastery.
Claim-specific sources
Reviewed 2026-10-02: this worked example and its explicit assumptions. Source links and notebook results do not establish production or hardware validation.
Problem, implementation and evidence
A coherent-looking animation can conceal time-step or seed effects.
The existing seeded oscillator model is paired with a separate fixed-fixture Euler refinement comparison and an ensemble extension task.
What the evidence establishes
Deterministic browser simulation plus executed CPU convergence fixture. No claim that the model reproduces fundamental physics or a physical universe.
Reproduce or extend
Hold initial conditions fixed for time-step checks, then compare multiple seeds at each coupling strength.
Open the related experiment or review surface →