Standard Kuramoto Production
Classical Kuramoto model with Euler/RK4/RK45 integrators. Sakaguchi phase lag. The foundation for all other engines. dθi/dt = ωi + ∑ Kij sin(θj − θi − α)
Stuart-Landau Production
Phase + amplitude dynamics. Hopf bifurcation, limit cycles, amplitude death. dzi/dt = (μ + iω)zi − |zi|²zi + K∑(zj − zi). Captures phenomena beyond pure phase models.
Inertial Kuramoto Production
Second-order dynamics with mass/damping. The swing equation for power grids. M d²θ/dt² + D dθ/dt = P − ∑ K sin(Δθ). Generator frequency dynamics.
Swarmalator Stable
Spatial + phase coupling (O’Keeffe 2017). Oscillators move in space AND synchronise in phase. 5 emergent patterns: static sync, static async, splintered, active, static phase wave.
Simplicial Stable
3-body interactions (Gambuzza 2023). Higher-order coupling on simplicial complexes. Explosive synchronisation transitions. Beyond pairwise coupling.
Stochastic Stable
Euler-Maruyama integration with noise. D* auto-tuning calibrates noise intensity from data. Stochastic resonance detection. Langevin dynamics for thermal systems.
Geometric Stable
Torus-preserving symplectic integrator. Conserves geometric structure of phase space. No artificial dissipation. Ideal for long-horizon simulations.
Delay Stable
Time-delayed coupling with circular buffer. Models signal propagation delays in neural circuits, power grids, and communication networks. Configurable per-link delay.
Ott-Antonsen Stable
Exact mean-field reduction. O(1) computational cost regardless of N. Analytical order parameter prediction. MPC supervisor uses this for 10-step lookahead.
Hypergraph Experimental
Higher-order interactions on hyperedges. Groups of 3+ oscillators coupled simultaneously. Captures phenomena invisible to pairwise models.
Variational/Adjoint Experimental
Free Energy Principle predictor. Adjoint gradient computation for parameter optimisation. Variational inference of coupling topology from observed dynamics.
Market Kuramoto Stable
Financial regime detection. Maps asset price oscillations to Kuramoto phases. Correlation breakdown → desynchronisation. Crash early warning via R(t) drop.