complex-kuramoto-control

name: complex-kuramoto-control description: "Unified control framework for synchronization in coupled oscillator networks using complex-valued Kuramoto extensions. Use when designing synchronization controllers, phase-locking mechanisms, oscillator network control, or when real-valued Kuramoto model fails. Keywords: Kuramoto, synchronization, complex-valued control, oscillator networks, phase locking, sliding-mode control."

Complex-Valued Kuramoto Networks: Unified Control Framework

Synchronization control in oscillator networks through complex-valued extensions that embed nonlinear phase dynamics into linear state space.

Problem: Classical Kuramoto Limitations

Real-valued Kuramoto model:

• Intrinsic nonlinearity limits analytical tractability
• Complicates control design
• Fails for heterogeneous networks with different natural frequencies

Solution: Complex-Valued Extension

Embed phase dynamics into higher-dimensional linear state space:

• Regulating complex-state moduli to common value → recovers Kuramoto phase behavior
• Enables linear control techniques for inherently nonlinear problem

Control Strategies

1. Switched Feedforward Law

• Ensures exact phase correspondence at all times
• No spectral gain tuning required

2. Feedforward + Sliding-Mode Law

• Finite-time convergence
• Robust to perturbations

3. Non-autonomous MIMO Sliding-Mode Controller

• Enforces phase locking at prescribed frequency in finite time
• Independent of natural frequencies and coupling strengths
• Works for heterogeneous networks where real-valued Kuramoto fails

Key Insight

Phase → Complex state → Control moduli → Recover phase

The transformation: $\theta \rightarrow z = e^{i\theta}$

Control $|z|$ to converge → $\theta$ synchronizes

Design Procedure

1. Map oscillator phases to complex plane
2. Design controller for complex state convergence
3. Verify phase behavior recovered from complex-state behavior
4. Handle heterogeneity through robust control design

Applications

• Power grid synchronization
• Neural network synchronization
• Circadian rhythm control
• Chemical oscillator networks
• Distributed clock synchronization
• Robot swarm coordination

When Classical Kuramoto Fails

Heterogeneous networks:

• Different natural frequencies
• Varying coupling strengths
• Non-uniform topology

Complex-valued approach handles these through robust control design.

Code Pattern (Conceptual)

# Classical Kuramoto: nonlinear phase dynamics
dθ/dt = ω - K * sin(θ_j - θ_i)  # Hard to control

# Complex-valued: linear complex dynamics
z = exp(iθ)  # Transform
dz/dt = iωz - K*(z_j - z_i)  # Now design control for z

# Control moduli
|z| → target  # Drive all |z| to common value
# Phase θ synchronizes as consequence

Performance Benefits

• Improved transient response
• Better steady-state accuracy
• Enhanced robustness
• Handles heterogeneity

References

• arXiv:2604.07249v1 - "Complex-Valued Kuramoto Networks: A Unified Control-Theoretic Framework"
• Kuramoto, Y. (1984) - Original model
• Acebrón et al. (2005) - Kuramoto review