A dynamical regime in which two or more oscillatory units maintain a constant (possibly integer-ratio) relationship between their phases, so that phase differences remain bounded and often constant in time.

Principle

Weak coupling or forcing can entrain oscillators when frequency detuning is compensated by interaction, producing stable fixed phase differences; stability follows from attracting phase dynamics on the torus.

Demonstration

Two weakly coupled limit-cycle oscillators with close natural frequencies adjust their instantaneous phases until a constant phase offset is reached and maintained despite small perturbations.

Misapplication

Concluding phase locking from transient alignment of oscillations without verifying persistence or robustness to perturbations, or confusing amplitude synchronization with genuine phase locking.

Consequence

Locked oscillators exhibit frequency entrainment and predictable relative timing, which enables coordinated behavior and reliable phase-coded signaling across the coupled system.

Reversal

Phase drift or slips occur when coupling is too weak or detuning too large: phase differences wander unboundedly or undergo sudden 2π slips, breaking the locked state.

Boundary

Applies to systems that possess a well-defined phase variable (limit cycles, oscillatory solutions); not meaningful for non-oscillatory transients or systems lacking a stable periodic orbit.

Semantic Tension

Closely related to resonance: resonance concerns amplitude response at matching frequencies, while phase locking concerns persistent phase relationships often regardless of amplitude changes.

Synthesis

Phase locking is the stable establishment of bounded, typically constant phase differences between oscillators produced by interaction or forcing, yielding coordinated timing without requiring amplitude equality.