Why stronger correction can become destabilising

Too little return cannot balance the mismatch

Consider a relative phase δ with uncoupled rate mismatch Δω, positive correction strength K and return delay τ. The candidate law is δ̇(t)=Δω−K sin δ(t−τ). A locked phase requires sin δ*=Δω/K.

If K is smaller than the magnitude of the mismatch, no locked phase can supply the needed balance. This is the lower limit: correction is insufficient for this particular relation.

Too much late correction can overshoot

On the principal locked branch, the small error obeys ė(t)=−a e(t−τ), where a=√(K²−Δω²). Its first stability crossing occurs at aτ=π/2. For positive delay, the strict local exponential-stability window therefore has two bounds.

For the declared example Δω=1 and τ=0.5, the window is 1<K<3.2969083095. The saved numerical comparison returns toward the lock at K=2 and loses that local lock at K=4. These are results of this equation and selected histories.

|Δω| < K < √(Δω² + [π/(2τ)]²)

Change the law and reassess the bound

With zero delay this model has no finite upper K bound. Additional instantaneous damping or a different delay placement changes the analysis. The lesson is to calculate return strength together with its timing rather than treating more reinforcement as universally better.

Loss of local phase lock is not automatically disappearance of the entire identity. A bounded oscillation, another closure or a transformed structure needs its own criterion.

Questions to carry forward

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