Why stronger correction can become destabilising

Why stronger correction can become destabilising

QuestionCan stronger correction lose a phase lock when its return is delayed?

AnswerYes, in the stated continuous delayed-phase model. Correction must exceed the mismatch to create a principal lock, while positive delay also imposes an upper bound for its local exponential stability. Those bounds belong to this equation; losing its local lock does not establish disappearance of the whole identity.

Explanation

Why stronger correction can become destabilising A delayed comparison can have both a lower coupling limit and an upper stability limit.

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.

Is this already the stability law of every Y-engine scene? It is a checked candidate benchmark. An engine scene needs its own actual equations and parameters mapped before this bound can be used for that scene.

Local exponential-stability result on the principal branch of the stated delayed phase model. Boundary cases and other initial histories require their own analysis.

How the explanations connect

Why stronger correction can become destabilising qualifies When correction restores, repeats, learns or destabilises. After the discrete recurrence distinguishes restoring, repeating and learning, a separate continuous delayed-phase equation shows how return timing bounds one particular local lock.

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