Delay-Coupled Oscillators
A time delay τ in the coupling between oscillators creates phase frustration, destabilizes synchronized states, and generates Hopf bifurcations. The stability boundary in (τ,κ) space partitions synchrony from chaos.
Synchronized
Theory
θ̇₁ = ω₁ + κ sin(θ₂(t−τ) − θ₁)
θ̇₂ = ω₂ + κ sin(θ₁(t−τ) − θ₂)
Hopf boundary:
κ_c = Δω/(2|cos(ω₀τ)|)
Phase-locked state:
Δφ* = ω₀τ + arcsin(Δω/2κ)
Delay→frustration→chaos