Delay-differential extension of the logistic two-gene oscillator undergoes a Hopf bifurcation at explicit critical total delay τ_c with closed-form ω_c, transversality bound, and supercritical amplitude laws, matching p53 data within 3%.
Wu, Symmetric functional differential equations and neural networks with memory,Transactions of the American Mathematical Society350(1998) 4799– 4838
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Sustained Limit Cycles in the Logistic Two-Gene Genetic Oscillator: A Delay-Driven Hopf Bifurcation
Delay-differential extension of the logistic two-gene oscillator undergoes a Hopf bifurcation at explicit critical total delay τ_c with closed-form ω_c, transversality bound, and supercritical amplitude laws, matching p53 data within 3%.