Feedback-delay dependence of the stability of cluster periodic orbits in populations of degrade-and-fire oscillators with common activator

The paper rigorously proves: (i) τ=0 implies instability to cluster-smearing for sufficiently large replication q, and (ii) any small positive delay τ yields a continued fixed point that is exponentially stable to smearing, via Lemmas 4.3 and 4.2 and the proof of Theorem 4.4 . The candidate’s Part (ii) broadly matches the paper’s mechanism (delay decouples self-impact; contraction factor 1+νȦ(·−τ) with Ȧ negative just before firing) , but Part (i) is flawed: it asserts all smearing-direction Jacobian diagonals exceed 1 for any nontrivial split, without a quantitative bound; it also misattributes the role of q (claiming q only ensures splitting), whereas the paper’s instability for τ=0 hinges on the Ȧ(0+) threshold becoming easier as q grows (denominator scales with total population) .