Increasing K, the two clusters merge when K reaches K J and the system is in a nonhomoge- neous stationary state when K > K J

in the range K c < K < K J (K c: critical point, K J: jump point), where the order parameter M is deﬁned by M = ||(Mx, M y)||

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nlin.PS · 2026-04-06 · unverdicted · novelty 1.0

The criticism of the original bifurcation analysis stems only from a definitional mismatch for the paramagnetic phase, with no other conflicts.

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Reply to: Comment on: Discontinuous codimension-two bifurcation in a Vlasov equation (arXiv:2603.22066) nlin.PS · 2026-04-06 · unverdicted · none · ref 1
The criticism of the original bifurcation analysis stems only from a definitional mismatch for the paramagnetic phase, with no other conflicts.