arXiv:2611.13031v1 Announce Type: new Abstract: We investigate the bifurcation of symmetric periodic solutions for differential systems with multiple distributed delays. For general multiple-delay systems with a quadratic potential, we establish a Hamiltonian formulation with a \(P_{n,m}\)-boundary condition and derive explicit Maslov index and nullity formulas. For the special case of two distributed delays, we extend the analysis to general nonlinear potentials. These results yield explicit resonance criteria, bifurcation conditions, and Rabinowitz type and Fadell-Rabinowitz type alternatives for $\mathbb{R} $-distinct solutions.
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