<p>We develop a rigorous framework for describing the period-doubling bifurcation of limit cycles in neutral functional differential equations. The approach is based on tools from functional analysis and singularity theory. We provide sufficient conditions for the bifurcation and derive explicit formulas for the normal form coefficients using derivatives of the defining operator. We also establish the stability exchange in the non-degenerate case. The analysis relies on Fredholm operators, the Lyapunov–Schmidt reduction, and the recognition problem for pitchfork bifurcation. Our results extend earlier work on delay differential equations.</p>