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We investigate the structure of the set of $T$-periodic solutions toperiodically perturbed coupled delay differential equations on differentiablemanifolds. By using fixed point index and degree-theoretic methods we prove theexistence of branches of $T$-periodic solutions to the considered equations. Asmain application of our methods, we study a generalized version of thesunflower equation.

Periodic perturbations with delay of coupled differential equations on manifolds with application to a sunflower-like equation