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We study conjugate points along homogeneous geodesics in generalized flag manifolds. This is done by analyzing the second variation of the energy of such geodesics. We also give an example of how the homogeneous Ricci flow can evolve in such way to produce conjugate points in the complex projective space \({\mathbb {C}}P^{2n+1} = \text {Sp}(n+1)/(\text {U}(1)\times \text {Sp}(n))\).

Variational aspects of homogeneous geodesics on generalized flag manifolds and applications