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We study the sub-Riemannian exponential for contact distributions onmanifolds of dimension greater or equal to 5. We compute an approximation ofthe sub-Riemannian Hamiltonian flow and show that the conjugate time can havemultiplicity 2 in this case. We obtain an approximation of the first conjugatelocus for small radii and introduce a geometric invariant to show that themetric for contact distributions typically exhibits an original behavior,different from the classical 3-dimensional case. We apply these methods to thecase of 5-dimensional contact manifolds. We provide a stability analysis of thesub-Riemannian caustic from the Lagrangian point of view and classify thesingular points of the exponential map.

Short geodesics losing optimality in contact sub-Riemannian manifolds and stability of the 5-dimensional caustic