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We prove a spectral flow formula for one‐parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of curves of Lagrangians induced by the stable and unstable subspaces, respectively. Finally, we deduce sufficient conditions for bifurcation of homoclinic trajectories of one‐parameter families of non‐autonomous Hamiltonian vector fields.

Spectral flow, crossing forms and homoclinics of Hamiltonian systems