SL₂-orbits and degenerations of mixed Hodge structure

We extend Schmid’s SL₂-orbit theorem to a class of variations of mixed Hodge structure which normal functions, logarithmic deformations, degenerations of 1-motives and archimedean heights. In particular, as a consequence of this theorem, we obtain a simple formula for the asymptotic behavior of the archimedean height of a flat family of algebraic cycles which depends only on the weight filtration and local monodromy.