We present an extension of the notion of infinitesimal Lyapunov function to singular flows, and from this technique we deduce a characterization of partial/sectional hyperbolic sets. In absence of singularities, we can also characterize uniform hyperbolicity. These conditions can be expressed using the space derivative DX of the vector field X together with a field of infinitesimal Lyapunov functions only, and are reduced to checking that a certain symmetric operator is positive definite at the tangent space of every point of the trapping region.

(Joint work with Luciana Salgado, being part of her PhD thesis.)

Reference:
Mathematische Zeitschrift (online april 2013)
DOI 10.1007/s00209-013-1163-8