Infinitesimal Lyapunov functions for singular flows
Abstract
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.
Cite
@article{arxiv.1201.2550,
title = {Infinitesimal Lyapunov functions for singular flows},
author = {Vitor Araujo and Luciana Salgado},
journal= {arXiv preprint arXiv:1201.2550},
year = {2015}
}
Comments
37 pages, 1 figure; corrected the statement of Lemma 2.2 and item (2) of Theorem 2.7; removed item (5) of Theorem 2.7 and its wrong proof since the statement of this item was false; corrected items (1) and (2) of Theorem 2.23 and their proofs. Included Example 6 on smooth reduction of families of quadratic forms. The published version in Math Z journal needs an errata