We consider a few applications of Mañé’s C2 Connecting Lemma. These are the C2 creation of homoclinic points associated to a basic set (i.e., isolated transitive hyperbolic set), a C2 locally generic criterion to know whether a given point belongs to the stable set of hyperbolic homoclinic classes, and that measurably hyperbolic diffeomorphisms (i.e., having the closure of supports of all invar...

In this paper we study the isotropic realizability of a given regular gradient field ∇u as an electric field, namely when ∇u is solution of the equation div (σ∇u) = 0 for some isotropic conductivity σ > 0. The case of a function u without critical point was investigated in [5] thanks to a gradient flow approach. The presence of a critical point needs a specific treatment according to the behavi...

A necessary and sufficient condition is given for all hyperbolic difference schemes that use up to nine mesh points to be of second-order accuracy. We also construct a new difference scheme for two-dimensional hyperbolic systems of conservation laws. The scheme is of second-order accuracy and requires knowledge of only seven mesh points. A stability condition is obtained and is utilized in nume...

We consider dynamical systems on compact manifolds, which are local dif-feomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that some (strong) integrability of the first hyperbo...

Generative network models play an important role in algorithm development, scaling studies, network analysis, and realistic system benchmarks for graph data sets. A complex network model gaining considerable popularity builds random hyperbolic graphs, generated by distributing points within a disk in the hyperbolic plane and then adding edges between points with a probability depending on their...

This paper provides a theoretical background for Lagrangian Descriptors (LDs). The goal of achieving rigourous proofs that justify the ability of LDs to detect invariant manifolds is simplified by introducing an alternative definition for LDs. The definition is stated for n-dimensional systems with general time dependence, however we rigorously prove that this method reveals the stable and unst...