We consider sliding motion, in the sense of Filippov, on a discontinuity surface Σ of co-dimension 2. We characterize, and restrict to, the case of Σ being attractive through sliding. In this situation, we show that a certain Filippov sliding vector field f_F (suggested in Alexander and Seidman, 1998 [2], di Bernardo et al., 2008 [6], Dieci and Lopez, 2011 [10]) exists and is unique. We also propose a characterization of first order exit conditions, clarify its relation to generic co-dimension 1 losses of attractivity for Σ, and examine what happens to the dynamics on Σ for the aforementioned vector field f_F . Examples illustrate our results.

In this paper we propose and implement numerical methods to detect exponential dichotomy on the real line. Our algorithms are based on the singular value decomposition and the QR factorization of a fundamental matrix solution. The theoretical justification for our methods was laid down in the companion paper: "Exponential Dichotomy on the real line: SVD and QR methods.".

The paper considers the rotation number for a family of linear nonautonomous Hamiltonian systems and its relation with the exponential dichotomy concept. We propose numerical techniques to compute the rotation number and we employ them to infer when a given system enjoys or not an exponential dichotomy. Comparisons with QR-based techniques for exponential dichotomy will give new insights on the structure of the spectrum for the one-dimensional quasi-periodic Schrödinger operator. Experiments on the two dimensional Schrödinger equation will be presented as well.

In this paper, we consider the class of sliding Filippov vector fields in R^3 on the intersection of two smooth surfaces: S= 1 \ 2, where S_i = {x : h_i(x) = 0}, and h_i : R^3 -> R, i = 1, 2, are smooth functions with linearly independent normals. Although, in general, there is no unique Filippov sliding vector field on , here we prove that –under natural conditions– all Filippov sliding vector fields determine the same solution trajectory on . In other words, the aforementioned ambiguity has no meaningful impact. We also examine the implication of this result in the important case of a periodic orbit a portion of which slides on S.