Shilnikov problem in Filippov dynamical systems

In this paper we introduce the concept of sliding Shilnikov orbits for $3$D Filippov systems. In short, such an orbit is a piecewise smooth closed curve, composed by Filippov trajectories, which slides on the switching surface and connects a Filippov equilibrium to itself, namely a pseudo saddle-focus. A version of the Shilnikov's Theorem is provided for such systems. Particularly, we show that sliding Shilnikov orbits occur in generic one-parameter families of Filippov systems, and that arbitrarily close to a sliding Shilnikov orbit there exist countably infinitely many sliding periodic orbits. Here, no additional Shilnikov-like assumption is needed in order to get this last result. In addition, we show the existence of sliding Shilnikov orbits in discontinuous piecewise linear differential systems. As far as we know, the examples of Fillippov systems provided in this paper are the first exhibiting such a sliding phenomenon

Fold-Saddle Bifurcation in Non-Smooth Vector Fields on the Plane

This paper presents results concerning bifurcations of 2D piecewise-smooth dynamical systems governed by vector fields. Generic three parameter families of a class of Non-Smooth Vector Fields are studied and its bifurcation diagrams are exhibited. Our main result describes the unfolding of the so called Fold-Saddle singularity

Information criterion on the change-point detection in water quality data

The number of studies using change-point methods to detect shifts has been increasing. The Schwarz Information Criterion SIC is used in this study to search for the change-point in time series of water quality variables

Change-point analysis in environmental time series

Change-points are present in many environmental time series. Time variations in environmental data are complex and they can hinder the identification of the so-called change-points when traditional models are applied to this type of problems. In this study, it is proposed an alternative approach for the application of the change-point analysis by taking into account this data structure (seasonality and autocorrelation) based on the Schwarz Information Criterion (SIC). The approach was applied to time series of surface water quality variables measured at eight monitoring site

Geometric singular perturbartion theory for non-smooth dynamical systems

In this article we deal with singularly perturbed Filippov systems Zε: (1) ˙x = ( F(x, y, ε) if h(x, y, ε) ≤ 0, G(x, y, ε) if h(x, y, ε) ≥ 0, εy˙ = H(x, y, ε), where ε ∈ R is a small parameter, x ∈ Rn, n ≥ 2, and y ∈ R denote the slow and fast variables, respectively, and F, G, h, and H are smooth maps. We study the effect of singular perturbations at typical singularities of Z0. Special attention will be dedicated to those points satisfying q ∈ {h(x, y, 0) = 0} ∩ {H(x, y, 0) = 0} where F or G is tangent to {h(x, y, 0) = 0}. The persistence and the stability properties of those objects are investigated.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP

On the birth of limit cycles for non-smooth dynamical systems

The main objective of this work is to develop, via Brower degree theory and regularization theory, a variation of the classical averaging method for detecting limit cycles of certain piecewise continuous dynamical systems. In fact, overall results are presented to ensure the existence of limit cycles of such systems. These results may represent new insights in averaging, in particular its relation with non smooth dynamical systems theory. An application is presented in careful detail

Gallic acid production with mouldy polyurethane particles obtained from solid state culture of Aspergillus niger GH1

Gallic acid production in a batch bioreactor was evaluated using as catalytic material the mouldy polyurethane solids (MPS) obtained from a solid-state fermentation (SSF) bioprocess carried out for tannase production by Aspergillus niger GH1 on polyurethane foam powder (PUF) with 5 % (v/w) of tannic acid as inducer. Fungal biomass, tannic acid consumption and tannase production were kinetically monitored. SSF was stopped when tannase activity reached its maximum level. Effects of washing with distilled water and drying on the tannase activity of MPS were determined. Better results were obtained with dried and washed MPS retaining 84 % of the tannase activity. Maximum tannase activity produced through SSF after 24 h of incubation was equivalent to 130 U/gS with a specific activity of 36 U/mg. The methylgallate was hydrolysed (45 %) in an easy, cheap and fast bioprocess (30 min). Kinetic parameters of tannase self-immobilized on polyurethane particles were calculated to be 5 mM and 04.1×102 mM/min for K M and V max, respectively. Results demonstrated that the MPS, with tannase activity, can be successfully used for the production of the antioxidant gallic acid from methyl-gallate substrate. Direct use of PMS to produce gallic acid can be advantageous as no previous extraction of enzyme is required, thus reducing production costs.Authors thank the National Council for Science and Technology (CONACYT-Mexico) for the financial support. The present work was performed as part of a cooperative agreement between DIA-Universidad Autonoma de Coahuila (Mexico) and IBB-Universidade do Minho (Portugal) within a specific training stay undertaken at the DEB-UM. Part of the research was funded by a project SEP-CONACYT-CB-2011