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

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

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

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

Agro-industrial wastewater treatment with Acacia dealbata coagulation/flocculation and photo-fenton-based processes

The removal of dissolved organic carbon (DOC) and total polyphenols (TPh) from agro-industrial wastewater was evaluated via the application of coagulation–flocculation–decantation (CFD) and Fenton-based processes. For the CFD process, an organic coagulant based on Acacia dealbata Link. leaf powder (LP) was applied. The results showed that the application of the LP at pH 3.0, with an LP:DOC ratio of 0.5:1 (w/w), achieved a high removal of turbidity, total suspended solids (TSS), and volatile suspended solids (VSS) of 84.7, 79.1, and 76.6%, respectively. The CFD sludge was recycled as fertilizer in plant culture (germination index ≥ 80%). Afterwards, the direct application of Fenton-based processes to raw WW was assessed. The Fenton-based processes (UV/Fenton, UV/Fenton-like, and heterogeneous UV/Fenton) showed high energy efficiency and a cost of 1.29, 1.31 and 1.82 €/g/L DOC removal, respectively. The combination of both processes showed the near complete removal of TPh and DOC after 240 min of reaction time, with high energy efficiency. In accordance with the results obtained, the combination of CFD with Fenton-based processes achieves the legal limits for the disposal of water into the environment, thus allowing the water to be recycled for irrigation.Fundação para a Ciência e a Tecnologia | Ref. UIDB/00616/2020Fundação para a Ciência e a Tecnologia | Ref. UI/BD/150847/2020CCRN - Comissão de Coordenação da Região Norte | Ref. NORTE-01-0145-FEDER-00004