Canard-like phenomena in piecewise-smooth Van der Pol systems

We show that a nonlinear, piecewise-smooth, planar dynamical system can exhibit canard phenomena. Canard solutions and explosion in nonlinear, piecewise-smooth systems can be qualitatively more similar to the phenomena in smooth systems than piecewise-linear systems, since the nonlinearity allows for canards to transition from small cycles to canards ``with heads." The canards are born of a bifurcation that occurs as the slow-nullcline coincides with the splitting manifold. However, there are conditions under which this bifurcation leads to a phenomenon called super-explosion, the instantaneous transition from a globally attracting periodic orbit to relaxations oscillations. Also, we demonstrate that the bifurcation---whether leading to canards or super-explosion---can be subcritical.Comment: 17 pages, 11 figure

On the use of blow up to study regularizations of singularities of piecewise smooth dynamical systems in R3\mathbb{R}^3

In this paper we use the blow up method of Dumortier and Roussarie \cite{dumortier_1991,dumortier_1993,dumortier_1996}, in the formulation due to Krupa and Szmolyan \cite{krupa_extending_2001}, to study the regularization of singularities of piecewise smooth dynamical systems \cite{filippov1988differential} in $\mathbb R^3$. Using the regularization method of Sotomayor and Teixeira \cite{Sotomayor96}, first we demonstrate the power of our approach by considering the case of a fold line. We quickly recover a main result of Bonet and Seara \cite{reves_regularization_2014} in a simple manner. Then, for the two-fold singularity, we show that the regularized system only fully retains the features of the singular canards in the piecewise smooth system in the cases when the sliding region does not include a full sector of singular canards. In particular, we show that every locally unique primary singular canard persists the regularizing perturbation. For the case of a sector of primary singular canards, we show that the regularized system contains a canard, provided a certain non-resonance condition holds. Finally, we provide numerical evidence for the existence of secondary canards near resonance.Comment: To appear in SIAM Journal of Applied Dynamical System

Why Children Obey the Law: Rethinking Juvenile Justice and Children's Rights in Europe through Procedural Justice

This article explores how the idea of procedural justice can help us to rethink juvenile justice and research children''s rights in Europe differently. To frame the following argument, we will question four implications of the procedural justice perspective: 1) the need to implement rights and not just proclaim them, 2) the need to investigate a double perspective'' on children''s rights implying both juvenile justice professionals and children in conflict with the law, 3) the child''s right to effectively participate and be involved in the process and 4) the idea that age matters in the judicial reaction to crime. The resulting conclusions and discussions revolve around the scientific consequences and challenges we must face when we take procedural justice perspective seriously

Where should we apply biochar?

The heating of biomass under low-oxygen conditions generates three co-products, bio-oil, biogas, and biochar. Bio-oil can be stabilized and used as fuel oil or be further refined for various applications and biogas can be used as an energy source during the low-oxygen heating process. Biochar can be used to sequester carbon in soil and has the potential to increase crop yields when it is used to improve yield-limiting soil properties. Complex bio-physical interactions have made it challenging to answer the question of where biochar should be applied for the maximum agronomic and economic benefits. We address this challenge by developing an extensive informatics workflow for processing and analyzing crop yield response data as well as a large spatial-scale modeling platform. We use a probabilistic graphical model to study the relationships between soil and biochar variables and predict the probability and magnitude of crop yield response to biochar application. Our results show an average increase in crop yields ranging from 4.7% to 6.4% depending on the biochar feedstock and application rate. Expected yield increases of at least 6.1% and 8.8% are necessary to cover 25% and 10% of US cropland with biochar. We find that biochar application to crop area with an expected yield increase of at least 5.3%–5.9% would result in carbon sequestration offsetting 0.57%–0.67% of US greenhouse gas emissions. Applying biochar to corn area is the most profitable from a revenue perspective when compared to soybeans and wheat because additional revenues accrued by farmers are not enough to cover the costs of biochar applications in many regions of the United States

Mixed-mode oscillations in a multiple time scale phantom bursting system

In this work we study mixed mode oscillations in a model of secretion of GnRH (Gonadotropin Releasing Hormone). The model is a phantom burster consisting of two feedforward coupled FitzHugh-Nagumo systems, with three time scales. The forcing system (Regulator) evolves on the slowest scale and acts by moving the slow nullcline of the forced system (Secretor). There are three modes of dynamics: pulsatility (transient relaxation oscillation), surge (quasi steady state) and small oscillations related to the passage of the slow nullcline through a fold point of the fast nullcline. We derive a variety of reductions, taking advantage of the mentioned features of the system. We obtain two results; one on the local dynamics near the fold in the parameter regime corresponding to the presence of small oscillations and the other on the global dynamics, more specifically on the existence of an attracting limit cycle. Our local result is a rigorous characterization of small canards and sectors of rotation in the case of folded node with an additional time scale, a feature allowing for a clear geometric argument. The global result gives the existence of an attracting unique limit cycle, which, in some parameter regimes, remains attracting and unique even during passages through a canard explosion.Comment: 38 pages, 16 figure

Singular perturbation analysis of a regularized MEMS model

Micro-Electro Mechanical Systems (MEMS) are defined as very small structures that combine electrical and mechanical components on a common substrate. Here, the electrostatic-elastic case is considered, where an elastic membrane is allowed to deflect above a ground plate under the action of an electric potential, whose strength is proportional to a parameter $\lambda$. Such devices are commonly described by a parabolic partial differential equation that contains a singular nonlinear source term. The singularity in that term corresponds to the so-called "touchdown" phenomenon, where the membrane establishes contact with the ground plate. Touchdown is known to imply the non-existence of steady state solutions and blow-up of solutions in finite time. We study a recently proposed extension of that canonical model, where such singularities are avoided due to the introduction of a regularizing term involving a small "regularization" parameter $\varepsilon$. Methods from dynamical systems and geometric singular perturbation theory, in particular the desingularization technique known as "blow-up", allow for a precise description of steady-state solutions of the regularized model, as well as for a detailed resolution of the resulting bifurcation diagram. The interplay between the two main model parameters $\varepsilon$ and $\lambda$ is emphasized; in particular, the focus is on the singular limit as both parameters tend to zero

Tratamiento quirúrgico en las cifosis congénitas: Revisión de 14 pacientes

Los autores efectúan una revisión de 14 pacientes con cifosis congénita, intervenidos entre los años 1979-1989, con un seguimiento medio de 7 años. La edad media preoperatoria fue de 11 años (todos ellos mayores de 5 años), con una cifosis media de 7 9 . En 6 casos se realizó una artrodesis posterior y en 8 una anterior combinada con una fusión posterior. Inicialmente obtuvieron una corrección media de la curva de 18° con la artrodesis posterior y 20° con la artrodesis combinada. La pérdida postoperatoria final fue de 10° y 8° respectivamente. En un caso, se produjo una pseudoartrosis por fusión corta. Como complicaciones postoperatorias en 5 pacientes, una radiculopatía, una infección superficial y cuatro protusiones de material que requirieron su extracción. Los autores analizan los factores que han podido influir en los resultados obtenidos, comparándolos posteriormente con los conseguidos por otros centros hospitalarios importantes.Fourteen patients with congenital kyphosis treated surgically between 1979- 1989 were reviewed. All had a follow-up of 2 years or more, with an average follow-up of 7 years. The average age at surgery was 11 and the average kyphosis was 79°. Six cases had posterior fusion only and eigth had combined anterior and posterior fusion. The results showed an average correction of the curve at surgery of 18° with posterior arthrodesis and 20° with combined arthrodesis. There was thus an average loss of 10° and 8° respectively from the time of surgery in both types of treatment. Pseudoarthrosis by short fusion ocurred in one case. Other complications after surgery were 1 radiculopathy, one wound infection and four rod protusion (six patients). The factors that have influence in this results were analysed. A comparison from the results of treatment at other medical centers was also carried out

Effects of a localized beam on the dynamics of excitable cavity solitons

We study the dynamical behavior of dissipative solitons in an optical cavity filled with a Kerr medium when a localized beam is applied on top of the homogeneous pumping. In particular, we report on the excitability regime that cavity solitons exhibits which is emergent property since the system is not locally excitable. The resulting scenario differs in an important way from the case of a purely homogeneous pump and now two different excitable regimes, both Class I, are shown. The whole scenario is presented and discussed, showing that it is organized by three codimension-2 points. Moreover, the localized beam can be used to control important features, such as the excitable threshold, improving the possibilities for the experimental observation of this phenomenon.Comment: 9 Pages, 12 figure