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

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

Hilbert's 16th problem for classical Liénard equations of even degree

AbstractClassical Liénard equations are two-dimensional vector fields, on the phase plane or on the Liénard plane, related to scalar differential equations x¨+f(x)x˙+x=0. In this paper, we consider f to be a polynomial of degree 2l−1, with l a fixed but arbitrary natural number. The related Liénard equation is of degree 2l. We prove that the number of limit cycles of such an equation is uniformly bounded, if we restrict f to some compact set of polynomials of degree exactly 2l−1. The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most l

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

Validity of the Brunet-Derrida formula for the speed of pulled fronts with a cutoff

We establish rigorous upper and lower bounds for the speed of pulled fronts with a cutoff. We show that the Brunet-Derrida formula corresponds to the leading order expansion in the cut-off parameter of both the upper and lower bounds. For sufficiently large cut-off parameter the Brunet-Derrida formula lies outside the allowed band determined from the bounds. If nonlinearities are neglected the upper and lower bounds coincide and are the exact linear speed for all values of the cut-off parameter.Comment: 8 pages, 3 figure

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

QTL analysis of high thermotolerance with superior and downgraded parental yeast strains reveals new minor QTLs and converges on novel causative alleles involved in RNA processing

Revealing QTLs with a minor effect in complex traits remains difficult. Initial strategies had limited success because of interference by major QTLs and epistasis. New strategies focused on eliminating major QTLs in subsequent mapping experiments. Since genetic analysis of superior segregants from natural diploid strains usually also reveals QTLs linked to the inferior parent, we have extended this strategy for minor QTL identification by eliminating QTLs in both parent strains and repeating the QTL mapping with pooled-segregant whole-genome sequence analysis. We first mapped multiple QTLs responsible for high thermotolerance in a natural yeast strain, MUCL28177, compared to the laboratory strain, BY4742. Using single and bulk reciprocal hemizygosity analysis we identified MKT1 and PRP42 as causative genes in QTLs linked to the superior and inferior parent, respectively. We subsequently downgraded both parents by replacing their superior allele with the inferior allele of the other parent. QTL mapping using pooled-segregant whole-genome sequence analysis with the segregants from the cross of the downgraded parents, revealed several new QTLs. We validated the two most-strongly linked new QTLs by identifying NCS2 and SMD2 as causative genes linked to the superior downgraded parent and we found an allele-specific epistatic interaction between PRP42 and SMD2. Interestingly, the related function of PRP42 and SMD2 suggests an important role for RNA processing in high thermotolerance and underscores the relevance of analyzing minor QTLs. Our results show that identification of minor QTLs involved in complex traits can be successfully accomplished by crossing parent strains that have both been downgraded for a single QTL. This novel approach has the advantage of maintaining all relevant genetic diversity as well as enough phenotypic difference between the parent strains for the trait-of-interest and thus maximizes the chances of successfully identifying additional minor QTLs that are relevant for the phenotypic difference between the original parents

Comparative analysis of inhaled particles contained in human bronchoalveolar lavage fluids, lung parenchyma and lymph nodes.

Translocation of inhaled particles from the alveolar spaces to lung parenchyma and lymph nodes is one of the mechanisms that determine the biopersistence of particles. This study compares the nonfibrous particulate burden in bronchoalveolar lavage (BAL) fluids, lung parenchyma, and thoracic lymph nodes and attempts to detect the degree of differentiation, if any, based on particle size or type. This comparison can only be done on BAL, lung parenchyma, and lymph node samples collected from the same subject over a short time. Patients undergoing surgical lung resection are suitable for this purpose. Particles recovered by digestion-filtration were counted, sized, and analyzed by analytical transmission electron microscopy. Total particle load ranges grossly between 10(5) to 10(7) p/ml in BAL, 10(9) to 10(10) p/g dry tissue in parenchyma and 10(10) to 10(11) p/g dry tissue in lymph nodes. Diameters are log-normally distributed and mean diameters range between 0.5 to 0.9 micron. Nonlamellar silicate particles have a significantly larger diameter in lymph nodes. Differences in particle type between the three sampling sites are small and nonsystematic