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

Chaos at the border of criticality

The present paper points out to a novel scenario for formation of chaotic attractors in a class of models of excitable cell membranes near an Andronov-Hopf bifurcation (AHB). The mechanism underlying chaotic dynamics admits a simple and visual description in terms of the families of one-dimensional first-return maps, which are constructed using the combination of asymptotic and numerical techniques. The bifurcation structure of the continuous system (specifically, the proximity to a degenerate AHB) endows the Poincare map with distinct qualitative features such as unimodality and the presence of the boundary layer, where the map is strongly expanding. This structure of the map in turn explains the bifurcation scenarios in the continuous system including chaotic mixed-mode oscillations near the border between the regions of sub- and supercritical AHB. The proposed mechanism yields the statistical properties of the mixed-mode oscillations in this regime. The statistics predicted by the analysis of the Poincare map and those observed in the numerical experiments of the continuous system show a very good agreement.Comment: Chaos: An Interdisciplinary Journal of Nonlinear Science (tentatively, Sept 2008

Investigating Organic Colorants Across Time: Interdisciplinary Insights into the use of Madder, Indigo/Woad, and Weld in Historical Written Sources, Archaeological Textiles, and Ancient Polychromy

Organic dyes have been used from the earliest times to provide color primarily to textiles, but also as a colorant in painting. Such organic dyes could create a wealth of colors, depending on the availability and know-how of resources. These dyes are usually organic in nature, and primarily obtained from different plant sources. Unfortunately, the characterization of natural organic colorants in textiles and artworks is still a challenge. The difficulty of analyzing these materials is sometimes allied to the frequent impossibility of micro-sampling, and the frailty of the objects. Many techniques, such as HPLC (High-Performance Liquid Chromatography) and SERS (Surface-Enhanced Raman Spectroscopy), require the use of a micro-sample, which cannot be recovered after analysis. Moreover, the portable technique Fibre Optic Reflectance Spectroscopy (FORS) can present some challenges in distinguishing between different dye sources belonging to the same molecular family, such as anthraquinone reds. Although no one technique alone can unravel the world of natural dyes, a multi-analytical approach has proven to be far more effective for their identification and characterization. In the present article, we intend to share insights into three different perspectives and types of source material for the study of the use of organic colorants in ancient and historical times: The first case study presents an 18th-century historical recipe for dyeing textiles, the second case study presents a study of preserved archaeological textiles from Nubia, while the third case study presents the use of organic colorants for the polychromy of ancient Greco-Roman iconography

On the approximation of the canard explosion point in singularly perturbed systems without an explicit small parameter

A canard explosion is the dramatic change of period and amplitude of a limit cycle of a system of nonlinear ODEs in a very narrow interval of the bifurcation parameter. It occurs in slow–fast systems and is well understood in singular perturbation problems where a small parameter epsilon defines the time-scale separation. We present an iterative algorithm for the determination of the canard explosion point which can be applied for a general slow–fast system without an explicit small parameter. We also present assumptions under which the algorithm gives accurate estimates of the canard explosion point. Finally, we apply the algorithm to the van der Pol equations, a Templator model for a self-replicating system and a model for intracellular calcium oscillations with no explicit small parameters and obtain very good agreement with results from numerical simulations.<br/

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

Painting the Palace of Apries I: ancient binding media and coatings of the reliefs from the Palace of Apries, Lower Egypt

This study gives an account of the organic components (binders and coatings) found in the polychromy of some fragmented architectural reliefs from the Palace of Apries in Memphis, Egypt (26th Dynasty, ca. 589-568 BCE). A column capital and five relief fragments from the collections of the Ny Carlsberg Glyptotek in Copenhagen were chosen for examination, selected because of their well-preserved polychromy. Samples from the fragments were first investigated using Fourier transform infrared (FTIR) spectroscopy to screen for the presence of organic materials and to identify the chemical family to which these materials belong (proteinaceous, polysaccharides or lipid). Only the samples showing the potential presence of organic binder residues were further investigated using gas chromatography with mass spectrometry detection (GC-MS) targeting the analysis towards the detection and identification of compounds belonging to the chemical families identified by FTIR. The detection of polysaccharides in the paint layers on the capital and on two of the fragments indicates the use of plant gums as binding media. The interpretation of the sugar profiles was not straightforward so botanical classification was only possible for one fragment where the results of analysis seem to point to gum arabic. The sample from the same fragment was found to contain animal glue and a second protein material (possibly egg). While the presence of animal glue is probably ascribable to the binder used for the ground layer, the second protein indicates that either the paint layer was bound in a mixture of different binding materials or that the paint layer, bound in a plant gum, was then coated with a proteinaceous material. The surface of two of the investigated samples was partially covered by translucent waxy materials that were identified as a synthetic wax (applied during old conservation treatments) and as beeswax, respectively. It is possible that the beeswax is of ancient origin, selectively applied on yellow areas in order to create a certain glossiness or highlight specific elements