Computation of saddle type slow manifolds using iterative methods

This paper presents an alternative approach for the computation of trajectory segments on slow manifolds of saddle type. This approach is based on iterative methods rather than collocation-type methods. Compared to collocation methods, that require mesh refinements to ensure uniform convergence with respect to $\epsilon$, appropriate estimates are directly attainable using the method of this paper. The method is applied to several examples including: A model for a pair of neurons coupled by reciprocal inhibition with two slow and two fast variables and to the computation of homoclinic connections in the FitzHugh-Nagumo system.Comment: To appear in SIAM Journal of Applied Dynamical System

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

Planck intermediate results. VIII. Filaments between interacting clusters

About half of the baryons of the Universe are expected to be in the form of filaments of hot and low density intergalactic medium. Most of these baryons remain undetected even by the most advanced X-ray observatories which are limited in sensitivity to the diffuse low density medium. The Planck satellite has provided hundreds of detections of the hot gas in clusters of galaxies via the thermal Sunyaev-Zel'dovich (tSZ) effect and is an ideal instrument for studying extended low density media through the tSZ effect. In this paper we use the Planck data to search for signatures of a fraction of these missing baryons between pairs of galaxy clusters. Cluster pairs are good candidates for searching for the hotter and denser phase of the intergalactic medium (which is more easily observed through the SZ effect). Using an X-ray catalogue of clusters and the Planck data, we select physical pairs of clusters as candidates. Using the Planck data we construct a local map of the tSZ effect centered on each pair of galaxy clusters. ROSAT data is used to construct X-ray maps of these pairs. After having modelled and subtracted the tSZ effect and X-ray emission for each cluster in the pair we study the residuals on both the SZ and X-ray maps. For the merging cluster pair A399-A401 we observe a significant tSZ effect signal in the intercluster region beyond the virial radii of the clusters. A joint X-ray SZ analysis allows us to constrain the temperature and density of this intercluster medium. We obtain a temperature of kT = 7.1 +- 0.9, keV (consistent with previous estimates) and a baryon density of (3.7 +- 0.2)x10^-4, cm^-3. The Planck satellite mission has provided the first SZ detection of the hot and diffuse intercluster gas.Comment: Accepted by A&

Planck early results XIV : ERCSC validation and extreme radio sources

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Planck early results XV : Spectral energy distributions and radio continuum spectra of northern extragalactic radio sources

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Planck early results XI : Calibration of the local galaxy cluster Sunyaev-Zeldovich scaling relations

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Planck intermediate results V : Pressure profiles of galaxy clusters from the Sunyaev-Zeldovich effect

This article has an erratum: http://dx.doi.org/10.1051/0004-6361/201220040ePeer reviewe

Planck early results IX : XMM-Newton follow-up for validation of Planck cluster candidates

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