Gevrey properties of real planar singularly perturbed systems

By applying geometric techniques to real analytic singularly perturbed vector fields on the plane, we develop a way to give a bound on the Gevrey type of the Taylor development of canard manifolds at degenerate planar turning points. By blowing up the phase space at the turning point, we find asymptotic estimates even when such expansions w.r.t. traditional phase space variables do not exist. The asymptotic estimates are then used to give a sufficient and necessary condition on the existence of (local) canard solutions.     

Unambiguous identification of three beta-decaying isomers in 70Cu

Using resonant laser ionization, beta-decay studies, and for the first time mass measurements, three beta-decaying states have been unambiguously identified in 70Cu. A mass excess of -62 976.1(1.6) keV and a half-life of 44.5(2) s for the (6-) ground state have been determined. The level energies of the (3-) isomer at 101.1(3) keV with T(1/2)=33(2) s and the 1+ isomer at 242.4(3) keV with T(1/2)=6.6(2) s are confirmed by high-precision mass measurements. The low-lying levels of 70Cu populated in the decay of 70Ni and in transfer reactions compare well with large-scale shell-model calculations, and the wave functions appear to be dominated by one proton-one neutron configurations outside the closed Z=28 shell and N=40 subshell. This does not apply to the 1+ state at 1980 keV which exhibits a particular feeding and deexcitation pattern not reproduced by the shell-model calculations.     

Canard cycle transition at a slow–fast passage through a jump point

We introduce transitory canard cycles for slow–fast vector fields in the plane. Such cycles separate “canards without head” and “canards with head”, like for example in the Van der Pol equation. We obtain optimal upper bounds on the number of periodic orbits that can appear near the cycle under whatever condition on the related slow divergence integral I  , including the challenging case I=0$I=0$.     

Slow-fast Bogdanov-Takens bifurcations

In this paper we study perturbations from planar vector fields having a line of zeros and representing a singular limit of Bogdanov-Takens (BT) bifurcations. We introduce, among other precise definitions, the notion of slow-fast BT-bifurcation and we provide a complete study of the bifurcation diagram and the related phase portraits. Based on geometric singular perturbation theory, including blow-up, we get results that are valid on a uniform neighborhood both in parameter space and in the phase plane.     

The period function of classical Liénard equations

In this paper we study the number of critical points that the period function of a center of a classical Liénard equation can have. Centers of classical Liénard equations are related to scalar differential equations , with f an odd polynomial, let us say of degree 2?−1. We show that the existence of a finite upperbound on the number of critical periods, only depending on the value of ?, can be reduced to the study of slow-fast Liénard equations close to their limiting layer equations. We show that near the central system of degree 2?−1 the number of critical periods is at most 2?−2. We show the occurrence of slow-fast Liénard systems exhibiting 2?−2 critical periods, elucidating a qualitative process behind the occurrence of critical periods. It all provides evidence for conjecturing that 2?−2 is a sharp upperbound on the number of critical periods. We also show that the number of critical periods, multiplicity taken into account, is always even.     

Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations

This paper deals with local bifurcations occurring near singular points of planar slow-fast systems. In particular, it is concerned with the study of the slow-fast variant of the unfolding of a codimension 3 nilpotent singularity. The slow-fast variant of a codimension 1 Hopf bifurcation has been studied extensively before and its study has lead to the notion of canard cycles in the Van der Pol system. Similarly, codimension 2 slow-fast Bogdanov–Takens bifurcations have been characterized. Here, the singularity is of codimension 3 and we distinguish slow-fast elliptic and slow-fast saddle bifurcations. We focus our study on the appearance on small-amplitude limit cycles, and rely on techniques from geometric singular perturbation theory and blow-up.     

Canard solutions at non-generic turning points

This paper deals with singular perturbation problems for vector fields on -dimensional manifolds. ``Canard solutions' are solutions that, starting near an attracting normally hyperbolic branch of the singular curve, cross a ``turning point' and follow for a while a normally repelling branch of the singular curve. Following the geometric ideas developed by Dumortier and Roussarie in 1996 for the study of canard solutions near a generic turning point, we study canard solutions near non-generic turning points. Characterization of manifolds of canard solutions is given in terms of boundary conditions, their regularity properties are studied and the relation is described with the more traditional asymptotic approach. It reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work. Since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve, we also study their contact and its relation to the ``control curve'.

     

Logical topology design for IP rerouting: ASONs versus static OTNs

IP-based backbone networks are gradually moving to a network model consisting of high-speed routers that are flexibly interconnected by a mesh of light paths set up by an optical transport network that consists of wavelength division multiplexing (WDM) links and optical cross-connects. In such a model, the generalized MPLS protocol suite could provide the IP centric control plane component that will be used to deliver rapid and dynamic circuit provisioning of end-to-end optical light paths between the routers. This is called an automatic switched optical (transport) network (ASON). An ASON enables reconfiguration of the logical IP topology by setting up and tearing down light paths. This allows to up- or downgrade link capacities during a router failure to the capacities needed by the new routing of the affected traffic. Such survivability against (single) IP router failures is cost-effective, as capacity to the IP layer can be provided flexibly when necessary. We present and investigate a logical topology optimization problem that minimizes the total amount or cost of the needed resources (interfaces, wavelengths, WDM line-systems, amplifiers, etc.) in both the IP and the optical layer. A novel optimization aspect in this problem is the possibility, as a result of the ASON, to reuse the physical resources (like interface cards and WDM line-systems) over the different network states (the failure-free and all the router failure scenarios). We devised a simple optimization strategy to investigate the cost of the ASON approach and compare it with other schemes that survive single router failures.     

Data-centric optical networks and their survivability

The explosive growth of data traffic-for example, due to the popularity of the Internet-poses important emerging network requirements on today's telecommunication networks. This paper describes how core networks will evolve to optical transport networks (OTNs), which are optimized for the transport of data traffic, resulting in an IP-directly-over-OTN paradigm. Special attention is paid to the survivability of such data-centric optical networks. This becomes increasingly crucial since more and more traffic is multiplexed onto a single fiber (e.g., 160×10 Gb/s), implying that a single cable cut can affect incredible large traffic volumes. In particular, this paper is tackling multilayer survivability problems, since a data-centric optical network consists of at least an IP and optical layer. In practice, this means that the questions "in which layer or layers should survivability be provided?" and "if multiple layers are chosen for this purpose, then how should this functionality in these layers be coordinated?" have to be answered. In addition to a theoretical study, some case studies are presented in order to illustrate the relevance of the described issues and to help in strategic planning decisions. Two case studies are studying the problem from a capacity viewpoint. Another case study presents simulations from a timing/throughput performance viewpoint