Search results for "Separatrix"

On invariant manifolds of saddle points for 3D multistable models

2017

In dynamical systems a particular solution is completely determined by the parameters considered and the initial conditions. Indeed, when the model shows a multistability, starting from different initial state, the trajectories can evolve towards different attractors. The invariant manifolds of the saddle points separate the vector field into the basins of attraction of different stable equilibria. The aim of this work is the reconstruction of these separation surfaces in order to know in advance the geometry of the basins. In this paper three-dimensional models with three or more stable fixed points is investigated. To this purpose a procedure for the detection of the scattered data lying …

Eurofusion-DEMO Divertor - Cassette Design and Integration

2020

International audience; The Eurofusion-DEMO design will complete the Pre Conceptual Design phase (PCD) with a PCD Gate, named G1, scheduled to take place in Q4 2020 that will focus on assessing the feasibility of the plant and its main components prior to entering into the Conceptual Design phase. In the paper first an overview is given of the Eurofusion-DEMO Divertor Assembly including design and interface description, systems and functional requirements, load specification, system classification, manufacturing procedures and cost estimate. Then critical issues are discussed and potential design solutions are proposed, e.g.:- Neutron material damage limits of the different (structural) mat…

Detecting tri‐stability of 3D models with complex attractors via meshfree reconstruction of invariant manifolds of saddle points

2018

In mathematical modeling it is often required the analysis of the vector field topology in order to predict the evolution of the variables involved. When a dynamical system is multi-stable the trajectories approach different stable states, depending on the initialmconditions. The aim of this work is the detection of the invariant manifolds of thesaddle points to analyze the boundaries of the basins of attraction. Once that a sufficient number of separatrix points is found a Moving Least Squares meshfree method is involved to reconstruct the separatrix manifolds. Numerical results are presented to assess the method referring to tri-stable models with complex attractors such as limit cycles o…

Minimal models of foliated algebraic surfaces

1999

MODELES MINIMAUX DES SURFACES ALGEBRIQUES FEUILLETEES. -Nous etudions les feuilletages holomorphes sur les surfaces algebriques du point de vue de la geometrie birationnelle. Apres avoir introduit une notion de modele minimal, nous classifions les feuilletages qui n'ont pas de modele minimal dans leur classe d'isomorphisme birationnel. Comme corollaire on obtient un resultat concernant la dynamique des diffeomorphismes polynomiaux.

ADVANCED MESHLESS NUMERICAL METHODS AND APPLICATIONS

On basins of attraction for a predator-prey model via meshless approximation

2016

Abstract. In this work an epidemiological predator-prey model is studied. It analyzes the spread of an infectious disease with frequency-dependent and vertical transmission within the predator population. In particular we consider social predators, i.e. they cooperate in groups to hunt. The result is a three-dimensional system in which the predator population is divided into susceptible and infected individuals. Studying the dynamical system and bifurcation diagrams, a scenario was identified in which the model shows multistability but the domain of attraction of one equilibrium point can be so small that it is almost the point itself. From a biological point of view it is important to anal…

EMERGENCE OF TRAVELLING WAVES IN SMOOTH NERVE FIBRES

2008

International audience; An approximate analytical solution characterizing initial condi- tions leading to action potential ¯ring in smooth nerve ¯bres is determined, using the bistable equation. In the ¯rst place, we present a non-trivial sta- tionary solution wave. Then, we extract the main features of this solution to obtain a frontier condition between the initiation of the travelling waves and a decay to the resting state. This frontier corresponds to a separatrix in the projected dynamics diagram depending on the width and the amplitude of the stationary wave.

Some Applications of the Poincaré-Bendixson Theorem

2021

We consider a C 1 vector field X defined on an open subset U of the plane, with compact closure. If X has no singular points and if U is simply connected, a weak version of the Poincaré-Bendixson Theorem says that the limit sets of X in U are empty but that one can defined non empty extended limit sets contained into the boundary of U. We give an elementary proof of this result, independent of the classical Poincaré-Bendixson Theorem. A trapping triangle T based at p, for a C 1 vector field X defined on an open subset U of the plane, is a topological triangle with a corner at a point p located on the boundary ∂U and a good control of the tranversality of X along the sides. The principal app…

Sur les feuilletages alg�briques de Rolle

1997

L'objet de ce travail est l'etude des feuilletages algebriques de Rolle dans \( \Bbb {R}^n \). On montre que leur restriction au complementaire d'un nombre fini de feuilles possede une structure de produit. On precise aussi la topologie de certaines de leurs feuilles.

Simulated effects of W dust ablation and deposition on the pedestal edge in JET D and DT experiments

2022

Abstract A modelling analysis is performed on JET D and DT discharges, where W dust influx across the separatrix, in the pedestal edge region may affect L–H–L mode transition. The experimental basis of the proposed approach stems from the observation that transient impurity events (TIEs) are often associated with the presence of a shower of particles seen in the camera images and with strong optical emission. If the localised source of radiation is a number of heated or ablated large dust particles, then the questions addressed here are: how far will the ablated dust material penetrate and what effect will this have on the edge of the pedestal in relevant JET D and in a high fusion yield D–…