International School on Dynamical Systems and Applications5-10 September, 2019

Monastir, Tunisia

Organizing Committee:

Pr. Nehla Abdellatif (FST, University of Mannouba, Tunisia) Pr. Zagharide Zine El Abidine (ISIMM, University of Monastir, Tunisia)Dr. Selma Negzaoui (IPEIM, TWMA President, Tunisia) Dr. Rim Jday (ISEAHM, General Secretary of TWMA, Tunisia) Dr. Khadija Mbarki (FSM, Treasurer of TWMA, Tunisia)

This Mathematical School is dedicated to Master Degree students to deepen andenrich their mathematical training and to PhD students who already are makinga thesis in the field of dynamical systems or who need to either discover orcomplement their mathematical training in this field. Three mini-courses indynamical systems will be presented during 5 days by Ms. Sandra Pinelas(Portugal), Shehrazad Selmane (Algeria) and Ms. Nahla Abdellatif (Tunisia).The courses include a significant proportion of time devoted to background andintroductory material. Different areas of dynamical systems will be presentedwith special emphasis in applications. We want this school to promote,encourage and bring together researchers in the field of dynamical systems. Wealso aim to motivate mathematician researchers in general and women inparticular who are very ambitious in this area.

Schedule5 September: arrival at 14h00

10 September: departure at 12h00

Monastir, Tunisia

06 September 07 September 08 September 09 September 10 September8h30-9h Welcome

Speech by thePresident of the

TWMA

Mini course

N. Abdellatif

Mini course

N. Abdellatif

Mini course

S.Selmane

Mini course

S. Selmane

9h-10h30 Mini courseN. Abdellatif

10h30-12h30 Mini courseS. Pinelas

Mini courseS. Pinelas

Mini courseS. Pinelas

Mini courseS. Pinelas

Mini courseS. Pinelas

12h30-14h30 Lunch Lunch Lunch Lunch Lunch14h30-15h15 Conference

Hasna RiahiConference

Ahlème BoukkazPosterssession

ConferenceHiba Hammedi

15h15-16h00 ConferenceZeineb

Ghardallou

ConferenceNadia Chouaib

Excursion

ConferenceIbtissem Ben

Aicha

16h-16h30 Coffee-Break Coffee-Break Coffee-Break

16h30-16h4516h45-17h0017h00-17h1517h15-17h30

Shortcommunications

Shortcommunications

Shortcommunications

18h-19h Round table TWMA AwardsCeremony

Mini-courses

Sandra Pinelas, Professor, Academia Militar, Portugal

This modulus is an introduction at some applications of mathematics inNatural Sciences, in particular in Ecology.

The students will learn how to construct a mathematical model inEcology not only for one specie as well for 2 species in competition andpredator-prey. After to construct the models will analyse the behaviour of thesolution of each model.

Real models will be analysed using difference and differential equations;in particular: butterflies in Holland, wild dogs in USA, elephants in Botswanaand others.

This modulus can be taught in 10 hours (more or less).

Program

1. Introduction2. Population models without regulation3. Structured population models4. Population models with regulation5. Metapopulation models6. Community models: predator-prey and competition7. Epidemiology

2

Schehrazad SELMANE, Science and Technology Houari Boumediene, Algeria

Vice President - North Africa: African Women in Mathematics Association (AWMA)

cselmane@usthb.dz

Mathematical modelling of infectious diseases

This mini-course, introduce the basic modelling skills and methodology for the study of infectious diseases.

We start with by introducing the basic process of setting up an epidemic model using differential equations.The fundamental concepts in infectious disease epidemiology such as the basic reproduction number,incidence rate, latency, immunity, demography, vaccination, route of transmission, and quarantine will bepresented. The local and global stability analysis will be explained in the context of epidemic models andapplications to selected classic epidemic models will be presented. In addition to a detailed analysis of atuberculosis model, numerical simulations will be provided.

3

Nahla Abdellatif, Ecole Nationale des Sciences de l’Informatique, Tunisia

Modèles de compétition entre espèces microbiennes dans un chemostatRésumé

La relation de compétition est une interaction mutuelle entre les espèces vivantes pour l'accèsaux ressources limitées du milieu, elle émerge lorsque deux individus ou populations ou plusdépendent de l’utilisation de la même ressource, réduisant ainsi la disponibilité de celle-ci pourchacun des compétiteurs, sans qu’il ait une interaction directe entre eux l’échelle despopulations, ces interactions peuvent causer une inhibition de la croissance ou mêmel’extinction des espèces. Le principe d’exclusion compétitive affirme qu’une espèce peutéliminer toutes les autres. Nous focalisons dans ce cours sur le phénomène de compétition dedeux ou de plusieurs espèces microbiennes. La compréhension de ce processus permetd’expliquer la coexistence de différentes populations observée dans le milieu naturel qui nousentoure. Nous étudions deux types de compétition la compétition interspécifique entredifférentes espèces de micro-organismes et la compétition intra-spécifique entre les individusde la même espèce. Nous étudions en particulier l’effet du type des fonctions de croissanceconsidérée sur l’issue de la compétition. L’étude d’un modèle de compétition intra-spécifiquelinéaire de espèces mettra en évidence l’existence d’un équilibre positif du systèmelocalement exponentiellement stable, correspondant la coexistence de toutes les espèces..

Plan

1- Introduction la compétition dans un chemostat2- Compétition entre deux espèces microbiennes

2.1 Compétition interspécifique2.2 Compétition intra-spécifique2.3 Le cas de fonctions de croissance monotones2.4 Le cas de fonctions de croissance non monotones

3- Compétition intra-spécifique linéaire entre plusieurs espèces microbiennes

Conferences

Nihed Trabelsi

nihed.trabelsi78@gmail.com

Singular limit solutions for 2-dimensional semilinear ellipticsystem of Liouville type

Although the real world seems in a muddle, many phenomena can be described byusing nonlinear differential equations. A fundamental goal in the study of non-linearinitial boundary value problems involving partial differential equations is todetermine whether solutions to a given equation develop a singularity. Resolving theissue of blow-up is important, in part because it can have bearing on the physicalrelevance and validity of the underlying model. We consider the existence ofsingular limit solutions for a nonlinear elliptic system of Liouville type with Dirichletboundary conditions. We use the nonlinear domain decomposition method.

Hasna Riahi

Solutions periodiques de certains problemes de type n-corps

HASNA RIAHIEcole Nationale d’ ingenieurs de Tunis,

BP 37, Le Belvedere, Tunis, 1002, Tunisie

On considere le systeme Hamiltonien suivantmiqi + ∂V

∂qi(t, q) = 0

q(t + T ) = q(t), ∀t ∈ <

ou qi ∈ <`, ` ≥ 3, 1 ≤ i ≤ n, q = (q1, ..., qn) et V : <× Fn(<`) −→ < avec

Fn(<`) = (q1, ..., qn) ∈ (<`)n/ qi 6= qj si i 6= j

et

V =n∑

i, j = 1i 6= j

Vij(t, qi − qj).

Si le potentiel verifie l’hypothese d’interaction forte, on etudie l’existence et lamultiplicite des solutions de ce systeme, en utilisant la theorie de Morse generaliseeau cas des points critiques a l’infini. Dans le cas ou cette hypothese n’est pas verifiee,on obtient des solutions faibles, admettant des collisions, et on donne une majorationdu nombre de collisions de ces solutions.

En effet, sous des hypotheses appropriees sur V , quand le probleme est pose soussa forme variationnelle, la fonctionnelle correspondante I, admet une suite non-borneede valeurs critiques si la singularite de V en 0 est de type interaction forte ( strongforce). Les points critiques de I sont des solutions T-periodiques classiques du systemeHamiltonien ci-dessus. Si on suppose que tous les points critiques de la fonctionnelleI sont non-degeneres, modulo les translations dans <`, alors on prouve des inegalitesde type Morse qui permettent de montrer que le nombre de points critiques a indicede Morse fixe k augmente exponentiellement avec k, quand k ≡ 0, 1 (mod l − 2). Lapreuve generalise les arguments topologiques utilises par A. Bahri et P. H. Rabinowitzdans leur travail sur le probleme des 3 corps.

Periodic Solutions for Classes of Nonlinear Functional DifferentialEquations with Delays Depending on State and Time

Ahlème Bouakkaz

Skikda University Algeria

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Abstract. The main bulk of the present work is to establish some new results on theexistence and uniqueness of periodic solutions for some classes of nonlinear functionaldifferential equations with delays depending on state and time.

The method used here is one of the most efficient techniques for studying these types ofequations since it combines some useful properties of Green's functions together with fixedpoint theorems to establish sufficient conditions for proving the desired results. The idea ofthis technique is based on the converting of the considered equation into an integral onewhose solutions are recourse to an appropriate fixed point theorem.

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Keywords. Fixed Point Theorem, Green’s Function, Iterative Differential Equations,Periodic solutions.

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References

[1] A. Bouakkaz, A. Ardjouni et A. Djoudi, Existence of Positive Periodic Solutions for aSecond-Order Nonlinear Neutral Differential Equation by the Krasnoselskii's Fixed PointTheorem, Nonlinear Dynamics and Systems Theory, 17 (3), 230-238, 2017.

[2] A. Bouakkaz, A. Ardjouni and A. Djoudi, Periodic Solutions for a Nonlinear IterativeFunctional Differential Equation, Electronic Journal of Mathematical Analysis andApplications, Vol. 7(1) Jan. 2019, pp. 156-166, (Accepted for publication).

[3] A. Bouakkaz, A. Ardjouni and A. Djoudi, Periodic Solutions for a Second OrderNonlinear Functional Differential Equation with Iterrative Terms by Schauder Fixed PointTheorem, Acta Math. Univ. Comenianae, Vol. LXXXVII, 2 , 223-235, 2018.

[4] R. Khemis, A. Ardjouni, A. Bouakkaz and A. Djoudi, Periodic Solutions of a Class of aThird-Order Differential Equations with two Delays Depending on Time and State,Commentationes Mathematicae, Univ. Carolinae (Accepted for publication).37/61 A. Bouakkaz Existence et Unicité des Solutions Périodiques d’une EDR

[5] H.Y. Zhao et M. Fe kan, Periodic solutions for a class of differential equations withdelays depending on state, Mathematical communication Math commun. 22, 1-14, 2017.

[6] H. Y. Zhao et J. Liu, Periodic solutions of an iterative functional differential equationwith variable coefficients. Math. Meth. Appl. Sci., 40 286-292, 2017.

Hiba HammediHigher Institute of Applied Sciences and Technology of Kairouan

University of Kairouan

hammedihiba@yahoo.fr

A spectral study of twisted perturbed waveguides

Abstract

In this presentation we focus on the study of the spectral properties of perturbed 3dquantum waveguides. We mainly consider two types of perturbation. The first one isgeometric. More precisely we study the Laplace operator with Dirichlet boundary conditionsdefined on a twisted tube (here we consider a repulsive twist). The second type ofperturbation is done by locally changing the considered boundary conditions. Actually, westudy the Laplace operator with Dirichlet conditions everywhere on the boundary of thetube except on a bounded part where we consider the Neumann boundary conditions type.On one hand we study the straight tubes (with no geometric perturbations) to figure out theeffect of the boundary conditions perturbation. On the other hand we study the twistedtubes to establish a comparison between the opposite effects of these two types ofperturbation.

Ibtissem Ben Aicha

ibtissembenaicha91@gmail.com

Optimal Stability for a first order coefficient in a non-self-adjointwave equation from Dirichlet-to-Neuman map

Abstract

This work is focused on the study of an inverse problem for a non-self-adjoint hyperbolicequation. More precisely, we attempt to stably recover a first order coefficient appearing ina wave equation from the knowledge of Neumann boundary data. We show in dimension ngreater than two, a stability estimate of Hölder type for the inverse problem underconsideration. The proof involves the reduction to an auxiliaryinverse problem for an electro-magnetic wave equation and the use of an appropriateCarleman estimate.

Solutions to a sublinear elliptic problems

Zeineb GhardallouFaculte des Sciences de Tunis,Universite El Manar, Tunisie

We study the existence of nonnegative continuous solutions to theequation

(1) Lu(x)− ϕ(x, u(x)) = 0, in Ω,

where Ω ⊂ Rd, (d ≥ 3), is a Greenian domain (bounded or un-bounded), L represents a second order elliptic operator with smoothcoefficients satisfying L1 ≤ 0. Under fairly general assumptions on ϕ,we establish a one to one correspondance betwween bounded positiveL−Harmoic functions in H+(Ω) and positive continuous bounded so-lutions of (1), we obtain a characterization of ϕ for which (1) has anonnegative nontrivial bounded solution and we prove that if there isa bounded non zero solution then there is no large solution.

1

This work is concerned with the analysis of the stability properties of the Hamiltonian system governing the dynamic of a Cosserat rods.The perturbed system around an equilibrium, satisfied a dynamical systemwith a particular properties.The analysis of the new dynamical system permits the characterization of the stability properties of equilibria such as uniform rods.

Stability of a Hamiltonian EquilibriumNadia Chouaieb

University of Tunis Elmanar , Tunisie