Thèses : « Bifurcation problems »

1

Duka, E. D. « Bifurcation problems in finite elasticity ». Thesis, University of Nottingham, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.384747.

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Melbourne, I. « Bifurcation problems with octahedral symmetry ». Thesis, University of Warwick, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.383295.

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3

Park, Jungho. « Bifurcation and stability problems in fluid dynamics ». [Bloomington, Ind.] : Indiana University, 2007. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3274924.

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Thesis (Ph.D.)--Indiana University, Dept. of Mathematics, 2007.
Source: Dissertation Abstracts International, Volume: 68-07, Section: B, page: 4529. Adviser: Shouhong Wang. Title from dissertation home page (viewed Apr. 22, 2008).

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McGarry, John Kevin. « Application of bifurcation theory to physical problems ». Thesis, University of Leeds, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.252925.

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Bougherara, Brahim. « Problèmes non-linéaires singuliers et bifurcation ». Thesis, Pau, 2014. http://www.theses.fr/2014PAUU3012/document.

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Cette thèse s’inscrit dans le domaine mathématique de l’analyse des équations aux dérivées partielles non linéaires. Précisément, nous nous sommes intéressés à une classe de problèmes elliptiques et paraboliques avec coefficients singuliers. Ce manque de régularité pose un certain nombre de difficultés qui ne permettent pas d’utiliser directement les méthodes classiques de l’analyse non-linéaire fondées entre autres sur des résultats de compacité. Dans les démonstrations des principaux résultats, nous montrons comment pallier ces difficultés. Ceci suppose d’adapter certaines techniques bien connues mais aussi d’introduire de nouvelles méthodes. Dans ce contexte, une étape importante est l’estimation fine du comportement des solutions qui permet d’adapter le principe de comparaison faible, d’utiliser la régularité elliptique et parabolique et d’appliquer dans un nouveau contexte la théorie globale de la bifurcation analytique. La thèse se présente sous forme de deux parties indépendantes. 1- Dans la première partie (chapitre I de la thèse), nous avons étudié un problème quasi-linéaire parabolique fortement singulier faisant intervenir l’opérateur p-Laplacien. On a démontré l’existence locale et la régularité de solutions faibles. Ce résultat repose sur des estimations a priori obtenues via l’utilisation d’inégalités de type log-Sobolev combinées à des inégalités de Gagliardo-Nirenberg. On démontre l’unicité de la solution pour un intervalle de valeurs du paramètre de la singularité en utilisant un principe de comparaison faible fondé sur la monotonie d’un opérateur non linéaire adéquat. 2- Dans la deuxième partie (correspondant aux Chapitres II, III et IV de la thèse), nous sommes intéressés à l’étude de problèmes de bifurcation globale. On a établi pour ces problèmes l’existence de continuas non bornés de solutions qui admettent localement une paramétrisation analytique. Pour établir ces résultats, nous faisons appel à différents outils d’analyse non linéaire. Un outil important est la théorie analytique de la bifurcation globale qui a été introduite par Dancer (voir Chapitre II de la thèse). Pour un problème semi linéaire elliptique avec croissance critique en dimension 2, on montre que les solutions le long de la branche convergent vers une solution singulière (solution non bornée) lorsque la norme des solutions converge vers l’infini. Par ailleurs nous montrons que la branche admet une infinité dénombrable de "points de retournement" correspondant à un changement de l’indice de Morse des solutions qui tend vers l’infini le long de la branche
This thesis is concerned with the mathematical study of nonlinear partial differential equations. Precisely, we have investigated a class of nonlinear elliptic and parabolic problems with singular coefficients. This lack of regularity involves some difficulties which prevent the straight-orward application of classical methods of nonlinear analysis based on compactness results. In the proofs of the main results, we show how to overcome these difficulties. Precisely we adapt some well-known techniques together with the use of new methods. In this framework, an important step is to estimate accurately the solutions in order to apply the weak comparison principle, to use the regularity theory of parabolic and elliptic equations and to develop in a new context the analytic theory of global bifurcation. The thesis presents two independent parts. 1- In the first part (corresponding to Chapter I), we are interested by a nonlinear and singular parabolic equation involving the p-Laplacian operator. We established for this problem that for any non-negative initial datum chosen in a certain Lebeque space, there exists a local positive weak solution. For that we use some a priori bounds based on logarithmic Sobolev inequalities to get ultracontractivity of the associated semi-group. Additionaly, for a range of values of the singular coefficient, we prove the uniqueness of the solution and further regularity results. 2- In the second part (corresponding to Chapters II, III and IV of the thesis), we are concerned with the study of global bifurcation problems involving singular nonlinearities. We establish the existence of a piecewise analytic global path of solutions to these problems. For that we use crucially the analytic bifurcation theory introduced by Dancer (described in Chapter II of the thesis). In the frame of a class of semilinear elliptic problems involving a critical nonlinearity in two dimensions, we further prove that the piecewise analytic path of solutions admits asymptotically a singular solution (i.e. an unbounded solution), whose Morse index is infinite. As a consequence, this path admits a countable infinitely many “turning points” where the Morse index is increasing

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Manoel, Miriam Garcia. « Hidden symmetries in bifurcation problems : the singularity theory ». Thesis, University of Warwick, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.327556.

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Menon, Shakti Narayana. « Bifurcation problems in chaotically stirred reaction-diffusion systems ». Thesis, The University of Sydney, 2008. http://hdl.handle.net/2123/3685.

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A detailed theoretical and numerical investigation of the behaviour of reactive systems under the influence of chaotic stirring is presented. These systems exhibit stationary solutions arising from the balance between chaotic advection and diffusion. Excessive stirring of such systems results in the termination of the reaction via a saddle-node bifurcation. The solution behaviour of these systems is analytically described using a recently developed nonperturbative, non-asymptotic variational method. This method involves fitting appropriate parameterised test functions to the solution, and also allows us to describe the bifurcations of these systems. This method is tested against numerical results obtained using a reduced one-dimensional reaction-advection-diffusion model. Four one- and two-component reactive systems with multiple homogeneous steady-states are analysed, namely autocatalytic, bistable, excitable and combustion systems. In addition to the generic stirring-induced saddle-node bifurcation, a rich and complex bifurcation scenario is observed in the excitable system. This includes a previously unreported region of bistability characterised by a hysteresis loop, a supercritical Hopf bifurcation and a saddle-node bifurcation arising from propagation failure. Results obtained with the nonperturbative method provide a good description of the bifurcations and solution behaviour in the various regimes of these chaotically stirred reaction-diffusion systems.

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Menon, Shakti Narayana. « Bifurcation problems in chaotically stirred reaction-diffusion systems ». University of Sydney, 2008. http://hdl.handle.net/2123/3685.

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Doctor of Philosophy
A detailed theoretical and numerical investigation of the behaviour of reactive systems under the influence of chaotic stirring is presented. These systems exhibit stationary solutions arising from the balance between chaotic advection and diffusion. Excessive stirring of such systems results in the termination of the reaction via a saddle-node bifurcation. The solution behaviour of these systems is analytically described using a recently developed nonperturbative, non-asymptotic variational method. This method involves fitting appropriate parameterised test functions to the solution, and also allows us to describe the bifurcations of these systems. This method is tested against numerical results obtained using a reduced one-dimensional reaction-advection-diffusion model. Four one- and two-component reactive systems with multiple homogeneous steady-states are analysed, namely autocatalytic, bistable, excitable and combustion systems. In addition to the generic stirring-induced saddle-node bifurcation, a rich and complex bifurcation scenario is observed in the excitable system. This includes a previously unreported region of bistability characterised by a hysteresis loop, a supercritical Hopf bifurcation and a saddle-node bifurcation arising from propagation failure. Results obtained with the nonperturbative method provide a good description of the bifurcations and solution behaviour in the various regimes of these chaotically stirred reaction-diffusion systems.

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Sallam, M. H. M. « Aspects of stability and bifurcation theory for multiparameter problems ». Thesis, University of Strathclyde, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.371969.

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Zhang, Tiansi. « Problems of homoclinic flips bifurcation in four-dimensional systems ». Lyon, École normale supérieure (sciences), 2007. http://www.theses.fr/2007ENSL0431.

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Freitas, Pedro S. C. de. « Some problems in nonlocal reaction-diffusion equations ». Thesis, Heriot-Watt University, 1994. http://hdl.handle.net/10399/1401.

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Bari, Rehana. « Local and global bifurcation theory for multiparameter nonlinear eigenvalue problems ». Thesis, Heriot-Watt University, 1995. http://hdl.handle.net/10399/755.

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13

Nishiyama, Takehiro. « Algorithm for Optimization Problems Based on Bifurcation Characteristics of Dynamical Systems ». 京都大学 (Kyoto University), 2001. http://hdl.handle.net/2433/150284.

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Sargent, Ethan. « Radial Solutions to Semipositone Dirichlet Problems ». Scholarship @ Claremont, 2019. https://scholarship.claremont.edu/hmc_theses/229.

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We study a Dirichlet problem, investigating existence and uniqueness for semipositone and superlinear nonlinearities. We make use of Pohozaev identities, energy arguments, and bifurcation from a simple eigenvalue.

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Stamic̆ar, Robert Nikola. « A free boundary problem modelling zoning in rocks / ». *McMaster only, 1998.

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ZONGO, WEND BENEDO EMMANUEL. « BOUNDARY VALUE PROBLEMS FOR QUASI-LINEAR AND HIGHER-ORDER ELLIPTIC OPERATORS AND APPLICATION TO BIFURCATION AND STABILIZATION ». Doctoral thesis, Università degli Studi di Milano, 2022. http://hdl.handle.net/2434/892091.

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In this thesis, we are interested in the study of nonlinear eigenvalue problem and the controllability of partial differential equations in a smooth bounded domain with boundary. The first part is devoted to the analysis of an eigenvalue problem for quasilinear elliptic operators involving homogeneous Dirichlet boundary conditions. We investigate the asymptotic behaviour of the spectrum of the related problem by showing on the one hand the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and bifurcation from infinity using the Leray-Schauder degree on the other hand. We also prove the existence of multiple critical points using variational methods and the Krasnoselski genus. At last, we show a stabilization result for the damped plate equation with logarithmic decay of the associated energy. The proof of this result is achieved by means of a proper Carleman estimate for the fourth-order elliptic operators involving the so-called Lopatinskii-Šapiro boundary conditions and a resolvent estimate for the generator of the damped plate semigroup associated with these boundary conditions.

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Battini, Jean-Marc. « Co-rotational beam elements in instability problems ». Doctoral thesis, KTH, Byggkonstruktion, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3284.

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The purpose of the work presented in this thesis is to implement co-rotational beam elements and branch-switching procedures in order to analyse elastic and elastoplasticinstability problems. For the 2D beam elements, the co-rotational framework is taken from Crisfield [23]. The main objective is to compare three different local elasto-plastic elements. The 3D co-rotational formulation is based on the work of Pacoste and Eriksson [73],with new items concerning the parameterisation of the finite rotations, the definitionof the local frame, the inclusion of warping effects through the introduction of aseventh nodal degree of freedom and the consideration of rigid links. Differenttypes of local formulations are considered, including or not warping effects. It isshown that at least some degree of non-linearity must be introduced in the localstrain definition in order to obtain correct results for certain classes of problems. Within the present approach any cross-section can be modelled, and particularly, the centroid and shear center are not necessarily coincident.Plasticity is introduced via a von Mises material with isotropic hardening. Numericalintegration over the cross-section is performed. At each integration point, theconstitutive equations are solved by including interaction between the normal andshear stresses. Concerning instabilities, a new numerical method for the direct computation of elasticcritical points is proposed. This is based on a minimal augmentation procedure asdeveloped by Eriksson [32–34]. In elasto-plasticity, a literature survey, mainly concernedwith theoretical aspects is first presented. The objective is to get a completecomprehension of the phenomena and to give a basis for the two branch-switchingprocedures presented in this thesis.A large number of examples are used in order to assess the performances of the elements and the path-following procedures.
QC 20100512

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SCAGLIA, MICHELE. « NONTRIVIAL SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS WITH MEASURE DATA ». Doctoral thesis, Università degli Studi di Milano-Bicocca, 2015. http://hdl.handle.net/10281/77845.

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The main subject of this thesis is the study, by a variational approach, of semilinear elliptic problems with measure data. Starting with a semilinear problem with unique solution, we introduce a parametrized perturbation and study the bifurcation phenomena giving rise to further solutions. The main feature is that we are able to use a direct variational approach, even when the semilinearity has no growth assumptions. In this setting, we prove bifurcation results in the line of classical results of Boehme-Marino and Rabinowitz and also global existence results.

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Lee, Yoon-Mee. « Hopf Bifurcation in a Parabolic Free Boundary Problem ». DigitalCommons@USU, 1992. https://digitalcommons.usu.edu/etd/7138.

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We deal with a free boundary problem for a nonlinear parabolic equation, which includes a parameter in the free boundary condition. This type of system has been used in models of ecological systems, in chemical reactor theory and other kinds of propagation phenomena involving reactions and diffusion. The main purpose of this dissertation is to show the global existence, uniqueness of solutions and that a Hopf bifurcation occurs at a critical value of the parameter r. The existence and uniqueness of the solution for this problem are shown by finding an equivalent regular free boundary problem to which existence results can be applied. We then show that as the bifurcation parameter r decreases and passes through a critical value rc, the stationary solution loses stability and a stable periodic solution appears. Several figures have been included, which illustrate this transistion. The pascal source program used in the numerical simulation is included in an appendix.

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Lambarki, El Allioui Saïd. « Resolution d'un probleme de bifurcation par une methode multigrilles ». Toulouse 3, 1986. http://www.theses.fr/1986TOU30102.

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On construit l'unique solution stationnaire asymptotiquement stable d'une equation modele du type burgers generalisee, dans un intervalle borne, en resolvant le probleme de cauchy par une methode multigrilles, a chaque pas du temps. On utilise pres de la bifurcation de la solution nulle une ocndition initiale correspondant a une approximation de galerkin. On construit la solution stationnaire asymptotiquement stable loin de la bifurcation a partir d'une initialisation definie pas a pas

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Lambarki, El Allioui Saïd. « Résolution d'un problème de bifurcation par une méthode multigrilles ». Grenoble 2 : ANRT, 1986. http://catalogue.bnf.fr/ark:/12148/cb375989145.

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GUYARD, FREDERIC. « Interactions de modes dans les problemes de bifurcation avec symetrie spherique ». Nice, 1994. http://www.theses.fr/1994NICE4807.

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Dans cette these, nous etudions les interactions de modes dans les problems de bifurcation avec symetrie spherique. Nous obtenons dans un premier temps une classification des treillis d'isotropie pour un couplage de deux modes 1 et 1' quelconques. Nous appliquons cette classification au cas du couplage de modes 1 et 1+1, ce type de couplage apparaissant de facon naturelle dans le probleme de la convection d'un liquide dans un domaine spherique. Nous definissons ensuite un type de cycle heterocline structurellement stable et force par symetrie. De tels cycles existent pour l'interaction de modes 1 et 1+1 et nous en donnons une classification pour tout 1. Enfin, nous etudions plus precisement quelques types d'interactions apparaissant dans divers problemes de bifurcation avec symetrie spherique

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Anson, David Kenneth. « Numerical bifurcation theory and its application to the Taylor vortex problem ». Thesis, University of Oxford, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.329902.

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Amdjadi, F. « Numerical methods for bifurcations and mode interactions in problems with symmetry ». Thesis, University of Surrey, 1994. http://epubs.surrey.ac.uk/843720/.

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The thesis studies two problems. The first is a problem with 0(2) symmetry which gives rise to a branch with Z2 symmetry. The Jacobian along the Z2-symmetric branch is always singular, due to the group orbit of solutions. The situation when a pair of complex eigenvalues cross the imaginary axis is considered and canonical coordinates are used to remove the degeneracy from the system so that the standard theory can be applied. This method is used in order to understand the type of solutions that occur. The method cannot be applied to partial differential equations and so a method involving a phase condition is employed which has also been used by Aston, Spence and Wu. Two examples have been considered. The first is a simple example on C3 and the second example is the Kuramoto Sivashinsky equation for which a multiple Hopf bifurcation on a D3 symmetric branch has been studied. A Hopf bifurcation on a so called strange fixed point has also been considered and results compared with the numerical results of Hyman and Nicolaenko. The second problem involves mode interactions between a steady state bifurcation and a Hopf bifurcation in problems with Z2 symmetry. Two different extended systems have been considered and it is shown that the mode interaction corresponds to a bifurcation of these systems. Finally a method for predicting the existence of a tertiary Hopf bifurcation arising from a mode interaction, using only the bifurcation diagrams, has also been considered. Initially a problem with Z2 ⊗ Z2 symmetry, which involves a steady state/steady state mode interaction, has been considered and then similar results are obtained for Hopf/steady state mode interactions.

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Lombardi, Eric. « Bifurcation d'orbites homoclines pour certains systemes reversibles. Applications au probleme des vagues ». Nice, 1996. http://www.theses.fr/1996NICE4931.

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On etudie la classe des champs de vecteurs reversibles, en dimension finie ou non, dont le spectre de la differentielle presente au voisinage de l'axe imaginaire la bifurcation suivante: superposition d'une dynamique oscillante induite par une paire de valeurs propres imaginaires pures d'ordre 1, simples et opposees, et d'une dynamique lente induite par une paire de valeurs propres passant du cas hyperbolique (+r et -r) au cas oscillant (+ir et -ir) ou r est la racine carree du module du parametre de bifurcation. Un tel champ de vecteurs en dimension infinie regit les ondes non lineaires a la surface libre d'un fluide parfait en presence de gravite et de tension superficielle. Pour les champs infiniment derivables, on prouve d'une part l'existence, pour chaque valeur du parametre de bifurcation, de solutions periodiques arbitrairement petites jusqu'a 0 et d'autre part l'existence de solutions reversibles homoclines a des orbites periodiques d'amplitude polynomialement petite par rapport a r. Pour les champs analytiques, on montre l'existence de solutions reversibles homoclines a des orbites periodiques exponentiellement petites (d'ordre exp(-c/r)). On ne peut se ramener dans ce cas a la dimension 4 via le theoreme de la variete centrale, car la demonstration s'appuie sur la construction de prolongements homolorphes des solutions pour obtenir des majorations exponentiellement petites d'integrales oscillantes. Enfin, on demontre qu'en dimension 4, les petites perturbations de la forme normale a l'ordre 2 sont en general singulieres: le systeme perturbe, a l'inverse de la forme normale n'admet pas de solutions reversibles homoclines a 0

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ROCHA, CARLOS ROGERIO RODRIGUES. « BIFURCATION THEORY IN ELECTRICAL POWER SYSTEMS : AN APPLICATION TO THE OPTIMIZATION PROBLEM ». PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 1995. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=8640@1.

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COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
Este trabalho visa estudar a teoria das bifurcações associada a sistemas de equações diferenciais não lineares com aplicação aos estudos de estabilidade em sistemas elétricos de potência. Como aplicação desta teoria analisamos o problema de operação ótima em sistemas de potência. Novas restrições são derivadas desta teoria que, incluídas às restrições tradicionais de um fluxo de potência ótimo, garantem a estabilidade do ponto de operação obtido.
This work aims at studying bufurcation theory associated to non-linear differential equations systems with application on stability of electrical power systems. As an application of this theory, the problem of optimal operation in power systems is analyzed. New restriction to optimum power flow are found and together with traditional ones, assure the stability of the operation point.

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BOSCH, VIVANIOS IGNACIO. « Bifurcations dans les problemes de l'hydrodynamique en symetries plane et spherique ». Nice, 1994. http://www.theses.fr/1994NICE4803.

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Dans cette these nous etudions le probleme dynamo a l'aide de l'outil mathematique des bifurcations avec symetries. Nous abordons la possible existence d'un champ magnetique auto-entretenu resultant d'une bifurcation secondaire a partir d'un ecoulement de base purement conductif. Dans le premier chapitre nous etudions le cas plan ou nous elargissons la theorie aux symetries d'un reseau bi-periodique. Ce premier travail est a l'origine du second chapitre consacre a la classification des motifs spatiaux qui peuvent apparaitre dans les systemes d'e. D. P possedant la symetrie euclidienne. Dans ce second chapitre est incluse la preuve que cette classification est desormais complete si on tient en compte les equations dites pseudo scalaires. Finalement au chapitre iii nous analysons le probleme dans le cas de la symetrie spherique

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EL, KOULANI ABDELMAJID POTIER-FERRY M. « CONTINUATION DANS LES PROBLEMES A FRONTIERES LIBRES DE TYPE BIFURCATIONS PLASTIQUES / ». [S.l.] : [s.n.], 1996. ftp://ftp.scd.univ-metz.fr/pub/Theses/1996/ElKoulani.Abdelmajid.SMZ9629.pdf.

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Yebari, Naji. « Sur un problème de diffusion-réaction avec frontière libre : un modèle de dissolution-croissance ». Saint-Etienne, 1988. http://www.theses.fr/1988STET4005.

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On présente une analyse mathématique d'un modèle simple de dissolution-croissance. Le système hétérogène est constitué d'un petit grain sphérique d'un seul composant au sein d'une solution liquide du même composant. L'échange de matière se fait à l'interface entre le solide et le liquide. Cette interface joue le rôle de la frontière libre. En régime stationnaire, on obtient des résultats d'existence et de non-existence correspondant aux données. L'étude de la stabilité concernant le problème d'évolution associé est menée par des simulations numériques

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Sathaye, Archana S. « BIFDE : a numerical software package for the hopf bifurcation problem in functional differential equations ». Thesis, Virginia Polytechnic Institute and State University, 1986. http://hdl.handle.net/10919/101145.

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A software package has been written to compute the Hopf bifurcation structure in functional differential equations. The package is modular, and consists of several routines which perform one or more tasks. In conjunction with the routines available in this package, the user is required to provide a few routines which describe the specific system under analysis. Three example systems (from epidemiology, biochemistry and aerospace engineering) have been analyzed to illustrate the use of this package.
M.S.

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Grdadolnik, Jake Matthew. « Supercritical and Subcritical Pitchfork Bifurcations in a Buckling Problem for a Graphene Sheet between 2 Rigid Substrates ». University of Akron / OhioLINK, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=akron1618585173630134.

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Chehab, Jean-Paul. « Méthode des inconnues incrémentales : application au calcul des bifurcations ». Paris 11, 1993. http://www.theses.fr/1993PA112031.

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Ce travail est consacré à l'élaboration de nouveaux schémas numériques en bifurcation. Ils résultent de plusieurs généralisations de la méthode de Marder et Weitzner (mW) à l'aide de la méthode des inconnues incrémentales introduite par R. Temam. Nous rappelons tout d'abord la construction et les principales propriétés des inconnues incrémentales d'ordre deux en dimension et un et deux d'espace. Le problème de Poisson donne une première illustration numérique des avantages de la nouvelle technique et nous proposons une famille de pré conditionneurs des matrices sous-jacentes. Ensuite, nous présentons l'algorithme de Marder et Weitzner (mW) qui est bien adapté au calcul de solutions instables. Nous construisons trois types de méthodes incrémentales. Elles sont basées sur une généralisation de la notion de relaxation et généralement mW dans des directions difficilement atteignables avec les techniques de discrétisation classiques. Les différents tests numériques portent sur le calcul de solutions instables de problèmes aux valeurs propres non linéaires. Les comparaisons avec la méthode usuelle de mW mettent en évidence une plus grande vitesse de convergence et un gain de temps cpu important

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Chalmers, Alexander David. « Mathematical Modelling of Atherosclerosis ». Thesis, The University of Sydney, 2015. http://hdl.handle.net/2123/14986.

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In atherosclerosis, the arterial lining undergoes a specific sequence of inflammatory responses to an injury to the cells that line the blood vessel and to low density lipoprotein (LDL) particles from the blood stream that penetrate through this injury into the arterial wall. We model the events that take place inside the blood vessel wall that occur immediately after such an injury with a system of partial differential equations that involve the LDL particles, two proinflammatory cytokines, monocyte-derived macrophages and their lipid-filled counterparts, foam cells. The model includes the chemical and physical interactions with the endothelial cells that line the arterial wall. These interactions are formulated as boundary conditions. Through numerical simulations, we show that different LDL concentrations in the blood stream and different immune responses can qualitatively affect the development of a plaque. Numerical bifurcation analysis at the quasi-steady state through AUTO shows that there exists of a fold bifurcation when the flux of LDL into the plaque from the blood is high. An atherosclerotic plaque that develops within the intima, deforms the intima locally as macrophages and foam cells accumulate. We model the structure of the developing plaque by cell pressure and cell sorting models to account for the limited space within the intima. We do this by modelling cell movement in crowded tissue in a discrete space and extend this to a spatial domain where cells also moves due to cell pressure and chemotaxis. We model the mechanics of the physical interactions on the two bounding interfaces, (the lumen-intima boundary and the intima-media boundary) and of the tissue inside the domain and add advective terms to ensure that the mechanics of the cellular species is consistent with the underlying tissue deformation. Using a finite element solver, we produce numerical results in one dimension across the intima and in two dimensions as a cross section of an artery. With appropriate parameter values, this moving boundary problem produces results in agreement with the current theory on compensatory enlargement in atherosclerotic remodelling

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Bagwell, Ross David. « Bifurcation and Stability of a Ring Problem Motivated by the Mechanics of Double-Walled Carbon Nanotubes ». University of Akron / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=akron1335288394.

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Diaz, Elkin Dario Cardenas. « Fenômeno de bifurcação no problema de Yamabe sobre variedades riemannianas com bordo ». Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-30082016-001339/.

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No presente trabalho consideramos o produto de uma variedade Riemanniana compacta sem bordo de curvatura escalar zero e uma variedade Riemanniana compacta com bordo, curvatura escalar zero e curvatura media constante no bordo, e fazemos uso da teoria de bifurcação para provar a existência de um numero infinito de classes conforme com, pelo menos, duas métricas Riemannianas não homotéticas de curvatura escalar zero e curvatura média constante no bordo, sobre a variedade produto.
In this work, we consider the product of a compact Riemannian manifold without boundary, null scalar curvature and a compact Riemannian manifold with boundary, null scalar curvature and constant mean curvature on the boundary and we use the bifurcation theory to prove the existence of a infinite number of conformal classes with at least two non homothetic Riemannian metrics of null scalar curvature and constant mean curvature of the boundary on the product manifold.

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Lui, Shiu-Hong Keller Herbert Bishop. « Part I. Multiple bifurcations. : Part II. Parallel homotopy method for the real nonsymmetric eigenvalue problem / ». Diss., Pasadena, Calif. : California Institute of Technology, 1992. http://resolver.caltech.edu/CaltechETD:etd-06152005-084230.

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Texier, Picard Rozenn Volpert Vitaly Marion Martine. « Problèmes de réaction-diffusion avec convection une étude mathématique et numérique / ». Villeurbanne : Université Claude Bernard (Lyon 1), 2002. http://tel.archives-ouvertes.fr/docs/00/04/50/62/PDF/tel-00002038.pdf.

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Rezende, Alex Carlucci. « Dois métodos para a investigação de ciclos limites que bifurcam de centros ». Universidade de São Paulo, 2011. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-11042011-113618/.

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Um dos mais investigados problemas na teoria qualitativa dos sistemas dinâmicos no plano é o XVI problema de Hilbert que trata dos ciclos limites. Mais precisamente, a segunda parte do referido problema questiona sobre o número máximo de ciclos limites de um sistema diferencial polinomial plano de grau n. Por ciclo limite entendemos uma órbita fechada isolada no conjunto de todas as órbitas periódicas de um sistema diferencial plano.Uma maneira clássica de obter um ciclo limite é perturbando um sistema com uma singularidade do tipo centro. Nesta dissertação apresentamos dois métodos utilizados para a análise do número de ciclos limites que bifurcam de um centro, a saber o método das integrais abelianas e o método do averaging
One of the most investigated problems in the qualitative theory of dynamical systems in the plane is the XVI Hilberts problem which deals with limit cycles. More precisely, the second part of the problem asks about the maximum number of limit cycles of a polynomial differential system of degree n. A limit cycle is a single closed orbit on the set of all periodic orbits of a differential planar system. A classic way to obtain a limit cycle is perturbing a system with a singularity of center type.In this work we discuss about two methods used to investigate the number of limit cycles which bifurcate from a center; they are known as Abelian integrals and averaging theory

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Zhang, Wei. « Study of Central Configurations and Relative Equilibria in the Problem of Four Bodies ». University of Cincinnati / OhioLINK, 2000. http://rave.ohiolink.edu/etdc/view?acc_num=ucin974472331.

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Sonnino, Giorgio. « Bifurcations and pattern selection in Rayleigh-Benard and Couette-Taylor problems : the role of the compressibility, boundary conditions and fluctuations ». Doctoral thesis, Universite Libre de Bruxelles, 1992. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/212913.

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Piqueira, José Roberto Castilho. « Aplicação da teoria qualitativa de equações diferenciais a problemas de sineronismo de fase ». Universidade de São Paulo, 1987. http://www.teses.usp.br/teses/disponiveis/3/3139/tde-31032017-082016/.

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Aplica-se a Teoria Qualitativa de Equações Diferenciais aos problemas de sincronismo de fase, associando às diversas regiões do espaço de parâmetros os tipos de atratores esperados.

Três casos básicos são estudados:

No caso (i), usando resultados clássicos da teoria de sistemas dinâmicos, discute-se os pontos de equilíbrio e os ciclos limite.

No caso (ii), usando o método de Melnikov propõem-se critérios para previsão de aparecimento de atratores caóticos.

No caso (iii), usando o teorema de bifurcações de Hopf, a estabilidade dos pontos de equilíbrio e a formação dos ciclos limite são analisadas
The Qualitative Theory of Differential Equations is applied to the phaselock problems, and the several parameters space regions are associated to the expected attractors.

Three basic cases are studied:

In case i), using classical results of dynamical systems theory, the equilibrium points and limit cycles are analyses.

In case ii), the Melnikov technique gives some criteria for chaotic attractors.

In case iii), Hopf bifurcation theorem provides propositions about equilibrium points and limit cycles.

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Guillet, Christophe. « INSTABILITE DE SYSTEMES HAMILTONIENS AU SENS DE CHIRIKOV ET BIFURCATION DANS UN PROBLEME D' EVOLUTION NON LINEAIRE ISSU DE LA PHYSIQUE ». Phd thesis, Université de Franche-Comté, 2004. http://tel.archives-ouvertes.fr/tel-00011975.

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Nous mettons en évidence une condition géométrico-dynamique minimale créant de l'hyperbolicité au voisinage d'un tore homocline transverse partiellement hyperbolique dans un système Hamiltonien presque intégrable à trois degrés de liberté. On en déduit une généralisation du théorème de dynamique symbolique d'Easton. Nous donnons ensuite une estimation optimale du temps de diffusion d'Arnold le long d'une chaîne de transition dans les systèmes Hamiltoniens initialement hyperboliques à trois degrés de liberté en utilisant une chaîne d'orbites périodiques hyperboliques sous-jacente.
Nous décrivons ensuite géométriquement à partir d'un système Hamiltonien presque intégrable à trois degrés de liberté à deux paramètres dû à Chirikov, un mécanisme de diffusion mettant en jeu un réseau de plans résonnants parallèles et voisins et un plan résonnant transversal au réseau. Ainsi, nous montrons qu'en dessous d'un certain seuil atteint par le paramètre prépondérant, on peut construire une orbite de transition dérivant en action à travers ce réseau modulationnel. Un des scénarii envisagés, le mécanisme de diffusion modulationnelle, basé sur l'existence de connexions hétéroclines entre tores partiellement hyperboliques issus de deux plans résonnants distincts est valide lorsqu'une condition de chevauchement est vérifiée.
Nous étudions enfin le modèle bidimensionnel décrivant un écoulement laminaire avec convection mixte entre deux plaques planes puis dans un tube vertical. Avec des conditions aux bords réduites, nous montrons via le théorème de la variété centrale qu'il existe dans le premier cas une bifurcation de pitchfork pour une valeur critique du nombre de Rayleigh.

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Moreira, Ana Claudia da Silva. « Técnicas de bifurcação para o problema de Yamabe em variedades com bordo ». Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-11032016-162814/.

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Apresentaremos alguns resultados de rigidez e de bifurcação para soluções do problema de Yamabe em variedades produto com bordo.
We will discuss some rigidity and bifurcation results for solutions of the Yamabe problem in product manifolds with boundary.

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LOPES, Juscelino Grigório. « Finitude genérica de classes de equilíbrios relativos no problema de quatro copos ». Universidade Federal de Pernambuco, 2016. https://repositorio.ufpe.br/handle/123456789/17992.

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CNPq
Neste trabalho, estudaremos o conjunto de equil brios relativos n~ao-colineares do problema de quatro corpos no plano complexo. Veremos que esse conjunto e uma subvariedade estrati cada maximal de certa variedade alg ebrica real e provaremos a unicidade do vetor massa normalizado associado a cada ponto dessa subvariedade. Por meio de transforma c~oes de regulariza c~ao, reduziremos a teoria de bifurca c~oes de equil brios relativos ao estudo de uma correspond^encia alg ebrica entre variedades reais. Atrav es dos teoremas de nitude para variedades alg ebricas reais, provaremos que existe uma cota para o n umero de classes de equil brios relativos n~ao-colineares v alida para todas as massas positivas no complementar de um subconjunto alg ebrico pr oprio no espa co das massas.
In this work, we study the set of non-collinear relative equilibria in the fourbody problem in the complex plane. We will see that this set is a maximal strati ed submanifold in a real algebraic variety and prove the uniqueness of the normalized vector mass associated with each point of this submanifold. By means of regularization transformations, we reduce the bifurcation theory to the study of an algebraic correspondence between real varieties. Through the theorems of niteness for real algebraic varieties, we prove that there is an upper bound for the number of a ne classes of non-collinear relative equilibria which holds for all positive masses in the complement of a proper, algebraic subset of all masses.

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Hassainia, Zineb. « Dynamique des tourbillons pour quelques modèles de transport non-linéaires ». Thesis, Rennes 1, 2015. http://www.theses.fr/2015REN1S016/document.

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Cette thèse est consacrée à l'étude théorique de quelques modèles d'évolution non-linéaires issus de la mécanique des fluides. Nous distinguons trois parties indépendantes. La première partie de la thèse traite essentiellement de l'existence des poches de tourbillon en rotation uniforme (appelées aussi V-states) pour un modèle quasi-géostrophique non visqueux. Notre étude est répartie sur deux chapitres où les poches présentent des structures topologiques différentes. Dans le premier chapitre nous étudions le cas simplement connexe et nous validons l'existence de ces structures dans un voisinage du tourbillon de Rankine en utilisant des techniques de bifurcation. Dans le deuxième chapitre nous abordons le cas doublement connexe où la poche admet un seul trou. Plus précisément, proche d'un anneau donné, nous décrivons cette famille par des branches dénombrables bifurquant de cet anneau à certaines valeurs explicites des vitesses angulaires liées aux fonctions de Bessel. Notre étude théorique a été complétée par des simulations numériques portant sur les V-states limites et un bon nombre de constatations ont été formulées ouvrant la porte à de nouvelles perspectives de recherche. La seconde partie concerne l'étude du problème de Cauchy pour le système de Boussinesq non visqueux 2D avec des données initiales de type Yudovich. Le problème est dans un certain sens critique à cause de quelques termes comportant la transformée de Riesz dans la formulation tourbillon-densité. Nous donnons une réponse positive pour une sous-classe comprenant les poches de tourbillon régulières et singulières. Dans la dernière partie nous analysons le problème de la limite incompressible pour les équations d'Euler isentropiques 2D associées à des données initiales très mal préparées et pour lesquelles les tourbillons ne sont pas forcément bornés mais appartiennent plutôt à des espaces de type ''BMO'' à poids. On utilise principalement deux ingrédients: d'un côté les estimations de Strichartz pour contrôler la partie acoustique. D'un autre côté, on se sert de la structure de transport compressible du tourbillon et on démontre une estimation de propagation linéaire dans l'esprit d'un travail récent de Bernicot et Keraani mené dans le cas incompressible
In this dissertation, we are concerned with the study of some non-linear evolution models arising in fluid mechanics. We distinguish three independent parts. The first part of the thesis deals with the existence of the rotating vortex patches (called also V-states) for an inviscid quasi-geostrophic model. Our study is divided into two chapters dealing with different topological structures of the V-states. In the first chapter we study the simply connected case and we prove the existence of such structures in a neighborhood of the Rankine vortices by using the bifurcation theory. In the second chapter we discuss the doubly connected case where the patches admit only one hole. More precisely, close to a given annulus we describe this family by countable branches bifurcating from this annulus at some explicit angular velocities related to Bessel functions of the first kind. Our theoretical study was completed by numerical simulations on the limiting V-states and a number of interesting numerical observation were formulated opening new research perspectives. The second part of the thesis concerns the local well-posedness theory for the inviscid Boussinesq system with rough initial data. The problem is in some sense critical due to some terms involving Riesz transforms in the vorticity-density formulation. We give a positive answer for a special sub-class of Yudovich data including smooth and singular vortex patches. In the last part we address the problem of the incompressible limit for the 2D isentropic fluids associated to ill-prepared initial data and for which the vortices are not necessarily bounded and belong to some weighted BMO spaces. We mainly use two ingredients: On one hand, the Strichartz estimates to control the acoustic part and prove that it does not contribute for low Mach number. On the other hand, we use the transport compressible structure of the vorticity and we establish a linear propagation estimate in the spirit of a recent work of Bernicot and Keraani conducted in the incompressible case. The first part of the thesis deals with the existence of the rotating vortex patches (called also V-states) for an inviscid quasi-geostrophic model. Our study is divided into two chapters dealing with different topological structures of the V-states. In the first chapter we study the simply connected case and we prove the existence of such structures in a neighborhood of the Rankine vortices by using the bifurcation theory. In the second chapter we discuss the doubly connected case where the patches admit only one hole. More precisely, close to a given annulus we describe this family by countable branches bifurcating from this annulus at some explicit angular velocities related to Bessel functions of the first kind. Our theoretical study was completed by numerical simulations on the limiting V-states and a number of interesting numerical observation were formulated opening new research perspectives. The second part of the thesis concerns the local well-posedness theory for the inviscid Boussinesq system with rough initial data. The problem is in some sense critical due to some terms involving Riesz transforms in the vorticity-density formulation. We give a positive answer for a special sub-class of Yudovich data including smooth and singular vortex patches. In the last part we address the problem of the incompressible limit for the 2D isentropic fluids associated to ill-prepared initial data and for which the vortices are not necessarily bounded and belong to some weighted BMO spaces. We mainly use two ingredients: On one hand, the Strichartz estimates to control the acoustic part and prove that it does not contribute for low Mach number. On the other hand, we use the transport compressible structure of the vorticity and we establish a linear propagation estimate in the spirit of a recent work of Bernicot and Keraani conducted in the incompressible case

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Hur, Seok. « Extension de la théorie de Bautin à des dimensions supérieures : notion de centre et structure de composition ». Paris 6, 2003. http://www.theses.fr/2003PA066162.

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Djabella, Karima. « Modélisation de l’activité électrique du coeur et de sa régulation par le système nerveux autonome ». Paris 11, 2008. http://www.theses.fr/2008PA112083.

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Nous avons développé un modèle de l'activité électrique cardiaque cellulaire dont la structure est moins complexe, sans pour autant affecter les caractéristiques essentielles (potentiel d’action, principaux courants ioniques, courbe de restitution). En outre, nous avons utilisé la même structure de modèle pour représenter l'activité électrique des divers types de cellules cardiaques. Cela permettra de déterminer des régions cardiaques de façon paramétrique et simplifiera l'identification des paramètres dans les modèles numériques de cœur. Par ailleurs, l'analyse de bifurcation nous a permis de retrouver l’origine du régime oscillatoire dans le cas pacemaker, et d’introduire un courant de fuite de calcium qui joue le rôle d'une entrée de commande pour le système nerveux autonome qui lui permet de modifier la fréquence cardiaque. D’autre part, il n’est plus possible de simplifier plus le modèle à l'aide de la méthode des perturbations singulières car ce n’est plus un système de Tikhonov. Le modèle permet une mise en œuvre en boucle fermée tenant compte du contrôle du système cardiovasculaire par l'arc baroréflexe et un couplage excitation-contraction tenant compte de l’effet de la fréquence sur la contractilité. Après avoir démontré la non-existence de solutions périodiques dans le modèle réduit à deux courants ioniques de Mitchell-Schaeffer, nous avons introduit une variante de ce dernier et ainsi étendu ses propriétés d'excitabilité. Le régime oscillatoire est obtenu, soit à travers une bifurcation de Hopf sous ou super critique, soit à travers une bifurcation nœud-col sur cercle invariant. Ce modèle réduit est utilisable dans des applications de traitement du signal ECG
We developed a cellular cardiac electrical activity model which is less complex, without affecting the essential characteristics (action potential, principal ionic currents, restitution curve). Besides, we used the same model’s structure in order to represent the electrical activity of various types of cardiac cells. This will allow the determining of cardiac regions in a parametric manner and will make easier the parametric identification in the numerical models of heart. Otherwise, the bifurcation analysis allowed us to determine the origin of oscillator regime in the pacemaker case, and to introduce a background calcium current that plays the role of a control input of the autonomic nervous system that allows him to modify the heart rate. On the other hand, it is not possible any more to simplify the model using singular perturbations method because it is not any more a Tikhonov's system. The model allows the implementation of the closed loop taking into account the control of the cardiovascular system by the baroreflex and the excitation-contraction coupling taking into account the effect of the frequency on the contractility. After having shown the non-existence of periodic solutions in the reduced two ionic currents model of Mitchell-Schaeffer, we introduced a variant of this last and thus we have extended its excitability properties. The oscillator regime is obtained, either through a sub or super critical Hopf bifurcation, or through a saddle-node bifurcation on invariant circle. This reduced model is usable in ECG signal processing applications

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Littig, Samuel. « The Eigenvalue Problem of the 1-Laplace Operator ». Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-161044.

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As a first aspect the thesis treats existence results of the perturbed eigenvalue problem of the 1-Laplace operator. This is done with the aid of a quite general critical point theory results with the genus as topological index. Moreover we show that the eigenvalues of the perturbed 1-Laplace operator converge to the eigenvalues of the unperturebed 1-Laplace operator when the perturbation goes to zero. As a second aspect we treat the eigenvalue problems of the vectorial 1-Laplace operator and the symmetrized 1-Laplace operator. And as a third aspect certain related parabolic problems are considered.

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SILVA, Kaye Oliveira da. « Existência e multiplicidade de soluções de problemas de contorno elípticos de quarta ordem via métodos topológicos ». Universidade Federal de Goiás, 2012. http://repositorio.bc.ufg.br/tede/handle/tde/1951.

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In this work, we employ topological methods in order to study existence and multiplicity of solutions, of nonlinear boundary value problems of the fourth order. More precisely, we make use of results on connected components of fixed points, as well as global bifurcation, to show existence and multiplicity of weak solutions of Partial Differential Equations, involving the Biharmonic operator under Navier boundary conditions. Proofs of the abstract results used, are presented in detail.
Neste trabalho, utilizamos métodos topológicos para estudar existência e multiplicidade de soluções de Problemas de Contorno Elípticos Não Lineares de 4a ordem. Mais precisamente, utilizamos resultados sobre componentes conexas de pontos fixos e tambem bifurcação global, para provar existência e multiplicidade de soluções fracas de Equações Diferenciais Parciais, envolvendo o Operador Binarmônico, sob condições de fronteira de Navier. As demonstrações dos resultados abstratos que utilizamos, são apresentadas em detalhes.

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Santos, Mayk Joaquim dos. « Estudo qualitativo de campos suaves por partes via problema de perturbação singular ». Universidade Federal de Goiás, 2017. http://repositorio.bc.ufg.br/tede/handle/tede/6852.

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In this work we will show that, given a piecewise smooth vector field, we can apply the regularization method and, from it, via blow-up, turn it into a singular perturbation problem. By doing that, we can use the tools from singular perturbation theory to perform a qualitative study of piecewise smooth vector fields. Finally, we will show that, through successive changes of coordinates, a singularity of a discontinuous submanifold of codimension k, where k=1 or k=2, can be transformed into a singularity of codimension 0 in order to study the qualitative behavior in this submanifold, where the Filippov’s convention holds.
Neste trabalho mostraremos que, dado um campo de vetores suaves por partes, podemos aplicar o método de regularização e, a partir deste, via “blow-up”, o transformamos em um problema de perturbação singular. Podemos, dessa forma, fazer uso das ferramentas da teoria de perturbação singular para realizar um estudo qualitativo dos campos de vetores suaves por partes. Por último, mostraremos que através de sucessivas mudanças de coordenadas podemos transformar uma singularidade de uma subvariedade de descontinuidade de codimensão k, onde k=1 ou k=2, em uma uma singularidade de codimensão 0 e estudar o comportamento qualitativo ao longo desta subvariedade, onde é válida a convenção de Filippov.

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