Orientador: Claudio Aguinaldo Buzzi / Banca: João Carlos da Rocha Medrado / Banca: Luciana de Fátima Martins / Resumo: Nesse trabalho são apresentados alguns resultados importantes sobre bifurcações de codimensão dois de campos de vetores. O resultado principal dessa dissertação e o teorema que d a o diagrama de bifurcação e os retratos de fase da Bifurcação de Bogdanov-Takens. Para a demonstracão são usadas algumas técnicas basicas de Sistemas Dinâmicos e Teoria das Singularidades, tais como Integrais Abelianas, desdobramentos de Sistemas Hamiltonianos, desdobramentos versais, Teorema de Preparação de Malgrange, entre outros. Outra importante bifurcação clássica apresentada e a Bifurca cão do tipo Hopf-Zero, quando a matriz Jacobiana possui um autovalor simples nulo e um par de autovalores imagin arios puros. Foram usadas algumas hipóteses que garantem propriedades de simetria do sistema, dentre elas, assumiuse que o sistema era revers vel. Assim como na Bifurcação de Bogdanov-Takens, foram apresentados o diagrama de bifurcao e os retratos de fase da Bifurcação Hopf-zero bifurcação reversível. As técnicas usadas para esse estudo foram a forma normal de Belitskii e o método do Blow-up polar. / Abstract: In this work is presented some important results about codimension two bifurcations of vector elds. The main result of this work is the theorem that gives the local bifurcation diagram and the phase portraits of the Bogdanov-Takens bifurcation. In order to give the proof, some classic tools in Dynamical System and Singularities Theory are used, such as Abelian Integral, versal deformation, Hamiltonian Systems, Malgrange Preparation Theorem, etc. Another classic bifurcation phenomena, known as the Hopf-Zero bifurcation, when the Jacobian matrix has a simple zero and a pair of purely imaginary eigenvalues, is presented. In here, is added the hypothesis that the system is reversible, which gives some symmetry in the problem. Like in Bogdanov-Takens bifurcation, the bifurcation diagram and the local phase portraits of the reversible Hopf-zero bifurcation were presented. The main techniques used are the Belitskii theory to nd a normal forms and the polar Blow-up method. / Mestre

Introduction : Les artères cérébrales de gros calibre du système antérieur, regroupées sous le nom de bifurcation carotidienne intracrânienne, sont les plus affectées par les accidents vasculaires cérébraux artériels néonatals, ces AVC étant localisés dans ce territoire dans 85% des cas. L'hypothèse pathophysiologique classique, mais non prouvée, postule que l'occlusion artérielle est causée par un embole d'origine placentaire. Cette croyance reste controversée par le débalancement de la distribution antérieure versus postérieure des infarctus cérébraux, et l'absence d'infarctus extra-cérébraux associés. Une nouvelle perspective pathophysiologique émerge de l'association épidémiologique entre l'inflammation gestationnelle et l'AVC néonatal. Nous postulons que l'inflammation materno-fœtale, induite par l'exposition gestationnelle aux pathogènes, mène à une vasculite affectant spécifiquement la bifurcation carotidienne puis provoque une thrombose focale. Méthodes : Des rates Lewis gestantes sont injectées avec de la saline ou du lipopolysaccharide (LPS) d'Escherichia coli (200 μg/kg/12h) entre les jours de gestation (G) 21 et 22. Les cerveaux de la progéniture sont prélevés à G21, G22 et au jour postnatal (P)1. Le sang maternel, les placentas et le sang fœtal sont échantillonnés à G21 ou G22. A P1, un stress prothrombotique (photothrombose transcutanée) combiné à une hypoxie (3h30, 8% O2) est appliqué sur les artères cérébrales moyennes pour comparer leur susceptibilité à la thrombose chez les ratons exposés au LPS et ceux non exposés. L'immunohistochimie, l'immunofluorescence et l'ELISA ont détecté les marqueurs inflammatoires maternels, placentaires et fœtaux. Résultats : Les artères intra-crâniennes les plus susceptibles à l'AVC expriment constitutivement plus de marqueurs inflammatoires en comparaison aux artères intra- ou extra-crâniennes non susceptibles à l'AVC périnatal. L'exposition gestationnelle au LPS provoque une inflammation maternelle, placentaire et fœtale associées à une production d'IL-1β, de TNF-α et de MCP-1 ainsi qu'une inflammation artérielle affectant le segment proximal des artères intra-crâniennes fortement susceptible aux AVC néonatals. Les ratons LPS + photothrombose + hypoxie présentent des AVC ischémiques ainsi que des déficits moteurs, qui n'étaient pas détectés lorsque la photothrombose + hypoxie était appliquée sans traitement préalable avec le LPS. Conclusion : les résultats acquits par le biais de notre nouveau modèle animal préclinique supportent notre hypothèse d'une augmentation de la susceptibilité à l'inflammation des artères cérébrales antérieures, et ouvrent une nouvelle avenue physiopathologique vasculitique pour les AVC néonatals.

published_or_final_version / abstract / Mechanical Engineering / Doctoral / Doctor of Philosophy

We have studied the stationary solutions to the two-dimensional Euler's equation. A highly accurate scheme, based on boundary integral techniques was used in investigating these steady-state configurations. Bifurcation analysis on the solution of a uniform vortex patch in an externally applied strain field, yield new non-elliptical steady-state solutions apart from the elliptical structures reported by Moore & Saffman. The elliptical solutions correspond to the points on the primary solution branch and the non-elliptical solutions correspond to points on the bifurcation branches. We also observe the presence of a turning point indicating the finite resistance of these uniform vortices. Some of these new solutions suggest the possibility of coalescence between neighboring vortices. This leads to a new problem of considering a vortex pair in a strain field and computing their steady-state solutions. Numerical computations suggest that this guess is indeed correct, as we see the solution branch corresponding to the vortex pair intersect the bifurcation branch of the single vortex at a unique strain rate. Furthermore, looking at the profiles on the other bifurcation branches, it appears that merger of neighboring vortices is a recurring phenomenon.

In this dissertation, first, the theoretical and experimental viewpoints of instability and bifurcation in mechanics are reviewed and discussed. The onset of instability of bifurcation depends on the constitutive assumptions, and is marked by the loss of ellipticity, singularity of the stiffness matrix, and negative or complex eigenvalues. Non-traditional regularization is necessary to obtain useful post-instability solutions. Based on dissipated energy and elastic potential, energy based instability criterion is considered and developed. The global instability criterion is concerned with global non-uniform deformation while the surface degradation instability criterion deals with near surface non-uniformities. In addition, the connection between surface degradation and size, shape effects for brittle materials is examined. The energy based stability theory is applied for some typical problems through analytical and numerical implementations. It is shown that the onset of both surface instability and global degradation instability occurs in the strain hardening stage, that is, before and close to the peak strength. The theoretical results are compared with experimental observations. Both strain gage tests and ultrasonic scanning tests are processed to study the degradation mechanisms of a brittle material. The surface effects are highlighted by the experiments. Ultrasonically dissipated energy shows a random distribution and it follows, in general, the initial non-homogeneity pattern. The relationship between the ultrasonically dissipated energy and mechanically dissipated energy is dependent on deformation and can be approximated by a power function of the factor of load level. The theory for surface degradation consideration involves a few material constants, and these constants are identified against experimental observations. The degradation mechanism and damage growth patterning of simulated rock under uniaxial load are modeled numerically by implementing the theory for damage and surface degradation with initial state consideration. The theoretical results are compared with experimental observations obtained through ultrasonic scanning tests. To extend the study to post-instability modelling by using various constitutive models, three alternative considerations are proposed to achieve so-called regularization of the problem.

L'étude de la fibration de Milnor prend une place très importante dans la Théorie des Singularités. Pour les polynômes holomorphes, on avait déjà beaucoup de résultats montrés par plusieurs spécialistes. Cependant la fibration de Milnor n'existe pas en général pour une application polynomiale. Dans cette thèse on s'intéresse aux propriétés des polynômes mixtes introduits par Oka, qui sont des polynômes C^n → C de variables complexes et leurs conjugées. En utilisant une condition de régularité à l'infini, on montre un théorème de fibration globale qui implique que l'ensemble de bifurcation pour un polynôme mixte est inclus dans un ensemble semi-algébrique fermé de dimension réelle inférieure ou égale à un. En particulier, on définit le polyèdre de Newton à l'infini pour un polynôme mixte et on étudie deux conditions de non-dégénéréscence à l'infini par rapport à ce polyèdre. Il s'avère que les deux conditions de "non-dégénéré" sont semi-algébriques ouvertes, et que la condition de "fortement non-dégénéré" n'est pas dense, donc non-connexe. Avec notre construction on généralise un théorème de Néméthi et Zaharia qui donne une approximation de l'ensemble de bifurcation pour un polynôme mixte non-dégénéré. On prouve la stabilité de la monodromie pour une famille de polynômes mixtes fortement non-dégénérés en supposant l'invariance des polyèdres de Newton. On établit aussi l'analogue à l'infini du théorème local d'Oka sur l'existence de la fibration de Milnor. Ceci étend considérablement des résultats dans le cas holomorphe.Enfin, on introduit une nouvelle définition de "non-dégénéré" pour des applications polynomiales mixtes et on trouve une extension du théorème de Bivia-Ausina en rapport avec la conjecture Jacobienne. / The study of the Milnor fibration plays an important role in Singularity Theory. In the holomorphic case there are plenty of results proved by many specialists. However, the Milnor fibration does not always exist for a polynomial application. In this thesis, we focus on the properties of mixed polynomials introduced by Oka which are in fact polynomials C^n → C of complex variables and their conjugates. By using a regularity condition at infinity, we prove a global fibration theorem which implies that the bifurcation set for a mixed polynomial is included in a semi-algebraic closed set of real dimension strictly less than two. In particular, we define the Newton polyhedron at infinity for a mixed polynomial and study two types of non-degenerate conditions at infinity with respect to this polyhedron. It turns out that these two non-degenerate conditions are semi-algebraic and open, and that the "strongly non-degenerat"e condition is neither dense nor connected. By our construction, we generalise the Néméthi and Zaharia's theorem which gives an approximation of the bifurcation set for a non- degenerate mixed polynomial. In addition, we show the stability of the monodromy in a family of strongly non-degenerate mixed polynomials supposing that their Newton polyhedron at infinity is constant. We also set up a global analogue at infinity of Oka's theorem on the existence of Milnor fibration which extends some results in the holomorphic case. In the end, we introduce a new definition of non-degenerate condition for mixed polynomial applications and find an extension of Bivia-Ausina's theorem which relates to the Jacobian conjecture.

Soit f:K^n-->K^p une application semi-algébrique de classe C^2 pour K=R, ou une application polynomiale pour K=C. Il est bien connu que f est une fibration localement triviale sur le complémentaire des valeurs de bifurcation B(f). Dans ce travail nous considérons la t-régularité et la rho-régularité dans l'étude de B(f). Nous démontrons que t-régularité est équivalent aux conditions de Rabier (1997), Gaffney (1999), Kurdyka, Orro, Simon (2000) et Jelonek (2003). On démontre que t-régularité implique rho-régularité. Avec la rho-régularité, on démontre un théorème de structure pour l'ensemble des valeurs non rho-régulières S(f). On démontre aussi que B(f) est inclus dans A_{rho}, où A_{rho} est l'union de f(Sing f) et S(f). Nous étudions aussi deux classes d'applications: les applications fair et les applications Newton non-dégénérées. Pour les fair, on obtient une interprétation de la t-régularité en termes de la clôture intégrale des modules, ce que étende le résultat de Gaffney (1999). Pour les Newton non dégénérées, nous obtenons une approximation de B(f), ce qui étende le résultat de Némethi et Zaharia (1990) et celui de Chen et Tibar (2012). Dans la dernière partie, on discute quelques conséquences:1).la t-régularité pour f:X --> K^p, où X est une variété lisse; 2).le problème de bijectivité des applications; 3).une formule pour calculer la caractéristique d'Euler des fibres régulières de f: R^n-->R^{n-1}. Les résultats présentés brièvement ci-dessus généralisent aussi certains résultats de Némethi et Zaharia (1990), Siersma et Tibar (1995), Paunescu et Zaharia (1997), Parusinski (1995) et Tibar (1998). / Let f:K^n-->K^p be a C^2 semi-algebraic mapping for K=R and a polynomial mapping for K=C. It is well-known that f is a locally trivial topological fibration over the complement of the bifurcation set B(f). In this work, we consider the t-regularity and rho-regularity to study B(f). We show that t-regularity is equivalent to regularity conditions of Rabier (1997), Gaffney (1999), Kurdyka, Orro, Simon (2000) and Jelonek (2003). We prove that t-regularity implies rho-regularity. From rho-regularity, we define the set of non rho-regular values S(f), and the set A_{rho}, which is the union of f(Sing f) and S(f). We prove a structure theorem for S(f) and A_{rho}. We also obtain that B(f) is contained in A_{rho}. We study also two classes of maps, the fair maps and the Newton non-degenerate maps. For fair maps, we give an interpretation of t-regularity in terms of integral closure of modules, which is a real counterpart of Gaffney's result (1999). For non-degenerate maps, we obtain an approximation for B(f) through a set which depends on the Newton polyhedron of f (results like this have been obtained by Némethi and Zaharia (1990) and by Chen and Tibar (2012)). To finish, we discuss some consequences of our work: the t-regularity for maps f: X-->K^p, where X is a smooth affine variety; the problem of bijectivity of semi-algebraic maps; and a formula to compute the Euler characteristic of regular fibers of f:R^n-->R^{n-1}. The above results are also extensions of some results obtained, for polynomial functions f:K^n-->K, by Némethi and Zaharia (1990), Siersma and Tibar (1995), Paunescu and Zaharia (1997), Parusinski (1995) and Tibar (1998).

Neste trabalho, estudamos a teoria de bifurcação para equações diferenciais ordinárias (escrevemos simplesmente EDOs), bem como a existência de ponto de bifurcação para soluções periódicas destas equações. Em seguida, desenvolvemos a teoria, até então inexistente, sobre bifurcação para equações diferenciais ordinárias generalizadas (EDOs generalizadas). Neste desenvolvimento, obtivemos para EDOs generalizadas, um resultado sobre existência de ponto de bifurcação para soluções periódicas. Em seguida, através da correspondência entre EDOs e EDOs generalizadas, obtivemos novos resultados sobre a existência de ponto de bifurcação para soluções periódicas para EDOs clássicas, agora sob a ótica das EDOs generalizadas, quando então, em vez de funções continuamente diferenciáveis, necessitamos, apenas, que as funções envolvidas na EDO sejam integráveis no sentido de Kurzweil-Henstock. Adicionamos, também, um resultado sobre a existência de soluções periódicas de EDOs generalizadas e aplicamos tal resultado para EDOs. A fim de obtermos os resultados que pretendíamos, utilizamos a teoria do grau coincidente. Finalmente, mencionamos que os resultados inéditos deste trabalho estão contidos em [6] / In this work, we study the bifurcation theory for ordinary dierential equations (we write simply ODEs), as well as the existence of a bifurcation point for periodic solutions of these equations. Then we develop the theory of bifurcation for generalized ordinary differential equations (we write generalized ODEs for short). Such theory is new. We obtained an existence result of a bifurcation point for periodic solutions of generalized ODEs. By means of the correspondence of classic ODEs and generalized ODEs, we were able to translate the results to classic ODEs, now in the framework of generalized ODE. This means that instead of the classic hypothesis that the functions involved in the differential equation are continuously differentiable, we only require that they are Kurzweil-Henstock integrable. We also added a result on the existence of a periodic solution of a generalized ODE and we applied such result to classic ODEs. In order to obtain our main results, we employed the coincidence degree theory. Finally, we point out that our results are contained in [6]

Neste trabalho, estudamos a teoria de bifurcação para equações diferenciais ordinárias (escrevemos simplesmente EDOs), bem como a existência de ponto de bifurcação para soluções periódicas destas equações. Em seguida, desenvolvemos a teoria, até então inexistente, sobre bifurcação para equações diferenciais ordinárias generalizadas (EDOs generalizadas). Neste desenvolvimento, obtivemos para EDOs generalizadas, um resultado sobre existência de ponto de bifurcação para soluções periódicas. Em seguida, através da correspondência entre EDOs e EDOs generalizadas, obtivemos novos resultados sobre a existência de ponto de bifurcação para soluções periódicas para EDOs clássicas, agora sob a ótica das EDOs generalizadas, quando então, em vez de funções continuamente diferenciáveis, necessitamos, apenas, que as funções envolvidas na EDO sejam integráveis no sentido de Kurzweil-Henstock. Adicionamos, também, um resultado sobre a existência de soluções periódicas de EDOs generalizadas e aplicamos tal resultado para EDOs. A fim de obtermos os resultados que pretendíamos, utilizamos a teoria do grau coincidente. Finalmente, mencionamos que os resultados inéditos deste trabalho estão contidos em [6] / In this work, we study the bifurcation theory for ordinary dierential equations (we write simply ODEs), as well as the existence of a bifurcation point for periodic solutions of these equations. Then we develop the theory of bifurcation for generalized ordinary differential equations (we write generalized ODEs for short). Such theory is new. We obtained an existence result of a bifurcation point for periodic solutions of generalized ODEs. By means of the correspondence of classic ODEs and generalized ODEs, we were able to translate the results to classic ODEs, now in the framework of generalized ODE. This means that instead of the classic hypothesis that the functions involved in the differential equation are continuously differentiable, we only require that they are Kurzweil-Henstock integrable. We also added a result on the existence of a periodic solution of a generalized ODE and we applied such result to classic ODEs. In order to obtain our main results, we employed the coincidence degree theory. Finally, we point out that our results are contained in [6]

In the deformation of layered materials such as geological strata, or stacks of paper, mechanical properties compete with the geometry of layering. Smooth, rounded corners lead to voids between layers, while close packing leads to geometrically induced curvature singularities. When creation of voids is penalized by external pressure, the system trades off these competing effects, leading to various accommodating formations. Three two dimensional energy based nonlinear models are presented to describe the formation of voids at areas of intense geological folding. For each model the layers are assumed to be flexible elastic beams under hard unilateral contact constraint; which are solved as quasi-static obstacle problems with a free boundary. In each case an application of Kuhn-Tucker theory leads to representation as a nonlinear fourth order differential equation. Firstly a single layered model for voiding is presented. An elastic layer is forced into a V-shaped singularity by a uniform overburden pressure, where the fourth order free boundary problem is shown to have a unique, convex, symmetric solution. Drawing parallels with the Kuhn-Tucker theory, virtual work and ideas of duality, the physical significance of this differential equation is emphasised. Finally, appropriate scaling of either the potential energy or the differential equation shows the solutions scale to a single parametric group, for which the size of the void scales inversely with the ratio of overburden pressure to bending stiffness of the layer. Common to structural geology, one or several especially thick layers can dominate the deformation process. As a result, the remaining weak layers must accommodate into the geometry imposed by these competent layers. The second model, extends the first by introducing a plastic hinge to replicate the geometry imposed by the competent layer, and also axial springs to resist the slip over the limbs. The equilibrium equations for the system are investigated using the mathematical techniques developed for the first model. Under rigid loading the system may snap from an initially flat state to a convex voiding solution, as seen in the first model. However, if resistance to slip is high, the slightest imperfection causes the system to jump to a convoluted up-buckled solution, following a de-stiffened path to a point of self contact. These solutions have similarities with the delamination of carbon fibre composites. Finally, we extend the two single layered models to a simple multilayered model, which describes the periodic formation of voids in a chevron fold. The model shows that in the limit of high overburden pressures solutions form voids every layer, producing straight limbs punctured by sharp corners. This analysis shows good agreement when compared with recent experiments. This work provides the basis for future work on the buckling of thin multilayer assemblies in which voids may develop, and emphasizes the importance of the intricate nonlinear constraints of layers fitting together in multilayered folds.