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We introduce numerical methods for the analysis of random dynamical systems. The subdivision and the continuation algorithm are powerful tools which will be demonstrated for a system from ship dynamics. With our software package we are able to show that the well known safe basin is a moving fractal set. We will also give a numerical approximation of the attracting invariant set (which contains a local attractor) and its evolution.

We study the classical nonlinear dynamics and the quantum vibrational energy eigenstates of the molecule HOCl. The classical vibrational dynamics, at energies below the HO+Cl dissociation energy, contains several saddle-center and period doubling bifurcations. The saddle-center bifurcations are shown to be due to a 2:1, and at higher energies a 3:1, nonlinear resonance between bend and stretch motions in various periodic orbits. The sequence of bifurcations takes the system from nearly integrable at low energies to almost completely chaotic at energies near the HO+Cl dissociation energy. At energies above dissociation we study the chaotic scattering of the Cl atom off the HO dimer. This scattering is governed by a homoclinic tangle formed by the stable and unstable manifolds of a parabolic periodic orbit at infinity. We construct the first three segments of the homoclinic tangle in phase space and use scattering functions to investigate its higher-order structure. For the quantum system we use a discrete variable representation to efficiently calculate the Hamiltonian matrix. We find 365 even and 357 odd parity eigenstates with energies below the dissociation energy. By plotting the eigenstates in configuration space we show that almost every quantum eigenstate can be associated with one or more of the classical periodic orbits. The classical bifurcations that give rise to new periodic orbits are manifest quantum mechanically through the sudden appearance of new classes of eigenstates. Despite the high degree of chaos in the classical dynamics at energies near the dissociation energy most quantum eigenstates remain highly ordered with recognizable nodal patterns. We use R-matrix theory together with a discrete variable representation to calculate quasibound states with energies above the dissociation energy. We find quasibound states with lifetimes ranging over 5 orders of magnitude. Using configuration space plots and Husimi distributions we show that the long-lived quasibound states are supported by unstable periodic orbits in the classical dynamics and medium-lived quasibound states are spread throughout the chaotic region of the classical phase space. Short-lived quasibound states show some similarity to unstable periodic orbits that stretch along the dissociation channel. / text

We study bifurcations of periodic solutions with spatio-temporal symmetries of functional differential equations (FDEs). The two main results are: (1) a centre manifold reduction around a periodic solution of FDEs with spatio-temporal symmetries, and (2) symmetry-breaking bifurcations for symmetric rings of delay-coupled lasers. For the case of ODEs, symmetry-breaking bifurcations from periodic solutions has already been studied. We extend this result to the case of symmetric FDEs using a Centre Manifold Theorem for symmetric FDEs which reduces FDEs into ODEs on an integral manifold around a periodic solution. We show that the integral manifold is invariant under the spatio-temporal symmetries which guarantees that the symmetry structure of the system of FDEs is preserved by this reduction. We also consider a problem in rings of delay-coupled lasers modeled using the Lang-Kobayashi rate equations. We classify the symmetry of bifurcating branches of solutions from steady-state and Hopf bifurcations that occur in 3-laser systems. This involves finding isotropy subgroups of the symmetry group of the system, and then using the Equivariant Branching Lemma and the Equivariant Hopf Theorem. We then utilize this result to find the bifurcating branches of solutions in DDE-Biftool. Symmetry often causes eigenvalues to have multiplicity, and in some cases, this could lead DDE-Biftool to incorrectly predict the bifurcation points. We address this issue by developing a method of finding bifurcation points which can be used for the general case of n-laser systems with unidirectional and bidirectional coupling. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2012-04-30 11:25:01.011

Nous présentons dans cette étude le développement de la convection à partir de différentes perturbations de l'état conductif d'une couche fluide confinée dans une cavité cylindrique, chauffée par le bas et avec une surface supérieure libre. La discrétisation spatiale du domaine repose sur la méthode des éléments spectraux et les itérations temporelles sont assurées par une méthode splitting.Au déclenchement de la convection, les structures convectives correspondent à des modes de Fourier, et les seuils critiques dépendent du rapport de forme de la cavité, et des nombres de Biotet de Marangoni qui caractérisent la surface libre. Les transitions d'écoulements au-delà du seuil primaire sont caractérisées quantitativement en fonction du nombre de Rayleigh pour différentes valeurs du nombre de Biot et Ma = 0. Les résultats présentés sont obtenus en résolvant l'ensemble des équations non-linéaires de conservation à travers une méthode de continuation. Lorsque la convection se déclenche sous la forme d'un mode axisymétrique m = 0, l'évolution non-linéaire montre la coexistence de différentes structures convectives, des structures axisymétriques avec écoulement montant ou descendant au centre de la cavité et des structures correspondant à des combinaisons de modes qui apparaissent sur des branches secondaires sous-critiques.L'action d'un champ magnétique constant est ensuite étudiée pour des fluides conducteurs dans une même configuration comprenant une surface supérieure libre. Nous montrons l'effet stabilisateur du champ magnétique sur les seuils primaires ainsi que son action sélective sur les différents modes de convection. Nous analysons l'évolution des structures convectives au delà de ces seuils et montrons comment le champ magnétique modifie les transitions entre ces structures.En soumettant le bain fondu à un champ magnétique tournant, le mouvement de rotation du fluide se superpose aux mouvements de convection thermique et on observe une diminution des fluctuations de température et un retard du déclenchement de l'instabilité de Rayleigh-Bénard(lorsque les deux parois haut/bas du bain sont rigides). La rotation influe sur ce déclenchement qui de stationnaire devient oscillatoire, à l'exception du mode m = 0 de Fourier, pour qui la transition reste stationnaire jusqu'à une certaine valeur critique du nombre de Taylor magnétique.La dynamique de l'écoulement axisymétrique de part et d'autre de cette valeur critique sera étudiée en détail.

The blood is a complex fluid that contains, in addition to water, cells, macro-molecules and a large number of smaller molecules. The physical properties of the blood are therefore the result of non-linear interactions of its constituents, which are influenced by the local flow field conditions. Hence, the local blood viscosity is a function of the local concentration of the blood constituents and the local flow field itself. This study considers the flow of blood-like fluids in generalised 90-degree bifurcating pipes and patient-specific arterial bifurcations relevant to the large aortic branches in humans. It is shown that the Red Blood Cell (RBC) distribution in the region of bifurcations may lead to large changes in the viscosity, with implications on the concentrations of the various cells in the blood plasma. This in turn implies that the flow in the near wall regions is more difficult to estimate and predict than that under the assumption of a homogeneous fluid. The rheological properties of blood are complex and are difficult to measure, since the results depend on the measuring equipment and the inherent flow conditions. We attempt to model the viscosity of water containing different volume fractions of non-deforming RBC-like particles in tubes. The apparent viscosities of the mixtures obtained from these model experiments have been compared to the predictions of the different rheological models found in the literature. The same rheological models have also been used in the different simulations, where the local RBC concentration and local shear rate are used in the viscosity models. The flow simulations account for the non-linearity due to coupling between the flow and fluid rheology. Furthermore, from a physiological perspective, it is shown that oscillatory wall shear stresses are affected by changes in RBC concentration in the regions of the bifurcation associated with atherogenesis. The intrinsic shear thinning rheological property of the blood, in conjunction with stagnation in separated flows, may be responsible for elevated temporal wall shear stress gradients (TWSSG) influencing endothelial cell behaviour, which has been postulated to play a role in the development of atherosclerosis. The blood-like fluid properties along with variations in the RBC concentration could also lead to variations in the developing flow structures in the larger arteries that could influence the work the heart has to bear. / <p>QC 20131206</p>

Cette thèse traite de la synchronisation et de la désynchronisation des systèmes dynamiques. Dans une première partie nous abordons, sous l'angle de la biologie systémique, le problème de la désynchronisation qui consiste à induire un comportement chaotique dans un système ayant une dynamique stable. Nous étudions ce problème sur un réseau génétique appelé V-système, inventé afin de coupler le plus simplement possible une bifurcation de Hopf et une hystérèse. Après avoir démontré qu'un champ de vecteurs de R^n présentant un tel couplage peut, sous certaines conditions, avoir un comportement chaotique, nous donnons un ensemble de paramètres pour lequel le V-système associé satisfait ces conditions et vérifions numériquement que le mécanisme responsable du chaos prend place dans ce système. Dans une deuxième partie, nous nous intéressons à la synchronisation de systèmes organisés hiérarchiquement. Nous commençons par définir une structure hiérarchique pour un ensemble de 2^n systèmes par une matrice représentant les étapes d'un processus de regroupement deux par deux. Cela nous amène naturellement au cas d'un ensemble de Cantor de systèmes, pour lequel nous obtenons un résultat de synchronisation globale généralisant le cas fini. Enfin nous traitons de la situation où certains défauts apparaissent dans la hiérarchie, i.e que certains liens entre les systèmes sont brisés. Nous montrons que l'on peut accepter un nombre infini de liens brisés, tout en gardant une synchronisation locale, à condition que ces liens soient uniquement présents aux N premiers étages de la hiérarchie (pour un N fixé) et qu'ils soient suffisamment espacés dans ces étages.

Ce travail propose différents modèles de mathématiques issus à des phénomènes naturels. L'outil indispensable à cette étude sont les inclusions différentielles, les équations (ou les systèmes d'équations) différentielles ou aux dérivées partielles et la théorie des bifurcations. La nature des ces équations dépend du problème traité: il peut s'agir d'équations de transport, de réaction-diffusion, d'équations non-locales, etc. Nous souhaitons apporter ici quelques informations et explications sur les différents modèles que nous allons étudier. Dans la première partie, il s'agit d'étudier l'existence des solutions, critère de compacité pour l'ensemble de solutions ainsi que la continuité de l'opérateur solution pour certaines classes d'inclusions différentielles impulsives de type neutre, un exemple d'application est traité à la fin de cet première partie, c'est une extension des résultats obtenus dans l'étude théorique. La seconde partie s'attache à l'analyse d'un autre modèle mathématique décrivant l'évolution de la maladie du cancer, il s'agit d'un système d'équations différentielles avec impulsions, les équations différentielles représentent l'évolution des cellules normales, cancéreuses sensibles et cancéreuses résistantes. Les impulsions représentent la chimiothérapie. On considère le cas de l'absence des cellules de la tumeur et on utilise un traitement préventif pour éradiquer la maladie, on étudie tout d'abord les conditions de stabilité des solutions triviales qui représentent l'éradication de la maladie, puis on traite le cas des bifurcations de solutions non triviales qui représentent le retour de maladie. On s'intéresse dans la dernière partie à la modélisation de la maladie d'Alzheimer. On construit un modèle qui décrit d'une part la formation de plaque amyloide {in vivo}, et d'autre part les interactions entre les oligomères A$\beta$ et la protéine prion qui induiraient la perte de mémoire. On mène l'analyse mathématique de ce modèle dans un cas particulier puis dans un cas plus général où le taux de polymérisation est une loi de puissance.

<p>The primary focus of this dissertation is the characterization of snap-through buckling of discrete and continuous systems. Snap-through buckling occurs as the consequence of two factors, first the destabilization, or more often the disappearance of, an equilibrium position under the change of a system parameter, and second the existence of another stable equilibrium configuration at a remote location in state space. In this sense snap-through buckling is a global dynamic transition as the result of a local static instability.</p><p> </p><p>In order to better understand the static instabilities that lead to snap-through buckling, the behavior of mechanical systems in the vicinity of various local bifurcations is first investigated. Oscillators with saddle-node, pitchfork, and transcritical bifurcations are shown analytically to exhibit several interesting characteristics, particularly in relation to the system damping ratio. A simple mechanical oscillator with a transcritical bifurcation is used to experimentally verify the analytical results. The transcritical bifurcation was selected since it may be used to represent generic bifurcation behavior. It is shown that the damping ratio may be used to predict changes in stability with respect to changing system parameters.</p><p>Another useful indicator of snap-through is the presence of chaos in the dynamic response of a system. Chaos is usually associated snap-through, as in many systems large amplitude responses are typically necessary to sufficiently engage the nonlinearities that induce chaos. Thus, a pragmatic approach for identifying chaos in experimental (and hence noisy) systems is also developed. The method is applied to multiple experimental systems showing good agreement with identification via Lyapunov exponents.</p><p>Under dynamic loading, systems with the requisite condition for snap-through buckling, that is co-existing equilibria, typically exhibit either small amplitude response about a single equilibrium configuration, or large amplitude response that transits between the static equilibria. Dynamic snap-through is the name given to the large amplitude response, which, in the context of structural systems, is obviously undesirable. This phenomenon is investigated using experimental, numerical, and analytical means and the boundaries separating safe (non-snap-through) from unsafe (snap-through) dynamic response in forcing parameter space are obtained for both a discrete and a continuous arch. Arches present an ideal avenue for the investigation of snap-through as they typically have multiple, often tunable, stable and unstable equilibria. They also have many direct applications in both civil engineering, where arches are a canonical structural element, and mechanical engineering, where arches may be used to approximate the behavior of curved plates and panels such as those used on aircraft.</p> / Dissertation

LIMA, Érico Oliveira de Araújo. Numa cama, numa festa, numa greve, numa revolução: o cinema se bifurca, o tempo se abre. 2014. 208f. – Dissertação (Mestrado) – Universidade Federal do Ceará, Programa de Pós-graduação em Comunicação Social, Fortaleza (CE), 2014. / Submitted by Márcia Araújo (marcia_m_bezerra@yahoo.com.br) on 2014-08-04T14:59:11Z No. of bitstreams: 1 2014_dis_eoalima.pdf: 1651935 bytes, checksum: 4bd50acc541319d070f4328aa52784d1 (MD5) / Approved for entry into archive by Márcia Araújo(marcia_m_bezerra@yahoo.com.br) on 2014-08-04T17:22:26Z (GMT) No. of bitstreams: 1 2014_dis_eoalima.pdf: 1651935 bytes, checksum: 4bd50acc541319d070f4328aa52784d1 (MD5) / Made available in DSpace on 2014-08-04T17:22:26Z (GMT). No. of bitstreams: 1 2014_dis_eoalima.pdf: 1651935 bytes, checksum: 4bd50acc541319d070f4328aa52784d1 (MD5) Previous issue date: 2014 / In a bed, a strike, a party or a revolution: that was how Glauber Rocha once thought that his last film, A Idade da Terra (1980), could be experienced. The work takes to the limit a research that turns making cinema a process more radical. It also invents news possibilities to aesthetic experience. It’s a film that I try to think with two others works from Glauber: Claro (1975), shot when he was in exile in Rome, and Di Cavalcanti (1977), a short film that he produced on an impulse, when he heard that his painter friend had died. Each of these films triggers singular procedures on occupying the world and on inventing paths to the cinema. They are taken here as bifurcations on configured forms of sensibility, as they try to proliferate ways of making a film. This is a broader concern in discussing the politics of films, singular ways in which cinema can inscribe in the world an operation of aesthetic fracture that is already political. In this sense, it is thinking with the works that we can propose some figures who claim cinema as field of resistances. One of the questions that guide the discussions of this work is the concept of a becoming minority, thought from Deleuze and Guattari, a movement in which cinema can engage to create tension with a visible and an audible that present themselves as major facts. In this process, it is common life that is at stake. Those operations of invention are taken in a entanglement of temporalities, which leads to an approach frankly anachronistic, on thinking what a work can shake in time and how it can open this time, in which I try to articulate with the thoughts of Benjamin, Warburg, Didi-Huberman and Agamben. Thus, the idea is to investigate how we could consider a contemporaneity in this cinema of Glauber, in that the proliferations researched by filmic forms are not restricted to a chronology of successive events, rather they spread in a rhizomatic way through temporalities, a proliferative and differentiating movement. Politics and aesthetics would be questions of how we bridge the temporalities, of which tracings are turn into possibilities. Or how the engagements in time can raise events and create resistances in becoming. / Numa cama, numa greve, numa festa ou numa revolução: era assim que Glauber Rocha imaginava que A Idade da Terra (1980), seu último filme, poderia ser experimentado. A obra leva ao limite um processo de pesquisa que radicalizava o fazer cinema e inventava novos possíveis para a experiência estética. É um trabalho que tento pensar junto a duas outras obras do realizador, Claro (1975), filmado durante o exílio em Roma, e Di Cavalcanti (1977), curta-metragem que ele produziu no impulso, ao saber da morte do amigo pintor. Cada um desses filmes desencadeia procedimentos singulares de ocupação do mundo e de invenção de caminhos para o cinema. Eles são entendidos aqui como bifurcações em formas configuradas de sensibilidade, na medida em que embarcam na proliferação de veredas para a fabricação fílmica. Trata-se de uma preocupação mais ampla em discutir as políticas dos filmes, maneiras singulares pelas quais o cinema pode inscrever no mundo uma operação de rotura estética que é já uma política. Nesse sentido, cabe pensar com as obras algumas figuras que afirmam o cinema como campo de resistências. Uma das questões que norteiam as discussões deste trabalho é a dimensão de um devir minoritário, pensado a partir de Deleuze e Guattari, um movimento no qual o cinema pode se engajar para tensionar com um visível e um audível que se apresentam como fatos majoritários. Nesse processo, é a própria vida em comum que está em jogo. Essas operações de invenção se dão em um emaranhado de temporalidades, o que leva a uma abordagem metodológica que se coloca como assumidamente anacrônica, na medida em que tenta pensar o que uma obra faz agitar no tempo e como ela pode fazer esse tempo se abrir, numa articulação que tento fazer com os pensamentos de Benjamin, Warburg, Didi-Huberman e Agamben. Dessa maneira, cabe investigar a dimensão de uma contemporaneidade desse cinema de Glauber, na medida em que as proliferações buscadas pelas formas fílmicas não estão circunscritas a um instante de uma cronologia de eventos sucessivos, mas se espalham rizomaticamente pelos tempos, em movimento proliferante e diferenciante. A política e a estética seriam questões de como fazemos pontes nas temporalidades, de quais traçados são arrancados como possíveis. Ou de como os engajamentos no tempo podem suscitar acontecimentos e instaurar resistências em devir.