Some modified bifurcation problems with application to imperfection sensitivity in buckling

The branching theory of solutions of certain nonlinear elliptic partial differential equations is developed, when the nonlinear term is perturbed from unforced to forced. We find families of branching points and the associated nonisolated solutions which emanate from a bifurcation point of the unforced problem. Nontrivial solution branches are constructed which contain the nonisolated solutions, and the branching is exhibited. An iteration procedure is used to establish the existence of these solutions, and a formal perturbation theory is shown to give asymptotically valid results. The stability of the solutions is examined and certain solution branches are shown to consist of minimal positive solutions. Other solution branches which do not contain branching points are also found in a neighborhood of the bifurcation point.

The qualitative features of branching points and their associated nonisolated solutions are used to obtain useful information about buckling of columns and arches. Global stability characteristics for the buckled equilibrium states of imperfect columns and arches are discussed. Asymptotic expansions for the imperfection sensitive buckling load of a column on a nonlinearly elastic foundation are found and rigorously justified.

Bifurcation theory of nonlinear boundary value problems

The theory of bifurcation of solutions to two-point boundary value problems is developed for a system of nonlinear first order ordinary differential equations in which the bifurcation parameter is allowed to appear nonlinearly. An iteration method is used to establish necessary and sufficient conditions for bifurcation and to construct a unique bifurcated branch in a neighborhood of a bifurcation point which is a simple eigenvalue of the linearized problem. The problem of bifurcation at a degenerate eigenvalue of the linearized problem is reduced to that of solving a system of algebraic equations. Cases with no bifurcation and with multiple bifurcation at a degenerate eigenvalue are considered.

The iteration method employed is shown to generate approximate solutions which contain those obtained by formal perturbation theory. Thus the formal perturbation solutions are rigorously justified. A theory of continuation of a solution branch out of the neighborhood of its bifurcation point is presented. Several generalizations and extensions of the theory to other types of problems, such as systems of partial differential equations, are described.

The theory is applied to the problem of the axisymmetric buckling of thin spherical shells. Results are obtained which confirm recent numerical computations.

Obtaining bifurcation diagrams with a thermoacoustic network model

Linear techniques can predict whether the non-oscillating (steady) state of a thermoacoustic system is stable or unstable. With a sufficiently large impulse, however, a thermoacoustic system can reach a stable oscillating state even when the steady state is also stable. A nonlinear analysis is required to predict the existence of this oscillating state. Continuation methods are often used for this but they are computationally expensive. In this paper, an acoustic network code called LOTAN is used to obtain the steady and the oscillating solutions for a horizontal Rijke tube. The heat release is modelled as a nonlinear function of the mass flow rate. Several test cases from the literature are analysed in order to investigate the effect of various nonlinear terms in the flame model. The results agree well with the literature, showing that LOTAN can be used to map the steady and oscillating solutions as a function of the control parameters. Furthermore, the nature of the bifurcation between steady and oscillating states can be predicted directly from the nonlinear terms inside the flame model. Copyright © 2012 by ASME.

Bifurcation of Pacific North Equatorial Current at the surface

The grid altimetry data between 1993 and 2006 near the Philippines were analyzed by the method of Empirical Orthogonal Function (EOF) to study the variation of bifurcation of the North Equatorial Current at the surface of the Pacific. The relatively short-term signals with periods of about 6 months, 4 months, 3 months and 2 months are found besides seasonal and interannual variations mentioned in previous studies. Local wind stress curl plays an important role in controlling variation of bifurcation latitude except in the interannual timescale. The bifurcation latitude is about 13.3A degrees N in annual mean state and it lies at the northernmost position (14.0A degrees N) in January, at the southernmost position (12.5A degrees N) in July. The amplitude of variation of bifurcation latitude in a year is 1.5A degrees, which can mainly be explained as the contributions of the signals with periods of about 1 year (1.2A degrees) and 0.5 year (0.3A degrees).

The Bifurcation Interpreter: A Step Towards the Automatic Analysis of Dynamical Systems

The Bifurcation Interpreter is a computer program that autonomously explores the steady-state orbits of one-parameter families of periodically- driven oscillators. To report its findings, the Interpreter generates schematic diagrams and English text descriptions similar to those appearing in the science and engineering research literature. Given a system of equations as input, the Interpreter uses symbolic algebra to automatically generate numerical procedures that simulate the system. The Interpreter incorporates knowledge about dynamical systems theory, which it uses to guide the simulations, to interpret the results, and to minimize the effects of numerical error.

Plankton populations at the bifurcation of the North Pacific Current

As the eastward-flowing North Pacific Current approaches the North American continent it bifurcates into the southward-flowing California Current and the northward-flowing Alaska Current. This bifurcation occurs in the south-eastern Gulf of Alaska and can vary in position. Dynamic height data from Project Argo floats have recently enabled the creation of surface circulation maps which show the likely position of the bifurcation; during 2002 it was relatively far north at 53 degrees N then, during early 2003, it moved southwards to a more normal position at 45 degrees N. Two ship-of-opportunity transects collecting plankton samples with a Continuous Plankton Recorder across the Gulf of Alaska were sampled seasonally during 2002 and 2003. Their position was dependent on the commercial ship's operations; however, most transects sampled across the bifurcation. We show that the oceanic plankton differed in community composition according to the current system they occurred in during spring and fall of 2002 and 2003, although winter populations were more mixed. Displacement of the plankton communities could have impacts on the plankton's reproduction and development if they use cues such as day length, and also on foraging of higher trophic-level organisms that use particular regions of the ocean if the nutritional value of the communities is different. Although we identify some indicator taxa for the Alaska and California currents, functional differences in the plankton communities on either side of the bifurcation need to be better established to determine the impacts of bifurcation movement on the ecosystems of the north-east Pacific.

Narrative Bifurcation, Cloud Computing Interfaces and Hitchcok

From a narratological perspective, this paper aims to address the theoretical issues concerning the functioning of the so called «narrative bifurcation» in data presentation and information retrieval. Its use in cyberspace calls for a reassessment as a storytelling device. Films have shown its fundamental role for the creation of suspense. Interactive fiction and games have unveiled the possibility of plots with multiple choices, giving continuity to cinema split-screen experiences. Using practical examples, this paper will show how this storytelling tool returns to its primitive form and ends up by conditioning cloud computing interface design.

Automaticité dans le coeur ischémique : analyse de bifurcation d’un modèle mathématique de cellule ventriculaire humaine

L’ischémie aigüe (restriction de la perfusion suite à l’infarctus du myocarde) induit des changements majeurs des propriétés électrophysiologique du tissu ventriculaire. Dans la zone ischémique, on observe une augmentation du potassium extracellulaire qui provoque l’élévation du potentiel membranaire et induit un "courant de lésion" circulant entre la zone affectée et saine. Le manque d’oxygène modifie le métabolisme des cellules et diminue la production d’ATP, ce qui entraîne l’ouverture de canaux potassique ATP-dépendant. La tachycardie, la fibrillation ventriculaire et la mort subite sont des conséquences possibles de l’ischémie. Cependant les mécanismes responsables de ces complications ne sont pas clairement établis. La création de foyer ectopique (automaticité), constitue une hypothèse intéressante expliquant la création de ses arythmies. Nous étudions l’effet de l’ischémie sur l’automaticité à l’aide d’un modèle mathématique de la cellule ventriculaire humaine (Ten Tusscher, 2006) et d’une analyse exhaustive des bifurcations en fonction de trois paramètres : la concentration de potassium extracellulaire, le "courant de lésion" et l’ouverture de canaux potassiques ATP-dépendant. Dans ce modèle, nous trouvons que seule la présence du courant de lésion peut entrainer une activité automatique. Les changements de potassium extracellulaire et du courant potassique ATP-dépendant altèrent toutefois la structure de bifurcation.

Sur un système de deux oscillateurs FitzHugh-Nagumo couplés

Ce mémoire consiste en l’étude du comportement dynamique de deux oscillateurs FitzHugh-Nagumo identiques couplés. Les paramètres considérés sont l’intensité du courant injecté et la force du couplage. Juqu’à cinq solutions stationnaires, dont on analyse la stabilité asymptotique, peuvent co-exister selon les valeurs de ces paramètres. Une analyse de bifurcation, effectuée grâce à des méthodes tant analytiques que numériques, a permis de détecter différents types de bifurcations (point de selle, Hopf, doublement de période, hétéroclinique) émergeant surtout de la variation du paramètre de couplage. Une attention particulière est portée aux conséquences de la symétrie présente dans le système.