David G. Schaeffer

Singularities and groups in bifurcation theory

This book has been written in a frankly partisian spirit-we believe that singularity theory offers an extremely useful approach to bifurcation prob- lems and we hope to convert the reader to this view. In this preface we will discuss what we feel are the strengths of the singularity theory approach. This discussion then Ieads naturally into a discussion of the contents of the book and the prerequisites for reading it. Let us emphasize that our principal contribution in this area has been to apply pre-existing techniques from singularity theory, especially unfolding theory and classification theory, to bifurcation problems. Many ofthe ideas in this part of singularity theory were originally proposed by Rene Thom; the subject was then developed rigorously by John Matherand extended by V. I. Arnold. In applying this material to bifurcation problems, we were greatly encouraged by how weil the mathematical ideas of singularity theory meshed with the questions addressed by bifurcation theory. Concerning our title, Singularities and Groups in Bifurcation Theory, it should be mentioned that the present text is the first volume in a two-volume sequence. In this volume our emphasis is on singularity theory, with group theory playing a subordinate role. In Volume II the emphasis will be more balanced. Having made these remarks, Iet us set the context for the discussion of the strengths of the singularity theory approach to bifurcation. As we use the term, bifurcation theory is the study of equations with multiple solutions.

Instability in the evolution equations describing incompressible granular flow

In this paper, equations governing the time dependent flow of granular material under gravity are derived and analyzed. Formally these equations bear a strong resemblance to the Navier-Stokes equations for the flow of an incompressible, viscous fluid. However, the main result of this paper is that, depending on geometric and material parameters, the equations governing granular flow may lead to a violent instability analogous to that for u, = u XI up ;

A two-current model for the dynamics of cardiac membrane

In this paper we introduce and study a model for electrical activity of cardiac membrane which incorporates only an inward and an outward current. This model is useful for three reasons: (1) Its simplicity, comparable to the FitzHugh-Nagumo model, makes it useful in numerical simulations, especially in two or three spatial dimensions where numerical efficiency is so important. (2) It can be understood analytically without recourse to numerical simulations. This allows us to determine rather completely how the parameters in the model affect its behavior which in turn provides insight into the effects of the many parameters in more realistic models. (3) It naturally gives rise to a one-dimensional map which specifies the action potential duration as a function of the previous diastolic interval. For certain parameter values, this map exhibits a new phenomenon—subcritical alternans—that does not occur for the commonly used exponential map.

Imperfect bifurcation in the presence of symmetry

Consider the familiar principle that typically (or generically) a system of m scalar equations in n variables where m>n has no solutions. This principle can be reformulated geometrically as follows. If S is a submanifold of a manifold X with codimension m (i.e. m = άimX — dimS) and iϊf:R-*X is a smooth mapping where m>n, then usually or generically Image /nS is empty. One of the basic tenets in the application of singularity theory is that this principle holds in a general way in function spaces. In the next few paragraphs we shall try to explain this more general situation as well as to explain its relevance to bifurcation problems. First we describe an example through which these ideas may be understood. Consider the buckling of an Euler column. Let λ denote the applied load and x denote the maximum deflection of the column. After an application of the Lyapunov-Schmidt procedure the potential energy function Ffor this system may be written as a function of x and λ alone and hence the steady-state configurations of the column may be found by solving

Boundary conditions and mode jumping in the buckling of a rectangular plate

We show that mode jumping in the buckling of a rectangular plate may be explained by a secondary bifurcation — as suggested by Bauer et al. [1] — when “clamped” boundary conditions on the vertical displacement function are assumed. In our analysis we use the singularity theory of mappings in the presence of a symmetry group to analyse the bifurcation equation obtained by the Lyapunov-Schmidt reduction applied to the Von Kármán equations. Noteworthy is the fact that this explanation fails when the assumed boundary conditions are “simply supported”.Mode jumping in the presence of “clamped” boundary conditions was observed experimentally by Stein [9]; “simply supported” boundary conditions are frequently studied but are difficult — if not impossible — to realize physically. Thus, it is important to observe that the qualitative post-buckling behavior depends on which idealization for the boundary conditions one chooses.

Bifurcation analysis near a double eigenvalue of a model chemical reaction

In this paper we analyze the steady-state bifurcations from the trivial solution of the reaction-diffusion equations associated to a model chemical reaction, the so-called Brusselator. The present analysis concentrates on the case when the first bifurcation is from a double eigenvalue. The dependence of the bifurcation diagrams on various parameters and perturbations is analyzed. The results of reference [2] are invoked to show that further complications in the model would not lead to new behavior.

Forces on a slowly rotating, rough cylinder in a Couette device containing a dry, frictional powder

In the present work, a fine, dry powder was sheared in a Couette device: i.e., sheared between concentric vertical cylinders. The torque generated on the rough, inner cylinder was measured as this inner wall was rotated. Our experiments provided evidence that, in a column of granular material undergoing continuous shearing, normal and shear stresses increase linearly with depth. In other words, Janssen’s analysis ceases to apply if granular material is continuously sheared.

Temporal control of self‐organized pattern formation without morphogen gradients in bacteria

Diverse mechanisms have been proposed to explain biological pattern formation. Regardless of their specific molecular interactions, the majority of these mechanisms require morphogen gradients as the spatial cue, which are either predefined or generated as a part of the patterning process. However, using Escherichia coli programmed by a synthetic gene circuit, we demonstrate here the generation of robust, self‐organized ring patterns of gene expression in the absence of an apparent morphogen gradient. Instead of being a spatial cue, the morphogen serves as a timing cue to trigger the formation and maintenance of the ring patterns. The timing mechanism enables the system to sense the domain size of the environment and generate patterns that scale accordingly. Our work defines a novel mechanism of pattern formation that has implications for understanding natural developmental processes.

Riemann problems for nonstrictly hyperbolic 2×2 systems of conservation laws

On resout le probleme de Riemann pour des systemes 2×2 de lois de conservation hyperboliques ayant des fonctions flux quadratiques

The Hopf Bifurcation

The term Hopf bifurcation refers to a phenomenon in which a steady state of an evolution equation evolves into a periodic orbit as a bifurcation parameter is varied. The Hopf bifurcation theorem (Theorem 3.2) provides sufficient conditions for determining when this behavior occurs. In this chapter, we study Hopf bifurcation for systems of ODE using singularity theory methods. The principal advantage of these methods is that they adapt well to degenerate Hopf bifurcations; i.e., cases where one or more of the hypotheses of the traditional theory fail. The power of these methods is illustrated by Case Study 2, where we present the analysis by Labouriau [1983] of degenerate Hopf bifurcation in the clamped Hodgkin-Huxley equations.