Phase topology of one system with separated variables and singularities of the symplectic structure

We consider an example of a system with two degrees of freedom admitting separation of variables but having a subset of codimension 1 on which the 2-form defining the symplectic structure degenerates. We show how to use separation of variables to calculate the topological invariants of non-degenerate singularities and singularities appearing due to the symplectic structure degeneration. New types of non-orientable 3-atoms are found.     

Spacelike submanifolds of codimension two in de Sitter space

We investigate the differential geometry of spacelike submanifolds of codimension two in de Sitter space and classify the singularities of lightlike hypersurfaces and lightcone Gauss maps in de Sitter 4-space.     

Regularity of harmonic maps with prescribed singularities

In this paper, we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds. This is motivated from one of Hawking's conjectures on the uniqueness of Kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum Einstein equation in general relativity.Research partially supported by a NSF grant DMS-8907849.Research partially supported by a NSF grant     

Hopf-pitchfork singularities in coupled systems

It is known from the literature that a family consisting of two brusselators linearly coupled by diffusion unfolds strange attractors due to the generic occurrence of a 4-dimensional nilpotent singularity of codimension 4. In this paper the attention is placed on the Hopf-pitchfork singularities which are unfolded by that coupled system. We will see that the associated map of bifurcations is very rich and includes configurations which could also play the role of organizing centers of chaotic dynamics. As it happens in the case of two brusselators, the occurrence of Hopf-pitchfork singularities is expected when Hopf bifurcations are coupled by a diffusion mechanism. On the other hand, one of the most interesting problems in the context of coupled systems is the understanding of processes of synchronization/desynchronization. We will also illustrate the role of Hopf-pitchfork singularities as organizing centers of these processes.     

Singularity theory study of overdetermination in models for L-H transitions

Two dynamical models that have been proposed to describe transitions between low- and high-confinement states in confined plasmas are analyzed using singularity theory and stability theory. It is shown that the stationary-state bifurcation sets have qualitative properties identical to standard normal forms for the pitchfork and transcritical bifurcations. The analysis yields the codimension of the highest-order singularities, from which we find that the unperturbed systems are overdetermined bifurcation problems and derive appropriate universal unfoldings. Questions of mutual equivalence and the character of the state transitions are addressed.     

The algebraic structure of cohomological field theory

The algebraic foundation of cohomological field theory is presented. It is shown that these theories are based upon realizations of an algebra which contains operators for both BRST and vector supersymmetry. Through a localization of this algebra, we construct a theory of cohomological gravity in arbitrary dimensions. The observables in the theory are polynomial, but generally non-local operators, and have a natural interpretation in terms of a universal bundle for gravity. As such, their correlation functions correspond to cohomology classes on moduli spaces of Riemannian connections. In this uniformization approach, different moduli spaces are obtained by introducing curvature singularities on codimension two submanifolds via a puncture operator. This puncture operator is constructed from a naturally occuring differential form of co-degree two in the theory, and we are led to speculate on connections between this continuum quantum field theory, and the discrete Regge calculus.     

Anti-Self-Duality of Curvature and Degeneration of Metrics with Special Holonomy

We study the structure of noncollapsed Gromov-Hausdorff limits of sequences, Mⁿ_i, of riemannian manifolds with special holonomy. We show that these spaces are smooth manifolds with special holonomy off a closed subset of codimension 4. Additional results on the the detailed structure of the singular set support our main conjecture that if the Mⁿ_i are compact and a certain characteristic number, C(Mⁿ_i), is bounded independent of i, then the singularities are of orbifold type off a subset of real codimension at least 6.The first author was partially supported by NSF Grant DMS 0104128 and the second by NSF Grant DMS 0302744.     

Anti-Self-Dual Instantons with Lagrangian Boundary Conditions II: Bubbling

We study bubbling phenomena of anti-self-dual instantons on where Σ is a closed Riemann surface. The instantons satisfy a global Lagrangian boundary condition on each boundary slice . The main results establish the energy quantization and removal of singularities near such boundary slices. This completes the analytic foundations for the definition of a new instanton Floer homology for 3-manifolds with boundary. In the interior case, for anti-self-instantons on our methods provide a new approach to the removable singularity theorem by Sibner-Sibner for codimension 2 singularities with a holonomy condition.     

Motion of Curves in Three Spatial Dimensions Using a Level Set Approach

The level set method was originally designed for problems dealing with codimension one objects, where it has been extremely succesful, especially when topological changes in the interface, i.e., merging and breaking, occur. Attempts have been made to modify it to handle objects of higher codimension, such as vortex filaments, while preserving the merging and breaking property. We present numerical simulations of a level set based method for moving curves in R³, the model problem for higher codimension, that allows for topological changes. A vector valued level set function is used with the zero level set representing the curve. Our results show that this method can handle many types of curves moving under all types of geometrically based flows while automatically enforcing merging and breaking.     

Structurally stable phase portraits for the five-dimensional Lorenz equations

We discuss the universal unfolding of a planar codimension four singularity which occurs in the five dimensional Lorenz equations. All structurally stable phase portraits are given and some representative bifurcation diagrams are displayed. These phase portraits have a rich structure including up to four limit cycles. The bifurcation sets in unfolding space — where the phase portraits undergo a qualitative change — are determined and new types of saddle loops are found. We show that the codimension four singularity occurs stably in a model for the laser with saturable absorber. Solution branches indicating birhythmicity and saddle loops for the pulsed mode of laser operation are found in bifurcation diagrams corresponding to the universal unfolding of the codimension four singularity. This explains numerical solutions of other authors which so far have not been related to a bifurcation analysis. Hints to other Lorenz models are given.     

Classification of singular Sobolev connections by their holonomy

For a connection on a principalSU(2) bundle over a base space with a codimension two singular set, a limit holonomy condition is stated. In dimension four, finite action implies that the condition is satisfied and an a priori estimate holds which classifies the singularity in terms of holonomy. If there is no holonomy, then a codimension two removable singularity theorem is obtained.Research partially supported by NSF Grant DMS-8701813Research partially supported by NSF Grant INT-8511481     

Regularization of the linearized gravitational self-force for branes

We discuss the linearized, gravitational self-interaction of a brane of arbitrary codimension in a spacetime of arbitrary dimension. We find that in the codimension two case the gravitational self-force is exactly zero for a Nambu-Goto equation of state, generalizing a previous result for a string in four dimensions. For the case of a 3-brane, this picks out the case of a six-dimensional brane-world model as having special properties that we discuss. In particular, we see that bare tension on the brane has no effect locally, suppressing the cosmological constant problem.     

Statistical Properties for Flows with Unbounded Roof Function,Including the Lorenz Attractor

For geometric Lorenz attractors (including the classical Lorenz attractor) we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors. We also obtain the functional central limit theorem and moment estimates, as well as iterated versions of these results. A consequence is deterministic homogenisation (convergence to a stochastic differential equation) for fast-slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two.     

The effect of spherical aberration on the phase singularities of focused dark-hollow Gaussian beams

The phase singularities of focused dark-hollow Gaussian beams in the presence of spherical aberration are studied. It is shown that the evolution behavior of phase singularities of focused dark-hollow Gaussian beams in the focal region depends not only on the truncation parameter and beam order, but also on the spherical aberration. The spherical aberration leads to an asymmetric spatial distribution of singularities outside the focal plane and to a shift of singularities near the focal plane. The reorganization process of singularities and spatial distribution of singularities are additionally dependent on the sign of the spherical aberration. The results are illustrated by numerical examples.     

Prescribing singularities of maximal surfaces via a singular Björling representation formula

We derive a proper formulation of the singular Björling problem for spacelike maximal surfaces with singularities in the Lorentz–Minkowski 3-space which roughly asks whether there exists a maximal surface that contains a prescribed curve as singularities, and then provide a representation formula which solves the problem in an affirmative way. As consequences, we construct many kinds of singularities of maximal surfaces and deduce some properties of the maximal surfaces related to the singularities due to the geometry of the Gauss map.     

Heteroclinic primary intersections and codimension one Melnikov method for volume-preserving maps

We study families of volume preserving diffeomorphisms in R(3) that have a pair of hyperbolic fixed points with intersecting codimension one stable and unstable manifolds. Our goal is to elucidate the topology of the intersections and how it changes with the parameters of the system. We show that the "primary intersection" of the stable and unstable manifolds is generically a neat submanifold of a "fundamental domain." We compute the intersections perturbatively using a codimension one Melnikov function. Numerical experiments show various bifurcations in the homotopy class of the primary intersections. (c) 2000 American Institute of Physics.     

Bifurcation analysis and amplitude equations for viscoelastic convective fluids

Summary The possible bifurcations of a convective instability in viscoelastic fluid are studied. The viscoelastic behaviour is modelized by means of the Oldroyd type fluid whose parameters can be adjusted to suit a large class of polymeric fluids. We analyse in some detail bifurcations of codimension one (stationary or oscillatory convection) and codimension two for such kind of fluids. By a weak nonlinear analysis, the coefficients of the amplitude equations corresponding to the different bifurcations are also determined. It has been found that the nature of the convective solution depends crucially on both the viscoelastic parameters and the constitutive equation used to describe the fluid.     

Singularities and spectral changes of Gaussian beams focused by a lens with spherical aberration

We address the effect of the truncation parameter and spherical aberration (SA) on the singularity transformation and spectral behavior of the polychromatic Gaussian beams focused by an aperture lens with SA in detail. The numerical simulation results, based on the derived equations of the intensity and the spectral density, are given. It is found that the axial singularities vanished with the change of the truncated parameter. The intensity and drastic spectral change fade away with an annihilation process of the phase singularities, and the drastic spectral change does not disappear immediately at the moment the phase singularity annihilates. The singularities in the focal region will redistribute with the increment of SA coefficient, some singularities will vanish, some will spilt into two new singularities, and other off-axial singularities will appear and split into two new singularities as well. When SA coefficient changed, we can find that the axial singularities disappear as well with the decreasing value of truncation parameter. These new splitted singularities due to the change of SA coefficient will converge into one singularity again and disappear gradually.