Elliptic Umbilic Catastrophes Research Articles

Tunable caustic phenomena in electron wavefields

Novel caustic phenomena, which contain fold, butterfly and elliptic umbilic catastrophes, are observed in defocused images of two approximately collinear oppositely biased metallic tips in a transmission electron microscope. The observed patterns depend sensitively on defocus, on the applied voltage between the tips and on their separation and lateral offset. Their main features are interpreted on the basis of a projected electrostatic potential model for the electron-optical phase shift.

Analysis of Shield Tunnel Stability during Construction Period Using Elliptic Umbilic Catastrophe Model

Using the monitoring data, an expression of ground settlement with associated coordinates above shield tunnel during construction was established, in which coefficients were obtained by the least square method. After transforming this expression into a standard form of elliptic umbilic catastrophe model, the instability criterion for the soil above shield tunnel was determined with the analysis of this model. Instability criterion values were thereafter obtained then the soil state would be judged according to these values. Three cases in different regions in China, referred to Bund Channel in Shanghai, metro line 1 in Chengdu City and metro line 4 in Beijing, were presented. They showed that the method developed in this paper was suitable in analyzing and forecasting the stabilities of soil above shield tunnels.

Wave dislocations in the diffraction pattern of a higher-order optical catastrophe

The paper explores a partial unfolding of the canonical three-dimensional diffraction field associated with the optical catastrophe X9 with modulus K = −6. A practical realization would be the focal region of a thin lens created by setting a drop of water on a horizontal glass slide and constraining its perimeter to be square. The pattern of caustics formed around the focus is a twisted and ribbed double trumpet with 4-fold symmetry. Like all diffraction catastrophes the essential structure is based on a pattern of line singularities (wave dislocations or optical vortices) on which the amplitude is zero and the phase is indeterminate. The caustic is encircled on the outside, and in the focal plane, by a highly puckered and non-circular ring and a forest of other dislocations. Far from the axis these are organized by the planar group 3m, despite the 4-fold symmetry. On the inside, the dislocation lines form a curved quasi-periodic lattice of small, nearly planar, nearly circular, rings based on the tetragonal space group I4mm. There are similarities to the pattern for the elliptic umbilic catastrophe, and, just as in that case, far from the focus the inner rings in lines close to the ribs of the caustic eventually join together to become the straight inner dislocations of the Pearcey pattern for the cusp. But the way in which this transition is accomplished, which involves four simultaneous reconnections, is quite different for the two catastrophes. Further, in the elliptic (and hyperbolic) umbilic catastrophes diffraction splits the focal spot longitudinally; in X9 with K = −6 it does not.

An atlas of predicted exotic gravitational lenses

Wide-field optical imaging surveys will contain tens of thousands of new strong gravitational lenses. Some of these will have new and unusual image configurations, and so will enable new applications: for example, systems with high image multiplicity will allow more detailed study of galaxy and group mass distributions, while high magnification is needed to super-resolve the faintest objects in the high-redshift universe. Inspired by a set of six unusual lens systems [including five selected from the Sloan Lens ACS survey (SLACS) and Strong Lens Legacy Survey (SL2S), plus the cluster Abell 1703], we consider several types of multi-component, physically motivated lens potentials, and use the ray-tracing code glamroc to predict exotic image configurations. We also investigate the effects of galaxy source profile and size, and use realistic sources to predict observable magnifications and estimate very approximate relative cross-sections. We find that lens galaxies with misaligned discs and bulges produce swallowtail and butterfly catastrophes, observable as ‘broken’ Einstein rings. Binary or merging galaxies show elliptic umbilic catastrophes, leading to an unusual Y-shaped configuration of four merging images. While not the maximum magnification configuration possible, it offers the possibility of mapping the local small-scale mass distribution. We estimate the approximate abundance of each of these exotic galaxy-scale lenses to be ∼1 per all-sky survey. In higher mass systems, a wide range of caustic structures are expected, as already seen in many cluster lens systems. We interpret the central ring and its counter-image in Abell 1703 as a ‘hyperbolic umbilic’ configuration, with total magnification ∼100 (depending on source size). The abundance of such configurations is also estimated to be ∼1 per all-sky survey.

Lensing by binary galaxies modelled as isothermal spheres

We consider the problem of lensing by binary galaxies idealized as two isothermal spheres. This is a natural extension of the problem of lensing by binary point masses first studied by Schneider & Weiss. In a wide binary, each galaxy possesses individual tangential, nearly astroidal, caustics and roundish radial caustics. As the separation of the binary is made smaller, the caustics undergo a sequence of metamorphoses. The first metamorphosis occurs when the tangential caustics merge to form a single six-cusped caustic, lying interior to the radial caustics. At still smaller separations, the six-cusped caustic undergoes the second metamorphosis and splits into a four-cusped caustic and two three-cusped caustics, which shrink to zero size (an elliptic umbilic catastrophe) before they enlarge again and move away from the origin perpendicular to the binary axis. Finally, a third metamorphosis occurs as the three-cusp caustics join the radial caustics, leaving an inner distorted astroid caustic enclosed by two outer caustics. The maximum number of images possible is 7. Classifying the multiple imaging according to critical isochrones, there are only eight possibilities: two three-image cases, three five-image cases and three seven-image cases. When the isothermal spheres are singular, the core images vanish into the central singularity. The number of images may then be 1, 2, 3, 4 or 5, depending on the source location, and the separation and masses of the pair of lensing galaxies. The locations of metamorphoses, and the onset of threefold and fivefold multiple imaging, can be worked out analytically in this case.

Inner caustics of cold dark matter halos

We prove that a flow of cold collisionless particles from all directions in and out of a region necessarily forms a caustic. A corollary is that, in cold dark matter cosmology, galactic halos have inner caustics in addition to the more obvious outer caustics. The outer caustics are fold catastrophes located on topological spheres surrounding the galaxy. To obtain the catastrophe structure of the inner caustics, we simulate the infall of cold collisionless particles in a fixed gravitational potential. The structure of inner caustics depends on the angular momentum distribution of the infalling particles. We confirm a previous result that the inner caustic is a ``tricusp ring'' when the initial velocity field is dominated by net overall rotation. A tricusp ring is a closed tube whose cross section is a section of an elliptic umbilic catastrophe. However, tidal torque theory predicts that the initial velocity field is irrotational. For irrotational initial velocity fields, we find the inner caustic to have a tentlike structure which we describe in detail in terms of the known catastrophes. We also show how the tent caustic transforms into a tricusp ring when a rotational component is added to the initial velocity field.

The reversible umbilic bifurcation

Hamiltonian systems with several degrees of freedom regularly lead to the investigation of bifurcating equilibria in reduced one-degree-of-freedom systems. This paper concerns equilibria with vanishing linearisation, a co-dimension two phenomenon in the reversible context. Under appropriate transversality conditions such equilibria have versal unfoldings related to the elliptic and hyperbolic umbilic catastrophes. This has applications to gyrostat motion and also helps to explain the dynamics defined by the normal form of the Hénon-Heiles system. The occurring unfoldings turn out to be versal even in the general reversible context of not necessarily Hamiltonian systems.

Equations of bifurcation sets of three-parameter catastrophes and the application in imperfection sensitivity analysis

In this paper, equations of bifurcation sets of swallowtail catastrophe, hyperbolic umbilic catastrophe and elliptic umbilic catastrophe are derived by alternative elimination method. The equations of bifurcation sets of elliptic/hyperbolic umbilic catastrophes are applied in imperfection sensitivity analysis of mono-symmetric bimodal interaction problems to determine the worst imperfection directions, the imperfection insensitive domain, the stability load contours and the allowable imperfection territories. Furthermore, new concepts such as imperfection insensitive structure, imperfection full-sensitive structure, passive structural stability control, and multiple objective structural stability optimization are advanced and discussed. Five representative cases of a longitudinally stiffened panel under compression are investigated numerically to demonstrate the application.

A normally elliptic Hamiltonian bifurcation

A universal local bifurcation analysis is presented of an autonomous Hamiltonian system around a certain equilibrium point. This central equilibrium has a double zero eigenvalue, the other eigenvalues being in general position. Main emphasis is given to the 2 degrees of freedom case where these other eigenvalues are purely imaginary. By normal form techniques and Singularity Theory unfoldings are obtained, having ‘integrable’ approximations related to the Elliptic and Hyperbolic Umbilic Catastrophes

Semiclassical analysis of H�non-Heiles coupled oscillators: quasi-periodic and chaotic quantum behavior and the resonance model of unimolecular decay

The quantum mechanics of the Henon-Heiles potential is analyzed using an adiabatic representation in polar coordinates and exploiting the asymptotic separability of the radius. The procedure allows us to establish a correlation between quasiperiodic and chaotic classical behavior, and regular or irregular quantum modes: It is found that irregularity can be attributed to nonadiabatic effects at the potential ridge. The resonance widths for this prototypic system of coupled oscillators are studied with reference to the lifetime in the quantum theory of unimolecular decay. The near separability of the radius of the polar coordinate representation is exploited for discussing energy dependence and symmetry effects on the widths. The relevance of this analysis for the characterization of quantum mechanical behavior near an elliptic umbilic catastrophe point is also briefly considered.

The transient wave fields in the vicinity of the elliptic, hyperbolic, and parabolic umbilic caustics

The caustics of high-frequency wave propagation may be classified using catastrophe theory. In this paper, we examine the transient wave fields in the vicinity of the elliptic, hyperbolic, and parabolic umbilic catastrophes. Analytical results are presented which describe the time-dependent wave field due to an impulsive source on and at large distances in each control direction away from the most singular point of each of these three catastrophes. Numerical results are presented which show the smooth variation of these transient wave fields as each control parameter is independently varied. These results are compared to previously derived results on the transient wave fields in the vicinity of the cuspoid catastrophes (the cuspoid and umbilic catastrophes have coranks 1 and 2, respectively). It is found that the transient wave fields in the vicinity of the cuspoid and umbilic catastrophes differ with regard to the temporal structure of the wave field on the most singular point of each catastrophe, the manner in which these temporal structures unfold, and the phase shifts which individual rays undergo as a result of touching one of the catastrophes.

A model of spatial flows and growth of capital and labor stocks

This article deals with optimal spatio-temporal development of capital and labour stocks in a production economy with spatial extension. Current stocks of capital and labour are used to produce a commodity, partly invested to replace worn capital, partly consumed. These stocks can be relocated in space, but relocation uses up some of the inputs themselves. Under these constraints the objective is to maximize a utility measure derived from per capita consumption and aggregated over individuals, space and time. The necessary conditions for optimum are derived as Euler equations of a continuous variational problem. They concern choice of production scale and technology, rate of reinvestment, and optimal flows of labour and produced commodities through space. The Lagrangian multipliers of the constraints are interpreted as imputed wages and commodity prices. The whole structure of optimum depends on these imputed wages and prices, and their solution can be derived from a pair of dependent non-linear partial differential equations. The spatial flow portrait at each moment depends on the time parameter and on the parameters of the model (net reproduction and capital depreciation rates). It can undergo sudden changes described by the elliptic and hyperbolic umbilic catastrophes.

Collisions and umbilic catastrophes

The theory of uniform approximations in semiclassical scattering theory is reviewed in order to emphasize the role of the unfolding of the relevant catastrophe type. Application of the theory to situations involving hyperbolic and elliptic umbilic catastrophes is shown to involve a mapping from four physically determined semiclassical phases {fk } to three control (a, b, c) and one translational parameter d. The paper is devoted to solution of this mapping problem. The final reduction is to a ninth degree polynomial equation in a 3, each root of which gives rise to six symmetry related combinations (b, c). Algebraic and numerical investigations demonstrate the general existence of three physically acceptable (real a, (b + c)) combinations for any set of four phases. Given the four associated jacobians {Jk }, this ambiguity may be definitively removed by following the evolution of (a, b, c) between members of a family of related scattering transitions. Numerical experiments indicate that the correct choice...

An algorithm for the nonlinear analysis of compound bifurcation

The paper presents a procedure for the localized analysis of compound bifurcations. A full range of phenomena are embraced, including loci of equilibria, secondary bifurcation, and imperfection sensitivity, the scheme showing how to generate the appropriate lowest-order non-trivial equations of interest. A number of aids to solution are presented, including the concepts of generalized imperfection and generalized loading parameter. The scheme is developed by using a general non-diagonalized format suitable for numerical analysis, but the special diagonalized form can also be used to good effect. This is illustrated in the application to the interactive buckling of stiffened plates and shells, when local and overall buckling occur simultaneously or nearly so. The modelling relies heavily on the elimination-of-passive-coordinates routine of the general scheme. The study shows that the parabolic umbilic catastrophe is the key phenomenon for most such problems. Finally, the branching analysis is fully illustrated for semi-symmetric branching, where one of the contributing bifurcations is symmetric and the other is asymmetric. In all, ten different loci are treated, including the full imperfection sensitivity at complete and near coincidence plotted in three-dimensional form; these relate to an earlier stiffened-plate formulation. The general scheme is thus made directly accessible for any problem that exhibits a bifurcational manifestation of either the elliptic or hyperbolic umbilic catastrophe.

Local diffeomorphisms in the bifurcational manifestations of the umbilic catastrophes

Typical imperfection-sensitivity plots for semi-symmetric points of bifurcation are presented in three-dimensional control space, giving rise to the universally-unfolded forms of the hyperbolic and elliptic umbilic catastrophes. The structurally-stable distorting of these surfaces is then observed with the addition of a fourth control parameter, which separates the two contributing bifurcations of the perfect system. A further consequence of structural stability is then explored. Simple transformations are presented that allow all the general forms of imperfection-sensitivity to be interpreted for specific problems, here a simple guyed cantilever model under a number of different initial conditions. This is only made possible by local diffeomorphisms which are known to be present, and which result directly from the underlying topological classifi­cation of catastrophe theory.

The elliptic umbilic diffraction catastrophe

We have made a detailed theoretical and experimental study of the three-dimensional diffraction pattern decorating the geometrical-optics caustic surface whose form is the elliptic umbilic catastrophe in Thom’s classification. This caustic has three sheets joined along three parabolic cusped edges (‘ribs’) which touch at one singular point (the ‘focus’). Experimentally, the diffraction catastrophe was studied in light refracted by a water droplet 'lens’ with triangular perimeter, and photographed in sections perpendicular to the symmetry axis of the pattern. Theoretically, the pattern was represented by a diffraction integral E(x,y,z) , which was studied numerically through computer simulations and analytically by the method of stationary phase. Particular attention was concentrated on the ‘dislocation lines’ where | E | vanishes, since these can be considered as a skeleton on which the whole diffraction pattern is built. Within the region bounded by the caustic surface the interference of four rays produces hexagonal diffraction maxima stacked in space like the atoms of a distorted crystal with space group R3m. The dislocation lines not too close to the ribs form hexagonally puckered rings. On receding from the focus and approaching the ribs, these rings approach one another and eventually join to form ‘hairpins’, each arm of which is a tightly wound sheared helix that develops asymptotically into one of the dislocations of the cusp diffraction catastrophe previously studied by Pearcey. Outside the caustic there are also helical dislocation lines, this time formed by interference involving a complex ray. There is close agreement, down to the finest details, between observation, exact computation of E(x,y,z) , and the four-wave stationary-phase approximation.

Optical caustics in the near field from liquid drops

When light passes through a small irregular droplet of water on a glass surface the envelopes of the refracted rays form a system of caustic surfaces. The caustics have been examined experimentally, and interpreted theoretically with the aid of Thom’s theorem. The main feature of interest is a unique plane of focus which contains typically several tens of elliptic umbilic catastrophes. These unfold and interact by beak-to-beak events and swallowtail catastrophes to give the many-cusped figures observed close to the drop and in the far field. The primary generic events produced by an irregular drop are analysed by considering them as the unfoldings of a symmetrical case having four control parameters and containing two elliptic umbilics and two butterfly catastrophes.

Umbilic points on Gaussian random surfaces

An umbilic point U on a surface Sigma is a place where the two principal curvatures of Sigma are equal. U is a singularity of Sigma in three different senses: (i) it is the source of elliptic (E) or hyperbolic (H) umbilic catastrophes in the envelope of normals ('focal surface') of Sigma ; (ii) it has index +or-1/2 depending on whether the principal curvature directions of Sigma (defining the lines of curvature) rotate by +or- pi during a circuit of U; (iii) it has a pattern of the 'star' (S), 'lemon' (L) or 'monstar' (M) type depending on the configuration of lines of curvature near U. We calculate the average density of umbilic points, and the fractions alpha of umbilics of the different sorts, for the case where Sigma is a surface whose (small) deviation from a plane is specified by a Gaussian random function. It is always the case that alpha S= alpha +or-12/=1/2. The other fractions depend on the degree of isotropy of the statistics of Sigma . In the isotropic case the elliptic umbilic fraction is alpha E=1- alpha H=0.268, and the monstar fraction is alpha M=1/2- alpha L=0.053.

A buckling model for the set of umbilic catastrophes

AbstractZeeman has drawn attention to the sequences in which catastrophes, or modes of instability, can be linked, and it is a common observation that sequences of catastrophes of low order are always found in the environment of catastrophes of higher order. In this paper, a simple buckling model is presented that generates in an elegant manner a complete sequence of the umbilic catastrophes as represented by the semi-symmetric branching points. The scan of a single fundamental parameter of this model is shown to trace a route through all regimes of the umbilic bracelet, giving in turn the hyperbolic, symbolic, elliptic, parabolic and hyperbolic umbilic catastrophes.

Imperfection-sensitivity of semi-symmetric branching

The general theory of elastic stability is extended to include the imperfection-sensitivity of twofold compound branching points with symmetry of the potential function in one of the critical modes (semi-symmetric points of bifurcation). Three very different forms of imperfection-sensitivity can result, so a subclassification into monoclinal, anticlinal and homeoclinal semi-symmetric branching is introduced. Relating this bifurcation theory to René Thom’s catastrophe theory, it is found that the anticlinal point of bifurcation generates an elliptic umbilic catastrophe, while the monoclinal and homeoclinal points of bifurcation lead to differing forms of the hyperbolic umbilic catastrophe. Practical structural systems which can exhibit this form of branching include an optimum stiffened plate with free edges loaded longitudinally, and an analysis of this problem is presented leading to a complete description of the imperfection-sensitivity. The paper concludes with some general remarks concerning the nature of the optimization process in design as a generator of symmetries, instabilities and possible compound bifurcations.