Hyperbolic Umbilic Catastrophe Research Articles

An asymptotic expansion of the hyberbolic umbilic catastrophe integral

We obtain an asymptotic expansion of the hyperbolic umbilic catastrophe integral Psi ^{(H)}(x,y,z):= int _{-infty }^{infty }int _{-infty }^{infty }exp (i(s^3+t^3+zst+yt+xs))mathrm{{d}}s,mathrm{{d}}t for large values of |x| and bounded values of |y| and |z|. The expansion is given in terms of Airy functions and inverse powers of x. There is only one Stokes ray at arg x=pi . We use the modified saddle point method introduced in (López et al. J Math Anal Appl 354(1):347–359, 2009). The accuracy and the asymptotic character of the approximations are illustrated with numerical experiments.

Distorted mirror images organised by cuspoid and umbilic caustics

The anatomy of the distorted and topologically disrupted images of an extended object after reflection by curved mirrors, and of their refraction analogues, e.g. in gravitational lensing, is determined by the caustic surfaces enveloping the ray family issuing from each point of the object. Simulations are presented of reflections of the same object when organised by each of the five caustics (elementary catastrophes) that are stable in three-dimensional space: the fold, cusp, swallowtail, elliptic and hyperbolic umbilic catastrophes. The object is a picture of a face, whose familiar features make the image distortions and disruptions easier to understand. An important distinction is between mirrors curved in one direction and those curved in two.

Instabilities of Elastic and Spinning Systems: Concepts and Phenomena

Bifurcational instabilities of increasing complexity are described, with instructive and intriguing applications. Multiple folds in the gravitational collapse of a star triggers a discussion about possible sequencing of folds. Columns, rings and a quirky floating log illustrate safe and dangerous pitchforks, with self-weight models as lecture demonstrations. Localized buckling is employed by biologists using magnetic tweezers to twist a DNA molecule and evaluate a cutting and de-knotting enzyme. Unfolding of complex bifurcations into “structurally-stable” forms by a number (codimension) of perturbations illuminates the imperfection-sensitivity of shells, and the interactive buckling of a guyed mast at a hyperbolic-umbilic catastrophe. Spinning of primitive planets introduces secondary bifurcations, while Kirchhoff’s analogy is displayed for rods and spinning tops. Examples of gyroscopic stabilization and levitation mimic the “traps” that are used in modern experimental physics to hold in place individual electrons and neutrons.

Spheroidal droplet measurements based on generalized rainbow patterns

The character of the rainbow from a droplet is investigated experimentally and theoretically. In the experiment, light scattering from spheroidal droplets in the vicinity of the primary rainbow region has been observed to contain a variety of characteristic interference patterns which are a generalization of the rainbow from a sphere. These patterns start from being a fold rainbow, change to transverse cusp caustics and then to hyperbolic umbilic catastrophe as the aspect ratio of the droplet increases. A comparison of the observed rainbow patterns in the horizontal equatorial plane with those of Airy theory reveals that these patterns can be used for characterizing droplet, in particular for determining the refractive index and the diameter of the droplet in the equatorial plane. The absolute error of the refractive index is smaller than 1.5×10−4. The absolute relative error of the equatorial diameter is less than 5%.

Duality between phase vortices and Argand-plane caustics

Singularities in the mappings of complex two-dimensional vortical wavefields to Argand space are used here to investigate the duality between vortices in the physical field and caustic-like structures in the Argand plane. Such map singularities are exemplified via a method of optical vortex generation based on low-pass filtering of two-dimensional noise. The presence of Argand-plane fold caustics crossing the Argand-plane origin is tied to changes in vortex numbers in the physical wavefield. Increasing the low-pass filter cutoff leads to higher-order Argand-plane caustic singularities, such as the hyperbolic umbilic catastrophe. This work presents a duality between vortices and caustics, explores the Argand plane as a tool in analysing optical wavefields, and provides a new environment for studying catastrophes.

Asymptotics of the Wigner 9j-symbol

We present the asymptotic formula for the Wigner 9j-symbol, valid when all quantum numbers are large, in the classically allowed region. As in the Ponzano–Regge formula for the 6j-symbol, the action is expressed in terms of lengths of edges and dihedral angles of a geometrical figure, but the angles require care in definition. Rules are presented for converting spin networks into the associated geometrical figures. The amplitude is expressed as the determinant of a 2 × 2 matrix of Poisson brackets. The 9j-symbol possesses caustics associated with the fold and elliptic and hyperbolic umbilic catastrophes. The asymptotic formula obeys the exact symmetries of the 9j-symbol.

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.

Evolution of the hyperbolic umbilic diffraction pattern from Airy rings

If the diffraction pattern of the hyperbolic umbilic diffraction catastrophe is produced by an optical system of increasing aperture, it passes continuously from the two-dimensional system of Airy rings in the focal plane, made by a very small aperture, to the full three-dimensional pattern corresponding to infinite aperture. The paper studies this transition by examining the truncated diffraction integral and following the evolution of the wave dislocation lines (phase singularities) on which the pattern is based. The seed of the evolution from a two-dimensional to a three-dimensional pattern turns out to be already present asymptotically even for the smallest aperture: namely, a column of small dislocation rings very close to the axis that stream in procession towards the focal plane, and become dislocations lying in the Airy fringe surfaces that run parallel to the main fold caustic, only to disappear ultimately by retreat to infinity. The evolution into the final dislocation pattern takes place by a sequence of primitive local topological events, such as reconnection (hyperbolic interchange) and ring creation.

Doubly focused backscattering from finite targets in an Airy caustic formed by a curved reflecting surface

Caustics can be formed in the water column when sound scatters off a curved-reflecting surface such as the ocean floor or surface. The simplest caustic is an Airy caustic formed by the merging of two rays. Small targets lying in or near Airy caustics have backscattered echoes that can be focused both to the target and upon return. For a point target, the doubly focused backscattering amplitude is proportional to the square of an Airy function whose argument depends on the target location through the changes in relative return times of contributing rays. For a finite sized target, the symmetry is broken and the amplitude unfolds into a hyperbolic umbilic catastrophe. The arguments for the hyperbolic umbilic function are calculated using the relative return times of transient echoes. These doubly focused echoes can lead to amplitudes larger than that of direct or singly focused echoes (echoes which focus once, either to the target or upon return). Experiments using a cylindrical half-pipe as a reflecting surface confirm these predictions.

Doubly-focused echos from spheres unfold into a hyperbolic umbilic diffraction catastrophe

An underwater spherical target resides in an Airy field formed by reflection off a curved surface representing the sea floor or sea surface. In prior work [B. Dzikowicz and P. L. Marston, J. Acoust. Soc Am. 110, 2778 (2001)] direct returns of a tone burst from the surface reflection focused toward the target were shown to have a dependence on the target position described by an Airy function. The return echo can also be focused again by the surface onto the source and receive transducer. This gives the square of an Airy function for the case of a point target. With a finite sized target (as in the experiment) this goes over to a hyperbolic umbilic catastrophe with symmetric arguments. The arguments of the hyperbolic umbilic function are derived from only the relative return times of a transient pulse. Experiment confirms the predicted merging of transient echoes in the time domain, as well as the hyperbolic umbilic diffraction integral amplitudes for a tone burst. This method would allow for the observation of a target at a greater distance in the presence of a focusing surface. [Research supported by ONR.]

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.

Light scattering from laser-damaged speckled surfaces

Scattering from an initially plane surface of a thin Plexiglas plate damaged by an impinging coherent light beam of a laser is considered. The morphology and the characteristic dimensions of the damaged speckles was studied during the evolution of the phenomenon by interferometry and the creation of diffraction patterns. The cusped interferogram created by superposition of evoluting two initial diffraction patterns, created during successive damage steps, forms caustics corresponding to hyperbolic umbilic catastrophes. An experimental study of the evolution of the damage phenomenon clarifies the mechanisms of elastic and plastic deformations of the affected zone.

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

Observation of elementary catastrophes in surface scattering

An experimental study is presented demonstrating elementary catastrophes in surface scattering for the first time. The system studied is 15–60 eV Na scattered at normal incidence from a clean and cesium-covered Ag(111) surface. Fold and hyperbolic umbilic catastrophes were detected in the intensity distributions measured as a function of the final polar and azimuthal angle. Because of the purely classical nature of the scattering, the origin of the catastrophes is easily traceable.

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.

Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci

An experiment by Marston and Trinh1, showed how an oblate spheroidal water drop illuminated by a parallel beam of light forms a caustic in the far field characterized by the hyperbolic umbilic catastrophe—a generalization of the primary rainbow formed by a spherical drop. As the axis-ratio of the drop was changed, they observed a singular section of the catastrophe appearing for D/H = 1.305 ± 0.016—D being the diameter in the horizontal equatorial plane and H the diameter along the vertical axis of rotational symmetry. I show here how this ratio may be calculated to confirm quantitatively the arguments1 that the hyperbolic–umbilic catastrophe is associated with skew rays, and predict the occurrence of novel ‘lips’ phenomena, yet to be observed.

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.

Critical Imperfection Territory∗

ABSTRACT The critical limit load of elastic structures can decrease due to the effect of unavoidable imperfections. The “critical imperfection territory” covers all imperfections resulting in a value of the critical load that is smaller than a prescribed value. These territories are determined with the help of a potential function and by using results of catastrophe theory. General rules for their determination are outlined and the specific critical imperfection territories are shown for the most important cases (fold, cusp, and elliptic and hyperbolic umbilic catastrophes). These territories give information similar to that given by imperfection-sensitivity surfaces, but they use a space of one less dimension.

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...