Elementary Catastrophe Research Articles

О классификации сингулярности в конфигурации деформации в модели жидкой капли

The deformation shape of the liquid-drop model of nuclear fission can be excellently represented by the Legendre-polynomial expansion method. Unfortunately, in the past, it has appeared necessary to include a large number of higher-order terms in order to describe the more deformed shapes encountered in the later stages of fission. In this paper, we classify the degenerate singularity that occurs in the Businaro-Gallone-Swiatecki representation of the Legendre-polynomial expansion of the deformation shape for the slightly distorted ground state. In our analysis, we found the singularity to be double-degenerate and strongly 3-determinate. As a result, we have classified it as a hyperbolic-umbilic elementary catastrophe. In order to make the potential function of the singularity a universal unfolding, we introduced two new control parameters for the liquid-drop model.

Bifurcation and hysteresis varieties for the thermal-chainbranching model with a negative modal parameter

The theory of unfoldings of singularities, or ‘elementary catastrophe theory’ (Thom(10), Poston and Stewart(9), Golubitsky and Guillemin(3), Gibson(2)) has been generalized by Golubitsky and Schaeffer(5,6), providing a powerful method for analysing imperfect bifurcation. One recent application by Golubitsky, Keyfitz, and Schaeffer(4) concerns the ‘explosion peninsula’ in chemical reactions such as that between hydrogen and oxygen. They show that the ‘thermal-chainbranching model’ of Gray and Yang(7,8) is capable of reproducing the necessary qualitative features, a result for which only heuristic arguments had previously been available.

A Lagrangian Klein bottle

Lagrangian theory (2) and elementary catastrophe theory (5) coincide locally but differ globally. Locally the singularities that occur in the two theories are the same, but globally the manifolds that occur are different. We shall show that Lagrangian manifolds are more general.

A global genericity theorem for bifurcations in variational problems

Let C be a Banach space, H a Hilbert space, and let F( C, H) be the space of C ∞ functions f: C × H → R having Fredholm second derivative with respect to x at each ( c, x) ϵ C × H for which Dƒ c(x) = 0 ; here we write ƒ c(x) for ƒ(c, x). Say ƒ is of standard type if at all critical points of ƒ c it is locally equivalent (as an unfolding) to a quadratic form Q plus an elementary catastrophe on the kernel of Q. It is proved that if f ϵ F ( A × B, H) satisfies a certain ‘general position’ condition, and dim B ⩽ 5, then for most a ϵ A the function f o ϵF( B, H) is of standard type. Using this it is shown that those fϵF( B, H) of standard type form an open dense set in F( B, H) with the Whitney topology. Thus both results are Hilbert-space versions of Thom's theorem for catastrophes in R n .

Elementary catastrophe theory modelling of an end-loading ring in a rigid cavity

The gas-tight integrity of the thin steel cylindrical liner shells of containment vessels is an important problem in nuclear engineering. To ensure the liner's integrity, it is required that buckling or deformation be avoided. Consequently, in order to estimate the load carrying capacity of the liner the load-displacement response must be investigated. To examine the instabilities of a circumferential slice of the liner shell, subjected to compressive end loading forces, elementary catastrophe theory (ECT) is used. ECT is a recently developed mathematical tool that provides conceptual insight into systems exhibiting discontinuous behavior changes. An ECT model for an elastic ring in a ridgid cavity is developed and a discussion of experimental data supporting the model is presented.

Optical bistability through nonlinear dispersion and absorption

A general theory of optical bistability for a homogeneously broadened two-level system in a Fabry-Perot interferometer is presented. Both absorptive and dispersive mechanisms are included, and in the mean-field approximation a cubic state equation is derived relating transmitted intensity to all physical parameters. From the perspective of catastrophe theory, this equation describes an elementary cusp catastrophe. We show that bistable switching may follow not only from sweeping the incident laser intensity, but also with continuous variation of atomic or cavity detuning, or atomic density, with the laser intensity fixed. For the case having incident intensity as the variable control, the domain of bistability is studied in some detail as a function of the remaining fixed parameters. Stability criteria are considered for the general case including both absorption and dispersion.

The Toy Airplane Catastrophe

A new demonstration of an elementary catastrophe using a simple machine, in the spirit of the Zeeman machine, is given.

Elementary catastrophe theory modelling of duffing's equation for seismic excitation of nuclear power facilities

Elementary catastrophe theory can provide conceptual insight into some aspects of a variety of problems in dynamics. It is a qualitative tool with some quantitative results. In this paper, it is applied to forced nonlinear vibrations of seismic disturbances, which may be approximated by Duffing's equation. The behavior of such a system fits naturally into ECT modelling, where changes in parameters of the system lead to “jump” type behavior. The important conclusion is that nonlinear oscillators can exhibit elementary catastrophes, but the design engineer may be able to manipulate characteristics of the system in order to avoid the “jump” behavior of the response.

Catastrophe of the molecular field

The molecular field theory of ferromagnetism is used as an example to illustrate the use of elementary catastrophe theory. It is shown that the molecular-field approach gives rise to a cusp catastrophe set. A brief examination is made of the reasons why the elementary catastrophe theory does not produce critical exponents which are in agreement with experimental values. It is suggested that the assumption of an equal areas rule denies the transversality upon which the generality of catastrophe theory is based. It is proposed that an approach to the problem of critical exponents through the much more sophisticated generalized catastrophes might prove fruitful.

Catastrophe theory and cellular determination, transdetermination, and differentiation.

Catastrophe theory can be used to illustrate the results of some of the events that occur during embryogenesis. Trajectories over the surface of catastrophe manifolds provide an extension of Waddington's concept of a description of embryogenesis based on chreodic paths on an epigenetic landscape. The development of Elementary Catastrophe Theory to describe time- and space-equivalent events provides a rich qualitative “language” that seems well suited for the description of cellular behavior. Space-equivalent catastrophes have been used to illustrate the processes of cellular determination, differentiation, and transdetermination with initial success. This use should be tested further, for example, by experimental manipulation of embryonic tissue at well defined intervals of time during its development. The relationship of this approach to some others and to possible future theoretical developments has been discussed.

An Introduction to Catastrophe Theory and Its Applications

This article is divided into two parts. In the first we give a description of the basic theorems of elementary catastrophe theory, along with heuristic explanations of why these theorems are valid....

Landscapes of change: Catastrophe theory and biological processes

Elementary catastrophe theory provides a method for the qualitative description of systems with associated potential energy functions. It can serve at least as an illustrative metaphor for the local description of biological development. It replaces earlier theories of development that utilized paths on an ad hoc epigenetic landscape, for example, with one that generates trajectories on a well-defined hypersurface, the catastrophe manifold. While the elementary theory does provide a much richer mathematical language than the earlier geometric theories, it suffers from several drawbacks, outlined in this paper, which prevent it from being a conclusive theory for biological development.

Reliability Analysis of the Elementary Catastrophe Theory Model of Muscle Contraction

Reliability Analysis of the Elementary Catastrophe Theory Model of Muscle Contraction