Showing papers on "Catastrophe theory published in 1980"

An Introduction to Catastrophe Theory

Abstract: Almost every scientist has heard of catastrophe theory and knows that there has been a considerable amount of controversy surrounding it. Yet comparatively few know anything more about it than they may have read in an article written for the general public. The aim of this book is to make it possible for anyone with a comparatively modest background in mathematics - no more than is usually included in a first year university course for students not specialising in the subject - to understand the theory well enough to follow the arguments in papers in which it is used and, if the occasion arises, to use it. Over half the book is devoted to applications, partly because it is not possible yet for the mathematician applying catastrophe theory to separate the analysis from the original problem. Most of these examples are drawn from the biological sciences, partly because they are more easily understandable and partly because they give a better illustration of the distinctive nature of catastrophe theory. This controversial and intriguing book will find applications as a text and guide to theoretical biologists, and scientists generally who wish to learn more of a novel theory.

Connectivity, Complexity, and Catastrophe in Large-Scale Systems

Abstract: This book represents an approach to large-scale system modeling that is a challenging synthesis for the systems analyst, the operations research worker, the system theorist, the policy analyst, and the student of social systems. After pointing out that the mathematical form of a system description dictates the types of questions that can be asked and answered by the model, the author declares that "there is no such thing as a model system: there are many models, each with its own characteristic mathematical features and each capable of addressing a certain subset of important questions about the system and its operation". The book supports this point with examples from a wide spectrum of contexts (such as physics, economic activity, water-resource management, ecology, transportation, and physiology) viewed from the points of view of various models and theories (such as general system theory, control theory, graph theory, linear and nonlinear system theory, and catastrophy theory). Against this broad background, the book then considers in depth the relations to large-scale systems of the theories of connectivity, complexity, stability, catastrophy, and resilience.

The Stokes set of the cusp diffraction catastrophe

Abstract: Stokes and anti-Stokes lines are familiar in the asymptotic approximation of functions of a complex variable. The author generalises this notion and define the Stokes and anti-Stokes sets of a complex function of many (possibly complex) variables, defined by a diffraction-type integral. They are subsets of the Maxwell set of catastrophe theory, extended to complex variables. On the Stokes set the number of complex stationary points contributing to the integral changes by one, whereas on the caustic the number of real stationary points changes by two. Knowledge of the location of the Stokes set is essential to perform a full stationary phase analysis of a diffraction integral, and it imposes a constraint upon the positions of wavefront dislocations. For the canonical cusp diffraction catastrophe the author finds the explicit equation of the Stokes set, which is a broadened mirror image of the cusp caustic.

On the existence of hysteresis in a transition to chaos after a single bifurcation

Abstract: We study and classify discontinuous transitions to chaotic behaviour occurring after a single bifurcation in real endomorphisms. This classification which can be extended to more realistic dynamical systems, relies on the possible existence of the hysteresis mechanism. We discuss the intermittency phenomenon as an evidence for transitions without hysteresis.

Bifurcation analysis of aircraft high angle-of-attack flight dynamics

Abstract: A new approach is presented for analyzing nonlinear and high-a dynamic behavior and stability of aircraft. This approach involves the application of bifurcation analysis and catastrophe theory methodology (BACTM) to s pecific phenomena such as stall, departure, spin entry, flat and steep s pin, nose slice, and wing rock. Quantitative results of a global nature are presented, using numerical techniques based on parametric continuation. It is shown how BACTM provides a complete representation of the aircraft equilibrium and bifurcation surfaces in the s tate-control space, u sing a rigid body model and aerosurface controls. Also presented is a particularly useful extension of continuation methods to the d etection and stability analysis of stable a ttracting orbits (limit cycles). The use of BACTM for understanding high-a phenomena, especially spin-related behavior, is discussed.

Estimation Theory for the Cusp Catastrophe Model

Abstract: The cusp model of catastrophe theory is very closely related to certain multiparameter exponential families of probability density functions. This relationship is exploited to create an estimation theory for the cusp model. An example is presented in which an independent variable has a bifurcation effect on the dependent variable.

The influence of external fluctuations on self-sustained temporal oscillations

Abstract: The influence of external fluctuations on the bifurcational behavior of two-dimensional dynamical systems exhibiting limit cycles is investigated. Studying both exactly and approximately solvable examples it is shown that the variances of the external fluctuations occur as additional bifurcation parameters. The threshold values for soft as well as for hard self-excitation of oscillations are affected by the external fluctuations. To classify bifurcations of dynamical systems in the presence of fluctuations some aspects of catastrophe theory are applied to the corresponding stationary probability distributions.

An introduction to Catastrophe Theory : Applications in the social sciences

Abstract: The applications in this chapter represent the opposite end of the spectrum to the physical systems of Chapter 5. When we are trying to analyse the behaviour of an individual, or of a group, we cannot write down a set of equations of motion for the system based on known quantitative laws, and then look to see what catastrophe theory has to say about the solutions of these equations. What we must do is quite different. If we observe in a system some or all of the features which we recognize as characteristic of catastrophes – sudden jumps, hysteresis, bimodality, inaccessibility and divergence – we may suppose, at least as a working hypothesis, that the underlying dynamic is such that catastrophe theory applies. We then choose what appear to be appropriate state and control variables and attempt to fit a catastrophe model to the observations. Right from the start, we see one of the advantages of catastrophe theory in this sort of problem. The data which are available are often not quantifiable. We can generally order our observations; for example, we can tell whether a person has become more angry or less angry. And we can usually say whether or not a variable is continuous and whether it changes smoothly. On the other hand, algebraic concepts such as addition and multiplication generally have no meaning: it makes little real sense to say that someone has become twice as angry.

A Geometrical Model of Piagetian Conservation

Abstract: A geometrical model is developed of behavior in a Piagetian conservation task. The model, which is based on Thom's catastrophe theory, encompasses the discontinuities in cognition found in such tasks and describes the changes in behavior as the subject progresses through the stages of intuitive, transitional, and concrete operational thinking.

The use of catastrophe theory to analyse the stability and toppling of icebergs

Abstract: As an iceberg melts, the resulting change of shape can cause it to list gradually or to become unstable and topple over suddenly. Similarly, when an iceberg breaks up some of the individual pieces may capsize. We have used Zeeman’s analysis of the stability of ships, which is based on catastrophe theory, to examine this problem. We deal only with statical equilibrium; dynamical effects induced by water motion are important for ships, but very large icebergs have correspondingly small oscillations and therefore dynamical aspects are ignored in this first study. The advantage of the catastrophe-theory approach over the conventional stability theory used by naval architects lies in the conceptual clarity that it provides. In particular, it gives a three-dimensional geometrical picture that enables one to see all the possible equilibrium attitudes of a given iceberg, whether they are stable or unstable, whether a stable attitude is dangerously close to an unstable one, and how positions of stable equilibrium can be destroyed as the shape of the iceberg evolves with time. By making two-dimensional computations we examine the stability of two different shapes of cross-section, rectangles and trapezia, with realistic density distributions. These shapes may list gradually or topple suddenly as a single parameter is changed. For example, we find that a conversion of the vertical sides of a rectangular section into the slightly inward-sloping sides of a trapezium has a comparatively large adverse effect on stability. The main purpose of this work is to suggest how the stability characteristics of any selected iceberg may be investigated systematically.

Catastrophe theory and dynamic games

Abstract: The notion of a dynamic system is fundamental in many of the natural sciences and in some social or behavioral sciences, such as ecology or economics. The qualitative theory of dynamic systems, sometimes called catastrophe theory, has recently been used to construct models of systems which can exhibit discontinuous behavior. Of more significance perhaps than the construction of simple models are the fundamental theoretical concepts of genericity and structural stability. Although simple dynamic models, such as the prey predator model, are well behaved \"almost always,\" more complex systems can display quite exotic behavior. Indeed simple difference equations, models of population growth, inter species competit ion and duopoly competition can display the form of indeterminacy known as chaos. The concern of this paper is to use the conceptual notions of genericity and structural stability to provide some hints as to tile possible structural features of a political economy. The behavior of a political economic system is the result o f optimization by collections, or coalitions, o f actors in the society. The emphasis of the theory of dynamic games presented here is on local behavior rather than tile global notions o f game theory. In voting games, for example, it is now known that the set o f natural equilibria, the core, will generally he empty. Instead we consider the phenomena of cycling. For voting games the cycle set will be dense, and consequently we may regard such games as chaotic. While the notion of natural equilibria is fundamental in economic and political theory, these results suggest the possibility that behavioral systems may in fact be highly unpredictable and indeterminate.

Catastrophe Theory and Human Evolution

Abstract: Thom's Theory of Catastrophes (1975) provides models for the potential resolution of several issues in the study of hominid evolution. This paper examines the problems of australopithecine divergence and of Neanderthal differentiation in terms of Thom's cusp catastrophe model. The cusp model provides for a resolution of the differences between theories of gradual or "instantaneous" evolutionary change and can account for apparent discontinuities between hominid forms.

Application of catastrophe theory to nuclear structure

Abstract: Three two-parameter models, one describing an A-body system (the atomic nucleus) and two describing many-body systems (the van der Waals gas and the ferroelectric (perovskite) system) are compared within the framework of catastrophe theory. It is shown that each has a critical point (second-order phase transition) when the two counteracting forces controlling it are in balance; further, each undergoes a first-order phase transition when one of the forces vanishes (the deforming force for the nucleus, the attractive force for the van der Waals gas, and the dielectric constant for the perovskite). Finally, when both parameters are kept constant, a kind of phase transition may occur at a critical angular momentum, critical pressure, and critical electric field. 3 figures, 1 table.

Discontinuity, decision and conflict

Abstract: The motivation for this paper arises out of the authors experiences in modelling real decision makers where the decisions show not only a continuous response to a continuously changing environment but also sudden or discontinuous changes. The theoretical basis involves a parametric characterisation of the environment, a decision makers perception of it in terms of a twice differentiable Distribution Function and a bounded Loss Function. Under a specified, minimizing dynamic, the resultant Expected Loss Function satisfies the conditions for a potential function and Thoms Catastrophe Classification Theorem may be used to assess the singularity points and the thresholds at which jump decisions are taken. The paper describes the theory, summarises some results on unimodal distributions illustrated by jump decisions and population polarisation. Mixture distributions are then examined and the E* models defined. These are then briefly illustrated by reference to models which have been constructed in relation to Prison Riots, Agricultural and Economic modelling.

Education and Development BIFURCATIONS IN DYNAMICAL SYSTEMS

Abstract: The concept of bifurcation in a dynamical system is illustrated through several examples. The elementary catastrophes are equilibrium sets of specific kinds of dynamical systems. The framework of bifurcating dynamical systems is put forward as an alternative to catastrophe theory models. Dynamical systems models require explicit accounting of all their underlying hypotheses. They offer unambiguous answers in return. The fuzzy transition from hypotheses to results so characteristic of the catastrophe theory literature is absent. The method is illustrated using a stock-market model.

Mathematical Preliminaries to Elementary Catastrophe Theory

Abstract: The purpose of this article is to introduce the nonspecialist to some of the mathematical concepts of Elementary Catastrope Theory, concepts which often occur in the form of jargon, or allusions, in the literature. This situation has helped fuel the controversy over the uses and misuses of catastrophe theory, and therefore it may be prudent to acknowledge immediately that, for this article, Elementary Catastrophe Theory is the study of singularities of parametrized families of C real-valued functions. We shall henceforth refer to these ideas as simply "Catastrophe Theory," although other interpretations are sometimes accorded the latter phrase, depending on the user's affiliations in the current controversy. Fortunately, the author of an introductory article need not take sides (there being little at stake) and can extol with impunity Christopher Zeeman's and Rene Thor's many interesting examples (if not "models") and speculations (see [1], [8], [9]). At the same time, the recent remarkable book by Tim Poston and Ian Stewart [6] need not compromise the most scrupulous mathematician, and contains a wealth of hard applications. The paper [5] by Martin Golubitsky is an excellent introduction to the subject at a more advanced level.

Estimation Theory for the Cusp Catastrophe Model

Abstract: The cusp model of catastrophe theory is very closely related to certain multiparameter exponential families of probability density functions. This relationship is exploited to create an estimation theory for the cusp model. An example is presented in which an independent variable has a bifurcation effect on the dependent variable.

Is Geothermal Simulation a "Catastrophe"?

Abstract: All numerical simulators of geothermal reservoirs depend upon an accurate representation of the thermodynamics of steam-water systems. These relationships are required to render tractable the system of balance equations derived from the physics of flow through porous media. While it is generally recognized that the steam-water system (i.e. two phase) is not in thermodynamic equilibrium, equihbrium thermodynamics are employed in its description. In this paper, we present an alternative view based on non-equilibrium thermodynamics. The underpinnings of this approach are found in a branch of topology generally referred to as "catastrophe theory". [Thom, 1975]