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A method, using Thom's classification of catastrophes, is described for the analysis of stability of systems whose static behavior is derived from a potential function. Examination of the stability of singular points of potential functions serves to illustrate the nature of the elementary catastrophes which also arise in nonconservative dynamical systems as well as in the static case of potential theory.

A method, using Thom's classification of catastrophes, is described for the analysis of stability of systems whose static behavior is derived from a potential function. Examination of the stability of singular points of potential functions serves to illustrate the nature of the elementary catastrophes which also arise in nonconservative dynamical systems as well as in the static case of potential theory.

A method, using Thom's classification of catastrophes, is described for the analysis of stability of systems whose static behavior is derived from a potential function. Examination of the stability of singular points of potential functions serves to illustrate the nature of the elementary catastrophes which also arise in nonconservative dynamical systems as well as in the static case of potential theory.

Scan of book An introduction to catastrophe theory by Peter Timothy Saunders b193910 [1980] {514.7--oclc} Chapter Material:- Catastrophe theory The Zeeman catastrophe machine A gravitational catastrophe machine Background [to catastrophe theory] The seven elementary catastrophes The fold The cusp The swallowtail The elliptic umbilic The hyperbolic umbilic The butterfly The parabolic umbilic Applications in physics Caustics Non-linear oscillations Applications in the social science Morphogenesis Catastrophe theory and explanation Elementary catastrophes of codimension ⫹ 5

Catastrophe theory is a recent and rich development in mathematics. However, its possible application to the behavioral sciences is uncertain and the source of considerable controversy. The theory was reviewed with emphasis on its usefulness in the behavioral sciences. Previous attempts to apply catastrophe theory were examined and criticized. A bibliography of publications in catastrophe theory was compiled, and three new investigations were designed to test the theory's applicability. A laboratory study of reversible apparent movement was the most successful of these; some aspects of the data showed properties that were in accord with expectations derived from a cusp catastrophe. Computer simulations of the behavior of hypothetical neural nets revealed selected catastrophic properties and suggested possible connections between mutually inhibitory systems and the phenomena of perceptual reversals. (Author)

Includes bibliographical references (p. 303-308) and index

Includes bibliographical references (p. 303-308) and index

Mathematical and statistical tools based on catastrophe theory have been developed and implemented as a fully functional software system which permits the capturing of data derived from analyst's perceptions of Operational Maneuver Groups related data elements and their analysis with a computer program based on statistical catastrophe theory. This system permits individuals with no mathematical background to undertake a rigorous analysis on non-linear I&W- related and other phenomena. These tools have been used in experiments with intelligence analysts involving the simulated detection of OMGs, one of the most difficult problems of tactical analysis. A set of unclassified notional indicators predicting the development of an OMG was developed and ten specific settings of these indicators were presented to intelligence analysts who were asked to assess the probability of OMG development. The resulting assessments were captured and analyzed and this revealed the existence of perceptual ambiguity as well as the potential for sudden gradual perceptual changes, perceptual hysteresis, and perceptual trapping.

We introduce a new notion of a curvature of a superconnection, different from the one obtained by a purely algebraic analogy with the curvature of a linear connection. The naturalness of this new notion of a curvature of a superconnection comes from the study of singularities of smooth sections of vector bundles (Catastrophe Theory). We demonstrate that the classical examples of obstructions to a local equivalence: exterior differential for 2-forms, Riemannian tensor, Weil tensor, curvature of a linear connection and Nijenhuis tensor can be treated in terms of one general approach. This approach, applied to the superconnection leads to a new notion of a curvature (proposed in this paper) of a superconnection.

In this work we classify the singularities obtained from the Gibbs potential of a lattice gas model with three components, two order parameters and five control parameters applying the general theorems provided by Catastrophe Theory. In particular, we clearly establish the existence of Landau potentials in two variables or, in other words, corank 2 canonical forms that are associated to the hyperbolic umbilic, D_{+4}, its dual the elliptic umbilic, D_{-4}, and the parabolic umbilic, D_5, catastrophes. The transversality of the potential with two order parameters is explicitely shown for each case. Thus we complete the Catastrophe Theory analysis of the three-component lattice model, initiated in a previous paper.

In this work we classify the singularities obtained from the Gibbs potential of a lattice gas model with three components, two order parameters and five control parameters applying the general theorems provided by Catastrophe Theory. In particular, we clearly establish the existence of Landau potentials in two variables or, in other words, corank 2 canonical forms that are associated to the hyperbolic umbilic, D_{+4}, its dual the elliptic umbilic, D_{-4}, and the parabolic umbilic, D_5, catastrophes. The transversality of the potential with two order parameters is explicitely shown for each case. Thus we complete the Catastrophe Theory analysis of the three-component lattice model, initiated in a previous paper.

Position papers and commentaries are presented covering some of the controversy surrounding applications of elementary catastrophe theory to social science in general and to organizational effectiveness in particular. Recommendations are made for research in the near and long term. (Author)

In spite of the widespread use of optical tweezers as a quantitative tool to measure small forces, there exists no unambiguous and simple experimental method for either validating its theoretically predicted form or empirically parameterizing it over the entire range. This problem is addressed by studying the transition of a colloidal particle between two spatially separated optical traps. The transition as a function of the relative intensity of the traps and the separation between them reveals a formal resemblance to the `butterfly catastrophe' which also maps onto to phase transitions observed, for example in ferroelectrics, on a phenomenological level. The method has been used to experimentally determine the force-displacement curve for an optical trap over its entire range.

The three-dimensional elastic-plastic deformation is considered. The catastrophe theory underlies the construction of this process model. It was shown that the variety of stable states consists on elastic states and can be depicted as a lattice on plane in coordinates shearing stress intensity and shearing strain intensity.

The three-dimensional elastic-plastic deformation is considered. The catastrophe theory underlies the construction of this process model. It was shown that the variety of stable states consists on elastic states and can be depicted as a lattice on plane in coordinates shearing stress intensity and shearing strain intensity.

We reanalyze the gravitating monopole and its black hole solutions in the Einstein-Yang-Mills-Higgs system and we discuss their stabilities from the point of view of catastrophe theory. Although these non-trivial solutions exhibit fine and complicated structures, we find that stability is systematically understood via a swallow tail catastrophe. The Reissner-Nordstr\"{o}m trivial solution becomes unstable from the point where the non-trivial monopole black hole appears. We also find that, within a very small parameter range, the specific heat of a monopole black hole changes its sign .

The main purpose of the current study was to provide a methodological framework for quick and efficient assessment of the vulnerability of urban spaces to earthquake. Most of the variables affecting seismic vulnerability in cities are spatial in nature, so using a geographic information system for integration of these variables can lead to desirable results. However, one of the most important subjects in overlaying of layers in a geographic information system is the uncertainty in weighting and decision making. In this study, attempts were made to modify such uncertainties to a large extent using the functions of catastrophe theory. Using the mentioned approach makes the results of the model repeatable, which is a considerable advantage. Regarding the catastrophe theory, an attempt was made to present the basic equations of this model in a concise and simple way. Varzeqan city was selected for evaluating the seismic vulnerability based on the catastrophe theory. Approximately 21 percent of the total area of Varzeqan lies in the high and very high vulnerability classes. Given that the area of the city's built-up lands is about 32 percent, considerable percentage of these lands in the city is in high and very high vulnerability classes. Distribution of these zones is impressive, especially in the center of the city. These zones can be considered the most critical regions of Varzeqan. Unstable geological formations, high residential and population density, fine grained building, low quality buildings and lack of urban open spaces have caused this region to have severe vulnerability to earthquake.

The main purpose of the current study was to provide a methodological framework for quick and efficient assessment of the vulnerability of urban spaces to earthquake. Most of the variables affecting seismic vulnerability in cities are spatial in nature, so using a geographic information system for integration of these variables can lead to desirable results. However, one of the most important subjects in overlaying of layers in a geographic information system is the uncertainty in weighting and decision making. In this study, attempts were made to modify such uncertainties to a large extent using the functions of catastrophe theory. Using the mentioned approach makes the results of the model repeatable, which is a considerable advantage. Regarding the catastrophe theory, an attempt was made to present the basic equations of this model in a concise and simple way. Varzeqan city was selected for evaluating the seismic vulnerability based on the catastrophe theory. Approximately 21 percent of the total area of Varzeqan lies in the high and very high vulnerability classes. Given that the area of the city's built-up lands is about 32 percent, considerable percentage of these lands in the city is in high and very high vulnerability classes. Distribution of these zones is impressive, especially in the center of the city. These zones can be considered the most critical regions of Varzeqan. Unstable geological formations, high residential and population density, fine grained building, low quality buildings and lack of urban open spaces have caused this region to have severe vulnerability to earthquake.

The process of the origin of life is analyzed on the basis of "self-assembly" of DNA and RNA molecules. The minimum amounts of various DNA/RNA molecules that are necessary for the nucleation and evolution of the biosphere as a whole are calculated, since life is an integral property. A cybernetic theory of evolution is presented, which considers DNA as a genetic program of an organism written in a binary system. The stages of evolution of the biosphere are given, with the calculation of the number of all species at each stage. It is shown that oxygen catastrophe is the main cause of the evolution of the biosphere and global glaciation. Moreover, the evolution of the biosphere inevitably leads to mass extinctions, which are a simple restructuring of the biosphere according to new conditions on the planet.

The process of the origin of life is analyzed on the basis of "self-assembly" of DNA and RNA molecules. The minimum amounts of various DNA/RNA molecules that are necessary for the nucleation and evolution of the biosphere as a whole are calculated, since life is an integral property. A cybernetic theory of evolution is presented, which considers DNA as a genetic program of an organism written in a binary system. The stages of evolution of the biosphere are given, with the calculation of the number of all species at each stage. It is shown that oxygen catastrophe is the main cause of the evolution of the biosphere and global glaciation. Moreover, the evolution of the biosphere inevitably leads to mass extinctions, which are a simple restructuring of the biosphere according to new conditions on the planet.

In the theory of optical gap solitons, slowly-moving finite-amplitude Lorentzian solutions are found to mediate the transition from bright to coexistent dark-antidark solitary wave pairs when the laser frequency is detuned out of the proper edge of a dynamical photonic bandgap. Catastrophe theory is applied to give a geometrical description of this strongly asymmetrical 'morphing' process.

In the theory of optical gap solitons, slowly-moving finite-amplitude Lorentzian solutions are found to mediate the transition from bright to coexistent dark-antidark solitary wave pairs when the laser frequency is detuned out of the proper edge of a dynamical photonic bandgap. Catastrophe theory is applied to give a geometrical description of this strongly asymmetrical 'morphing' process.

In the theory of optical gap solitons, slowly-moving finite-amplitude Lorentzian solutions are found to mediate the transition from bright to coexistent dark-antidark solitary wave pairs when the laser frequency is detuned out of the proper edge of a dynamical photonic bandgap. Catastrophe theory is applied to give a geometrical description of this strongly asymmetrical 'morphing' process.

In the theory of optical gap solitons, slowly-moving finite-amplitude Lorentzian solutions are found to mediate the transition from bright to coexistent dark-antidark solitary wave pairs when the laser frequency is detuned out of the proper edge of a dynamical photonic bandgap. Catastrophe theory is applied to give a geometrical description of this strongly asymmetrical 'morphing' process.

In the theory of optical gap solitons, slowly-moving finite-amplitude Lorentzian solutions are found to mediate the transition from bright to coexistent dark-antidark solitary wave pairs when the laser frequency is detuned out of the proper edge of a dynamical photonic bandgap. Catastrophe theory is applied to give a geometrical description of this strongly asymmetrical 'morphing' process.

Scan of book Catastrophe theory with Mathematica, a geometric approach by Werner Sanns b1950 [2000] {514.744--jrmul}

Scan of book Catastrophe theory with Mathematica, a geometric approach by Werner Sanns b1950 [2000] {514.744--jrmul}

We investigate neutron stars in scalar-tensor theories. We examine their secular stability against spherically symmetric perturbations by use of a turning point method. For some choices of the coupling function contained in the theories, the number of the stable equilibrium solutions changes and the realized equilibrium may change discontinuously as the asymptotic value of the scalar field or total baryon number is changed continuously. The behaviour of the stable equilibrium solutions is explained by fold and cusp catastrophes. Whether the cusp catastrophe appears or not depends on the choices of the coupling function. These types of the catastrophes are structurally stable. Recently discovered spontaneous scalarization, which is non-perturbative strong-field phenomenon due to the presence of the gravitational scalar field, is well described in terms of the cusp catastrophe.

We investigate neutron stars in scalar-tensor theories. We examine their secular stability against spherically symmetric perturbations by use of a turning point method. For some choices of the coupling function contained in the theories, the number of the stable equilibrium solutions changes and the realized equilibrium may change discontinuously as the asymptotic value of the scalar field or total baryon number is changed continuously. The behaviour of the stable equilibrium solutions is explained by fold and cusp catastrophes. Whether the cusp catastrophe appears or not depends on the choices of the coupling function. These types of the catastrophes are structurally stable. Recently discovered spontaneous scalarization, which is non-perturbative strong-field phenomenon due to the presence of the gravitational scalar field, is well described in terms of the cusp catastrophe.

We investigate neutron stars in scalar-tensor theories. We examine their secular stability against spherically symmetric perturbations by use of a turning point method. For some choices of the coupling function contained in the theories, the number of the stable equilibrium solutions changes and the realized equilibrium may change discontinuously as the asymptotic value of the scalar field or total baryon number is changed continuously. The behaviour of the stable equilibrium solutions is explained by fold and cusp catastrophes. Whether the cusp catastrophe appears or not depends on the choices of the coupling function. These types of the catastrophes are structurally stable. Recently discovered spontaneous scalarization, which is non-perturbative strong-field phenomenon due to the presence of the gravitational scalar field, is well described in terms of the cusp catastrophe.

We investigate neutron stars in scalar-tensor theories. We examine their secular stability against spherically symmetric perturbations by use of a turning point method. For some choices of the coupling function contained in the theories, the number of the stable equilibrium solutions changes and the realized equilibrium may change discontinuously as the asymptotic value of the scalar field or total baryon number is changed continuously. The behaviour of the stable equilibrium solutions is explained by fold and cusp catastrophes. Whether the cusp catastrophe appears or not depends on the choices of the coupling function. These types of the catastrophes are structurally stable. Recently discovered spontaneous scalarization, which is non-perturbative strong-field phenomenon due to the presence of the gravitational scalar field, is well described in terms of the cusp catastrophe.

We investigate neutron stars in scalar-tensor theories. We examine their secular stability against spherically symmetric perturbations by use of a turning point method. For some choices of the coupling function contained in the theories, the number of the stable equilibrium solutions changes and the realized equilibrium may change discontinuously as the asymptotic value of the scalar field or total baryon number is changed continuously. The behaviour of the stable equilibrium solutions is explained by fold and cusp catastrophes. Whether the cusp catastrophe appears or not depends on the choices of the coupling function. These types of the catastrophes are structurally stable. Recently discovered spontaneous scalarization, which is non-perturbative strong-field phenomenon due to the presence of the gravitational scalar field, is well described in terms of the cusp catastrophe.

Elementary catastrophe theory. Monografii Matematice 24, Universitatea din Timisoara, (1985). Lecture Notes Collection FreeScience.info ID1090 Obtained from http://www.mat.univie.ac.at/~michor/catastrophes.pdf http://www.freescience.info/go.php?pagename=books&id=1090

Elementary catastrophe theory. Monografii Matematice 24, Universitatea din Timisoara, (1985). Lecture Notes Collection FreeScience.info ID1090 Obtained from http://www.mat.univie.ac.at/~michor/catastrophes.pdf http://www.freescience.info/go.php?pagename=books&id=1090

Bibliography: p. 18-19

Bibliography: p. 18-19

A nonlinear differential equation model and its associated catastrophe is shown to model the simplest version of the sexual response of humans. The mathematical model is derived via well-known and non-controversial aspects of sexual orgasm as can be found in the literature. Two different and independent derivations of the equation are given; the equilibrium-oscillation model and the slow-fast dual equation model.

Cover title

We make an analysis of Q-balls and boson stars using catastrophe theory, as an extension of the previous work on Q-balls in flat spacetime. We adopt the potential $V_3(\phi)={m^2\over2}\phi^2-\mu\phi^3+\lambda\phi^4$ for Q-balls and that with $\mu =0$ for boson stars. For solutions with $|g^{rr}-1|\sim 1$ at its peak, stability of Q-balls has been lost regardless of the potential parameters. As a result, phase relations, such as a Q-ball charge versus a total Hamiltonian energy, approach those of boson stars, which tell us an unified picture of Q-balls and boson stars.

We make an analysis of Q-balls and boson stars using catastrophe theory, as an extension of the previous work on Q-balls in flat spacetime. We adopt the potential $V_3(\phi)={m^2\over2}\phi^2-\mu\phi^3+\lambda\phi^4$ for Q-balls and that with $\mu =0$ for boson stars. For solutions with $|g^{rr}-1|\sim 1$ at its peak, stability of Q-balls has been lost regardless of the potential parameters. As a result, phase relations, such as a Q-ball charge versus a total Hamiltonian energy, approach those of boson stars, which tell us an unified picture of Q-balls and boson stars.

We re-analyze the globally neutral non-Abelian black holes and present a unified picture, classifying them into two types; Type I (black holes with massless non-Abelian field) and Type II (black holes with ``massive" non-Abelian field). For the Type II, there are two branches: The black hole in the high-entropy branch is ``stable" and almost neutral, while that in the low entropy branch, which is similar to the Type I, is unstable and locally charged. To analyze their stabilities, we adopt the catastrophe theoretic method, which reveals us a universal picture of stability of the black holes. It is shown that the isolated Type II black hole has a fold catastrophe structure. In a heat bath system, the Type I black hole shows a cusp catastrophe, while the Type II has both fold and cusp catastrophe.

A close relationship is demonstrated between the catastrophe theory, Petrov's algebraic classification of the gravitational fields and the theory of phase transitions.

Various scenarios of the initial inflation of the universe are distinguished by the choice of a scalar field {\em potential} $U(\phi)$ which simulates a {\it temporarily} non--vanishing {\em cosmological term}. Our new method, which involves a reparametrization in terms of the Hubble expansion parameter $H$, provides a classification of allowed inflationary potentials and of the stability of the critical points. It is broad enough to embody all known {\it exact} solutions involving one scalar field as special cases. Inflation corresponds to the evolution of critical points of some catastrophe manifold. The coalescence of its nondegenerate critical points with the creation of a degenerate critical point corresponds the reheating phase of the universe. This is illustrated by several examples.

We analyze the thermodynamical potential of a lattice gas model with three components and five parameters using the methods of Catastrophe Theory. We find the highest singularity, which has codimension five, and establish its transversality. Hence the corresponding seven-degree Landau potential, the canonical form Wigwam or $A_6$, constitutes the adequate starting point to study the overall phase diagram of this model.

Chebyshev maps in the complex plane are typical chaotic maps. Veselov generalized these map. We consider a class of those maps and view them as holomorphic endomorphisms on the 3-dimensional complex projective space and make use of the theory of complex dynamics in higher dimension. We determine Julia sets and external rays of those maps, exactly. In the Julia sets, Moebius strip and a special ruled surface appear. We also show some relation between those maps and binary quartic forms.

The uniaxial elastic-plastic deformation process is considered. Mathematical model of this process was built. According to this model all stable static states form the lattice, which is called the delta-lattice.

We introduce the basic concepts of catastrophe theory needed to derive analytically the phase diagram of the proton-neutron interacting boson model (IBM-2). Previous studies [1,2,3] were based on numerical solutions. We here explain the whole IBM-2 phase diagram including the precise order of the phase transitions in terms of the cusp catastrophe.

In diffraction catastrophes such as the rainbow the wave nature of light resolves ray singularities and draws delicate interference patterns. In quantum catastrophes such as the black hole the quantum nature of light resolves wave singularities and creates characteristic quantum effects related to Hawking radiation. The paper describes the theory behind a recent proposal [U. Leonhardt, arXiv:physics/0111058, Nature (in press)] to generate a quantum catastrophe of slow light.

An analytical description of arbitrary strongly aberrated axially symmetric focusing is developed. This is done by matching the solution of geometrical optics with a wave pattern which is universal for the underlying ray structure. The corresponding canonical integral is the Bessoid integral, which is a three dimensional generalization of the Pearcey integral that approximates the field near an arbitrary two-dimensional cusp. We first develop the description for scalar fields and then generalize it to the vector case. As a practical example the formalism is applied to the focusing of light by transparent dielectric spheres with a few wavelengths in diameter. The results demonstrate good agreement with the Mie theory down to Mie parameters of about 30. Compact analytical expressions are derived for the intensity on the axis and the position of the diffraction focus both for the general case and for the focusing by microspheres. The high intensity region is narrower than for an ideal lens of the same aperture at the expense of longitudinal localization and has a polarization dependent fine structure, which can be explained quantitatively. The results are relevant for aerosol and colloid science where natural light focusing occurs and can be used in laser micro- and nano-processing of materials.