Catastrophe Point Research Articles

Thin current sheets in the magnetotail and the loss of equilibrium

Using two‐dimensional magnetohydrostatic equilibium theory, together with flux, entropy, and topology conservation, we demonstrate that a finite boundary deformation of magnetotail equilibria can lead to locally infinite current density enhancement, that is, the formation of a singular current sheet. Equilibrium configurations that satisfy the constraints cease to exist when the boundary deformation exceeds the critical value. This provides a strong argument for the onset of instability or the loss of equilibrium, regardless of the dissipation mechanism. The major feature causing the current density increase appears to be the localized increase of the boundary flaring angle, rather than the absolute magnitude of the boundary deformation. Such boundary perturbations might be caused by a solar wind pressure variation or suitable changes of the boundary electric field. Catastrophe points can be reached even for relatively small perturbations of the boundary. In those cases the excess energy associated with the thin current sheet is small. Nevertheless, a major disruption may be triggered when the underlying configuration becomes unstable upon the local breakdown of ideal MHD.

Chaos, catastrophe, and critical frequency points on elastic shells

Exact equations exist for the determination of the backscattered signals excited by acoustic signals on submerged elastic shells. In the absence of resonances these signals may be cast in terms of conservation principles of inertial components of the interaction of the signal with the object. This indeed leads to an adequate description of the acoustic background of such targets. What is determined to be critical is the inclusion of an entrained mass related to the displacement of the mass equivalent of the fluid due to the movement of the object. But what happens to such inertial effects in the presence of elastic resonances? We explore this issue and demonstrate that the effective or inertial component ranges from zero to infinity over a small frequency range that characterizes the resonance width. Over this range there is also a phase shift of 1800. We may as well examine the effect of radiation loading in this range. It is also possible to investigate a phase plot of the real and imaginary components of the effective mass factor which leads to closed trajectories for certain classes of resonances and to hyperbolic trajectories for other types. The meaning of these observations is discussed. [Work supported by NRL and ONR.]

On the application of numerical continuation methods to the calculation of magnetostatic equilibria

Sequences of magnetostatic equilibria can often be used to model the quasi-static pre-eruptive energy storage phase of eruptive phenomena in e.g. Earth's magnetosphere or the solar corona. During these phases the systems evolve only due to slow changes in their environment, being practically in equilibrium on large scales. The eruption onset would then be identified with a bifurcation or catastrophe point in the solution diagram. Different energy storage mechanisms can be studied by different parameterizations of e.g. the boundary conditions. Also from the more fundamental point-of-view of the theory of dynamical systems, studying the possible stationary states and the bifurcation properties of plasma systems should be the first step towards a more thorough understanding of their full dynamical behaviour. In any case one will have to solve highly non-linear partial differential equations with the possibility of the existence of multiple solutions (or of none at all) for a given set of boundary conditions. Such problems can, in general, only be solved numerically. The most appropriate class of numerical algorithms for this type of problem are continuation methods which can calculate complete solution branches and detect bifurcation points. In this work a numerical bifurcation code based on a continuation method is presented. In addition to solving the non-linear magnetohydrostatic equations, the code can check a sufficient linear stability criterion for each solution. Some preliminary results for simple magnetohydrostatic equilibria are presented and potential future applications are discussed.

From weak discontinuity to gradient catastrophe

The Cauchy problem for a quasilinear parabolic equation with small parameter at the highest derivative is considered in the case when the solution of the degenerate equation has a weak discontinuity subsequently turning into a strong discontinuity. The singularities that the coefficients of the asymptotic formula representing the solution in the boundary layer of the weak discontinuity develop on approaching the point of the gradient catastrophe are analysed.

Equatorial pacific SSTA-related decadal variations of potential predictability of ENSO and interannual climate

Based on analysis of NCEP reanalysis data and SST indices of the recent 50 years, decadal changes of the potential predictability of ENSO and interannual climate anomalies were investigated. Autocorrelation of Nino3 SST anomalies (SSTA) and correlation between atmospheric anomalies fields and Nino3 SSTA exhibit obvious variation in different decades, which indicates that Nino3 SSTA-related potential predictability of ENSO and interannual climate anomalies has significant decadal changes. Time around 1977 is not only a shift point of climate on the interdecadal time scale but also a catastrophe point of potential predictability of ENSO and interannual climate. As a whole, ENSO and the PNA pattern in boreal winter are more predictable in 1980s than in 1960s and 1970s, while the Nino3 SSTA-related potential predictability of the Indian monsoon and the East Asian Monsoon is lower in 1980s than in 1960s and 1970s.

Catastrophes, bifurcations and hysteretic loops in torsional potentials of internal rotations in molecules

Three approaches to the calculation of torsional potentials V ϕ of internal rotations in molecules, namely rigid rotation, relaxed rotation and reaction path, are discussed. The rigid-rotation approach produces V ϕ Values that are anomaly free but insufficiently accurate in general. In systems in which the hindrances that impede the internal rotation can be relieved or entirely avoided at relatively low cost in energy by relaxation of the other degrees of freedom, catastrophe points and hysteretic loops may be present in the relaxed-rotation V ϕ. Although V ϕ obtained within the reaction-path approach are guaranteed to be continuous and to encompass all the relevant minima and transition states, they may possess bifurcation points. These anomalies all stem from the projection of the multivariate total energy U(R) onto the univariate V ϕ. Their presence in V ϕ of such simple systems as 2,6,2',6'-tetrabromobiphenyl and oxamide raises questions about the general validity and usefulness of the concept of torsional potentials.

Regulation of wound healing from a connective tissue perspective

Tissue injury resulting in irreversible tissue loss initiates the repair process. The restoration of dermal loss is by scarring, where a new cell population resides in a new connective tissue matrix. The chemical composition of a scar is similar to normal dermis, but the organization of that tissue is altered. The inability of the organism to reassemble collagen into a normal dermal pattern is an attribute of a scar, but in most cases it restores normal function. With impaired scarring, wound dehiscence or chronic wounds arise, whereas the overproduction of scar tissue results in keloid or hypertrophic scarring. In both situations a catastrophic end point occurs. The volume of scar tissue deposited, as well as its organization, is critical for determining the scar's integrity, stability, and restoration of function. The maturation of scar depends on the character of its resident cell populations, the quality of deposited connective tissue, and the interactions between those components. In this Perspective article, the focus will be on the repair process in terms of collagen fiber organization. As our knowledge of cell-cell and cell-matrix interactions in the repair process increases, we may be able to direct the pattern of scar collagen fibers to resemble that of dermis and thereby provide better wound care to the patient of the future.

Discontinuous Adoption Paths With Dynamic Scale Economies

This paper examines the equilibrium adoption patch for two innovations when there are “network externalities”. We show that the existence of significant increasing returns in system-scale can give the result that one variant will drive out the other one and so emerges as the unique standard for the industry. Moreover, if “network externalities” are sufficiently strong, then the equilibrium adoption path is discontinuous, that is, it includes a catastrophe point.

Discontinuous Adoption Paths with Dynamic Scale Economies

This paper examines the equilibrium adoption patch for two innovations when there are “network externalities”. We show that the existence of significant increasing returns in system-scale can give the result that one variant will drive out the other one and so emerges as the unique standard for the industry. Moreover, if “network externalities” are sufficiently strong, then the equilibrium adoption path is discontinuous, that is, it includes a catastrophe point.

Evolution and Revolution

This article considers the problem of how cooperative norms can be established by modeling the evolution of a system of corruption. A fixed population of players is assumed that play a series of supergames with randomly chosen opponents. Each stage game in the supergames is a prisoner's dilemma. This article exhibits the conditions under which an equilibrium of corruption exists and is stable. There are two types of players, adaptive and nonadaptive ones. Among the nonadaptive players, there is a small proportion that always chooses to be conditionally honest in every new supergame. Furthermore, corruption generates small but cumulative social costs. This article shows that the joint presence of a small group of “honest” players and of cumulative social costs is sufficient to drive the system to a critical (i.e., catastrophis) point in which the stable equilibrium of corruption suddenly becomes unstable. When the system has reached such a catastrophic point, a small perturbation is enough to drive it toward a different equilibrium. The new equilibrium is cooperative, in that all players choose to be conditionally honest and that a cooperative equilibrium is always stable under the model's conditions. Thus the catastrophic transition to the new equilibrium exemplifies a sudden and spontaneous establishment of a cooperative pattern of behavior.

Observational support for the current sheet catastrophe model of substorm current disruption

It has recently been found that a one‐dimensional current sheet equilibrium with a non‐zero confection electric field, Ey, and a non‐zero normal magnetic field component, Bn, can reach a point of catastrophe through either the reduction of the drift velocity, υD = cEy/Bn or the increase of Bn. This point of catastrophe coincides with a value of κA ≃ 0.7, where κA is the self‐consistent value of κ = (Rmin/ρA max)½ corresponding to ions of average energy. (Here Rmin is the minimum field‐line radius of curvature and ρA max is the maximum gyroradius for ions of average energy.) The point of catastrophe was found to be preceded by a twisting of the current sheet field‐lines into the dawnward direction, i.e. by the development of a y‐component of B, with odd symmetry in z, and a sign opposite Bx, where positive x is earthward, positive y is in the dawn‐to‐dusk direction and positive z is northward. Since the loss of the current sheet would cause the local configuration to become more dipolar, it was suggested that the catastrophic loss of the local current sheet equilibrium could correspond to local current disruption and dipolarization. In this paper, observations of some of the signatures predicted by theory are presented.

Input buffer limiting: behavior analysis of a node throughout the range of blocking probabilities

A numerical method is suggested for estimating the performance parameters of a node with input buffer limiting, throughout the range of blocking probabilities. Congestion is attributed to the sudden increase in the effective service time when the input rate increases. This sudden increase, resulting in the thrashing of the throughput rate, can be related to changes in the shape of the potential function of the node. The methodology of deriving the potential function of the node from its gradient dynamics and dynamic flow conservation considerations is presented. Using the results from catastrophe theory, it is shown that the abrupt changes in the throughput can be expected due to the existence of a catastrophe point in the parameter space. The behavior of a node in a homogeneous network can be characterized by a fold catastrophe. The simulation results are found to agree with the results obtained by the numerical method. >

Thermal model of the catastrophic degradation of high-power stripe-geometry GaAs/(AlGa)As double-heterostructure diode lasers

In the present work, the three-dimensional thermal analysis of the process of the catastrophic mirror damage in double-heterostructure stripe-geometry GaAs/(AlGa)As diode lasers has been carried out for the case of their high-power operation. In the analysis, the temperature dependencies of both the thermal conductivity and the thermal diffusivity have been taken into account. The analytical solution has been obtained with the aid of the Green’s function method. The catastrophic mirror damage has been explained from the thermal point of view as a result of a local overheating of the active area close to or at the mirror surface. It has been proved that a determination of an exact value of the critical mirror temperature from the catastrophic degradation point of view is of minor importance because the considered temperature increase is of an avalanche type. The analytical solution obtained in the work has enabled us to show a dependence of a catastrophic-degradation time, i.e., a permissible length of current pulses from a point of view of the catastrophic mirror damage, on the amplitude of the pulses for the standard stripe diode lasers. Experimentally observed catastrophic mirror damage in diode lasers caused by an increase in their facet reflectivity has been also investigated. It has been confirmed that an increase in the reflectivity of an unprotected mirror may be accompanied by a very rapid facet failure. It has been also shown in which way one can avoid the above phenomenon.

A hierarchy of catastrophes as a succession of stability limits for the crystalline state

MANY theories of melting have focused on some form of instability or catastrophe delimiting the range of stability of the crystalline state. These are exemplified by the vibrational instability proposed by Lindemann1 and the elastic shear instability of Born2. Fecht and Johnson3 suggested that an entropy catastrophe delimits the stability range of both a superheated crystal and a supercooled liquid. Drawing on Kauzmann's idea4 that the glass transition pre-empts a catastrophe point at which the entropy of supercooled liquid equals that of the crystal, they suggest that a second catastrophe point occurs when the entropy of the superheated crystal equals that of the liquid phase, and that this is pre-empted by melting. Here I argue that these entropy catastrophes are an outer bound on the stability limits of the crystalline and liquid phases, and that a hierarchy of inner catastrophes occurs at reduced superheatings or supercoolings.

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.

Critical points in the potential of a set of point charges

The use of a direct search as a method of finding the critical points of a function of three variables is advocated. It is illustrated with the potential of a set of point charges. A personal computer program enables the results to be obtained efficiently. The tetrahedral set displays a catastrophe point and introduces new considerations which might otherwise be overlooked. The example is designed as an exercise for students.

The effect of gravity on the stability of a line-tied coronal magnetohydrostatic equilibrium

Abstract The magnetohydrodynamic stability of a class of magnetohydrostatic equilibria is investigated. The effect of gravity is included as well as the stabilising influence of the dense photospheric line-tying. Although the two-dimensional equilibria exhibit a catastrophe point, when the ratio of plasma pressure to magnetic pressure exceeds a critical value, arcade structures, with both footpoints connected to the photosphere, become unstable to three-dimensional disturbances before the catastrophe point is reached. Numerical results for field lines that are open into the solar corona suggest that they are completely stable. Although there is no definite proof of stability, this would allow the point of non-equilibrium to be reached.

Analysis of intramolecular hydrogen bonding in terms of the topological properties of the charge density. The protonated fluoroacetones

A new method for the analysis of intramolecular hydrogen bonding is proposed and applied to the equilibrium structures of the protonated fluoroacetones. The method, based on a theory of molecular structure due to Bader, uses the topological properties of the charge density to elucidate the types of interactions within a molecule of interest. The calculations show a strong basis set dependence. In particular, the STO-3G optimized geometries exhibit intramolecular hydrogen bonding between a fluorine substituent and the carbonyl proton, whereas some calculations with geometries optimized at the 4-31G and 6-31G* levels indicate a direct bonding interaction between the fluorine and oxygen atoms. Two types of catastrophe point, namely conflict and bifurcation, are possible in protonated monofluoroacetone. Only the latter was located in the present calculations.

VLASIC: A Catastrophic Fault Yield Simulator for Integrated Circuits

This paper describes the yield simulator VLASIC (VLSI Layout Simulation for Integrated Circuits). VLASIC is a Monte Carlo simulator that uses defect models and statistics to place random catastrophic point defects on a chip layout and determine what circuit faults, if any, have occurred. The defect models are described in tables, and so are readily extended to new processes or defect types. The defect statistical model is based on actual fabrication line data, and has not appeared before in the literature. The circuit fault information generated by VLASIC can be used to predict yield, optimize design rules, generate test vectors, evaluate redundancy, etc. A redundancy analysis system which uses these data is described, and an example of its use given.

The Orientations and Distortions of Caustics in Geometrical Optics

The caustics of geometrical optics appear in experiments as scaled and sheared versions of the canonical forms given by catastrophe theory. However, because the longitudinal direction has a physically different status from the transverse directions, not all orientations and linear distortions are possible. Apart from the basic restriction that the caustic has to be tangential to the ray through the catastrophe point, the cuspoid catastrophes lack C − 2 degrees of freedom and the umbilics lack C − 3, where C is the co-dimension. Thus, for example, it is not possible to have a non-degenerate unsheared swallowtail caustic (C = 3) unless it is sideways on. These results hold for non-dissipative propagation in a general anisotropic and inhomogeneous medium, provided certain exceptional cases are excluded.