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Concept: Catastrophe theory


By means of the joint use of electron localization function (ELF) and Thom’s catastrophe theory, a theoretical analysis of the energy profile for the hetero-Diels-Alder reaction of 4-methoxy-1,2-benzoquinone 1 and methoxyethylene 2 has been carried out. The 12 different structural stability domains obtained by the bonding evolution theory have been identified as well as the bifurcation catastrophes (fold and cusp) responsible for the changes in the topology of the system. This analysis permits finding a relationship between the ELF topology and the evolution of the bond breaking/forming processes and electron pair rearrangements through the reaction progress in terms of the different ways of pairing up the electrons. The reaction mechanism corresponds to an asynchronous electronic flux; first, the O1C5 bond is formed by the nucleophilic attack of the C5 carbon of the electron rich ethylene 2 on the most electrophilically activated carbonyl O1 oxygen of 1, and once the σ bond has been completed, the formation process of the second O4C6 bond takes place. In addition, the values of the local electrophilicity and local nucleophilcity indices in the framework of conceptual density functional theory accounts for the asychronicity of the process as well as for the observed regioselectivity. © 2012 Wiley Periodicals, Inc.

Concepts: Scientific method, Electron, Greek loanwords, Chemical bonding, Density functional theory, Quantum chemistry, Nucleophile, Catastrophe theory


This study aimed (a) to address the evidence for situational specificity in the connection between safety climate to occupational accidents, (b) to resolve similar issues between anxiety and accidents, © to expand and develop the concept of safety climate to include a wider range of organizational con-structs, (d) to assess a cusp catastrophe model for occupational accidents where safety climate and anxiety are treated as bifurcation variables, and environ-mental hazards are asymmetry variables. Bifurcation, or trigger variables can have a positive or negative effect on outcomes, depending on the levels of asymmetry, or background variables. The participants were 1262 production employees of two steel manufacturing facilities who completed a survey that measured safety management, anxiety, subjective danger, dysregulation, stressors and hazards. Nonlinear regression analyses showed, for this industry, that the accident process was explained by a cusp catastrophe model in which safety management and anxiety were bifurcation variables, and hazards, age and experience were asymmetry variables. The accuracy of the cusp model (R2 = .72) exceeded that of the next best log-linear model (R2 = .08) composed from the same survey variables. The results are thought to generalize to any industry where serious injuries could occur, although situationally specific effects should be anticipated as well.

Concepts: Regression analysis, Linear regression, Injury, Thought, Logic, Safety, Accident, Catastrophe theory


Self-regulation shift theory (SRST) is a threshold theory explaining self-regulation following trauma that utilizes nonlinear dynamics to capture systemic shifts in trauma adaptation. Cusp catastrophe modeling tests nonlinear changes in an outcome (e.g., posttraumatic distress) based on an identified bifurcation factor under specific conditions (i.e., asymmetry variables). We evaluated two cusp models in a motor vehicle accident (MVA) database and then confirmed findings within a similar dataset. Based on SRST, we tested coping self-efficacy (CSE) as the bifurcation factor and a set of asymmetry controlling variables. Results demonstrated significant cusp models with CSE as a consistent bifurcation factor in all models. When participants reported lower peritraumatic dissociation, early lower CSE was a significant bifurcation factor for 3-month trauma symptoms in Sample 1, R(2) = .18. The cusp model for changes in symptoms from 30 days to 3 months showed CSE as a significant bifurcation variable with higher levels of avoidant coping (R(2) = .27). In a separate sample, early lower CSE was again a significant bifurcation variable with lower injury severity (R(2) = .52). Results support the importance of self-regulatory appraisals in nonlinear shifts in posttraumatic stress symptoms 3 months post-MVA. Theoretical and practical implications are discussed.

Concepts: Scientific method, Educational psychology, Psychological trauma, Tram accident, Model, Theory, Road accidents, Catastrophe theory


Catastrophic and sudden collapses of ecosystems are sometimes preceded by early warning signals that potentially could be used to predict and prevent a forthcoming catastrophe. Universality of these early warning signals has been proposed, but no formal proof has been provided. Here, we show that in relatively simple ecological models the most commonly used early warning signals for a catastrophic collapse can be silent. We underpin the mathematical reason for this phenomenon, which involves the direction of the eigenvectors of the system. Our results demonstrate that claims on the universality of early warning signals are not correct, and that catastrophic collapses can occur without prior warning. In order to correctly predict a collapse and determine whether early warning signals precede the collapse, detailed knowledge of the mathematical structure of the approaching bifurcation is necessary. Unfortunately, such knowledge is often only obtained after the collapse has already occurred.

Concepts: Scientific method, Mathematics, Greek loanwords, Logic, Formal verification, Formal methods, Catastrophe theory, Catastrophe


Because the collapse of massive stars occurs in a few seconds, while the stars evolve on billions of years, the supernovae are typical complex phenomena in fluid mechanics with multiple time scales. We describe them in the light of catastrophe theory, assuming that successive equilibria between pressure and gravity present a saddle-center bifurcation. In the early stage we show that the loss of equilibrium may be described by a generic equation of the Painlevé I form. This is confirmed by two approaches, first by the full numerical solutions of the Euler-Poisson equations for a particular pressure-density relation, secondly by a derivation of the normal form of the solutions close to the saddle-center. In the final stage of the collapse, just before the divergence of the central density, we show that the existence of a self-similar collapsing solution compatible with the numerical observations imposes that the gravity forces are stronger than the pressure ones. This situation differs drastically in its principle from the one generally admitted where pressure and gravity forces are assumed to be of the same order. Moreover it leads to different scaling laws for the density and the velocity of the collapsing material. The new self-similar solution (based on the hypothesis of dominant gravity forces) which matches the smooth solution of the outer core solution, agrees globally well with our numerical results, except a delay in the very central part of the star, as discussed. Whereas some differences with the earlier self-similar solutions are minor, others are very important. For example, we find that the velocity field becomes singular at the collapse time, diverging at the center, and decreasing slowly outside the core, whereas previous works described a finite velocity field in the core which tends to a supersonic constant value at large distances. This discrepancy should be important for explaining the emission of remnants in the post-collapse regime. Finally we describe the post-collapse dynamics, when mass begins to accumulate in the center, also within the hypothesis that gravity forces are dominant.

Concepts: Fundamental physics concepts, Fluid dynamics, Mass, Fluid mechanics, Viscosity, Liquid, Fluid, Catastrophe theory


We analyse the behaviour of a non-linear model of coupled stock and bond prices exhibiting periodically collapsing bubbles. By using the formalism of dynamical system theory, we explain what drives the bubbles and how foreshocks or aftershocks are generated. A dynamical phase space representation of that system coupled with standard multiplicative noise rationalises the log-periodic power law singularity pattern documented in many historical financial bubbles. The notion of ‘ghosts of finite-time singularities’ is introduced and used to estimate the end of an evolving bubble, using finite-time singularities of an approximate normal form near the bifurcation point. We test the forecasting skill of this method on different stochastic price realisations and compare with Monte Carlo simulations of the full system. Remarkably, the approximate normal form is significantly more precise and less biased. Moreover, the method of ghosts of singularities is less sensitive to the noise realisation, thus providing more robust forecasts.

Concepts: Scientific method, Mathematics, Simulation, Monte Carlo, Dynamical system, Dynamical systems, Catastrophe theory, Period-doubling bifurcation


The critical points analysis of electron density, i.e. ρ(x), from ab initio calculations is used in combination with the catastrophe theory to show a correlation between ρ(x) topology and the appearance of instability that may lead to transformations of crystal structures, as a function of pressure/temperature. In particular, this study focuses on the evolution of coalescing non-degenerate critical points, i.e. such that ∇ρ(xc) = 0 and λ1, λ2, λ3≠ 0 [λ being the eigenvalues of the Hessian of ρ(x) at xc], towards degenerate critical points, i.e. ∇ρ(xc) = 0 and at least one λ equal to zero. The catastrophe theory formalism provides a mathematical tool to model ρ(x) in the neighbourhood of xcand allows one to rationalize the occurrence of instability in terms of electron-density topology and Gibbs energy. The phase/state transitions that TiO2(rutile structure), MgO (periclase structure) and Al2O3(corundum structure) undergo because of pressure and/or temperature are here discussed. An agreement of 3-5% is observed between the theoretical model and experimental pressure/temperature of transformation.

Concepts: Scientific method, Mathematics, Fundamental physics concepts, Thermodynamics, Ab initio, Model theory, Philosophy of science, Catastrophe theory


Pursuit of sustainability requires a systematic approach to understand a system’s specific dynamics to adapt and enhance from disturbances in social-environmental systems. We developed a systematic resilience assessment of social-environmental systems by connecting catastrophe theory and probability distribution equilibrium. Catastrophe models were used to calculate resilience shifts between slow and fast variables; afterwards, two resilience transition modes (“Less resilient” or “More resilient”) were addressed by using probability distribution equilibrium analysis. A tipping point that occurs in “Less resilient” system suggests that the critical resilience transition can be an early warning signal of approaching threshold. Catastrophic shifts were explored between the interacting social-environmental sub-systems of land use and energy (fast variables) and environmental pollution (slow variables), which also identifies the critical factors in maintaining the integrated social-environmental resilience. Furthermore, the early warning signals enable the adaptability of urban systems and their resilience to perturbations, and provide guidelines for urban social-environmental management.

Concepts: Systems theory, Sustainability, Random variable, Environmentalism, Catastrophe theory, Catastrophe, Singularity theory


In recent years, research in organizational psychology has witnessed a shift in attention from a mostly variable-focused approach, to a mostly person-focused approach. Indeed, it has been widely recognized that the study of worker’s heterogeneity is a meaningful and necessary task of researchers dealing with human behavior in organizational contexts. As a consequence, there has been growing interest in the application of statistical analyses able to uncover latent sub-groups of workers. The present contribution was conceived as a tutorial for the application of one of these statistical analyses, namely second-order growth mixture modeling, and to illustrate its inner links with concepts from non-linear dynamic models. Throughout the paper, we provided (a) a discussion on the relationships between growth mixture modeling and the cusp catastrophe model; (b) Mplus syntaxes and output excerpts of a longitudinal analysis conducted on job performance (N = 420 employees rated once a year for four consecutive years);

Concepts: Scientific method, Psychology, Sociology, Model, Employment, Behavior, Human behavior, Catastrophe theory


The mechanochemistry of ring-opening reactions of cyclopropane derivatives turns out to be unexpectedly rich and puzzling. After showing that a rare socalled uphill bifurcation in the case of trans-gem-difluorocyclopropane turns into a downhill bifurcation upon substitution of fluorine by chlorine, bromine and iodine in the thermal activation limit, the dichloro derivative is studied systematically in the realm of mechanochemical activation. Detailed exploration of the force-transformed potential energy surface in terms of Dijkstra path analysis unveils a hitherto unknown topological catastrophe where the global shape of the energy landscape is fundamentally changed. From thermal activation up to moderately large forces, it is an uphill bifurcation that decides about dis- versus conrotatory ring-opening followed by separate transition states along both pathways. Above a critical force, the two distinct transition states merge to yield a single transition state such that the decision about the dis- versus conrotatory ring-opening process is taken at a newly established downhill bifurcation. The discovery of a force-induced qualitative change of the topology of a reaction network vastly transcends the previous understanding of the ring-opening reaction of this species. It would be astonishing to not discover a wealth of such catastrophes for mechanochemically activated reactions which will greatly extend the known opportunities to manipulate chemical reaction networks.

Concepts: Energy, Chemical reaction, Catalysis, Transition state, Chemistry, Force, Potential energy, Catastrophe theory