Showing papers on "Catastrophe theory published in 2020"

Catastrophe theory classification of Fermi surface topological transitions in two dimensions

Abstract: This paper classifies point singularities that occur in two dimensional bands using catastrophe theory. Further, the connection between lattice symmetries and singularities is brought out, leading to the classification of singularities that can occur at high symmetry points in the Brillouin zone

Exploration-Exploitation in Multi-Agent Learning: Catastrophe Theory Meets Game Theory

Abstract: Exploration-exploitation is a powerful and practical tool in multi-agent learning (MAL), however, its effects are far from understood. To make progress in this direction, we study a smooth analogue of Q-learning. We start by showing that our learning model has strong theoretical justification as an optimal model for studying exploration-exploitation. Specifically, we prove that smooth Q-learning has bounded regret in arbitrary games for a cost model that explicitly captures the balance between game and exploration costs and that it always converges to the set of quantal-response equilibria (QRE), the standard solution concept for games under bounded rationality, in weighted potential games with heterogeneous learning agents. In our main task, we then turn to measure the effect of exploration in collective system performance. We characterize the geometry of the QRE surface in low-dimensional MAL systems and link our findings with catastrophe (bifurcation) theory. In particular, as the exploration hyperparameter evolves over-time, the system undergoes phase transitions where the number and stability of equilibria can change radically given an infinitesimal change to the exploration parameter. Based on this, we provide a formal theoretical treatment of how tuning the exploration parameter can provably lead to equilibrium selection with both positive as well as negative (and potentially unbounded) effects to system performance.

On the phase-space catastrophes in dynamics of the quantum particle in an optical lattice potential.

Abstract: We have investigated the dynamics of a quantum particle in the optical lattice potential. Initially, the quantum particle was represented by a Gaussian wave packet, located in the center of the well. The corresponding Schrodinger equation was solved explicitly by the method of the Chebyshev global propagation. Obtained solutions were also used for the construction of the Wigner functions. We found a great number of local abrupt changes of the solution shape. To explain this behavior, we used the fact that structurally stable systems, which form the largest class of the low dimensional dynamical systems, can be modeled and classified according to the catastrophe theory. All important features of the exact solution were explained on the basis of the mathematical properties of the catastrophic model. Such an approach enabled us to extract relevant information out of numerical solutions without employing any kind of approximations. We have investigated the influence of the Wigner catastrophes on the details of the quantum-classical correspondence breakdown. The wave packet was found to expand rapidly, filling the whole classically available area of the phase space. It was found that its self-interference pattern saturates quickly. A region of the phase space emerges in which the Wigner function oscillations transform into the singularity driven fluctuations. Once this region covers the whole area of the phase space, a wave packet dynamics enters into the new regime where its Wigner function fluctuates around the ergodic average. It will be shown that all mentioned processes are caused by the proliferation of the catastrophes and their mutual interactions.

Using catastrophe theory to analyze subway fire accidents

Abstract: Catastrophe theory can describe a continuous process that is undergoing abrupt changes. A dynamic process can be considered to be a swallowtail catastrophe if it has the following six qualities: bimodality, divergence, sudden transitions, hysteresis, inaccessibility and irreversibility. In this paper, the swallowtail catastrophe model is applied to describe the changing dynamic process of subway fire accidents. This dynamic process is also proved to have the six qualities of a swallowtail catastrophe. By using the swallowtail catastrophe model, we construct a model for the subway fire accidents, and we present analyses of subway fire accidents. On the basis of the model and analyses, the dynamic changes in the subway fire accident evolution process can be described with a novel approach. The causes of fire accidents in subways are also discussed, from the perspective of the fire triangle and four elements of an accident. We hope that this study’s theoretical descriptions and discussion of subway fire accidents will facilitate a profound analysis of subway safety.

Numerical simulation and analysis of five-stage lifting pump by improved turbulence model and catastrophe theory

Abstract: How to theoretically describe the process of turbulence formation emains an unsolved problem. A new method is presented in this paper for quantitatively researching the turbulence phase transition ...

Analysis of labor strike based on evolutionary game and catastrophe theory

Abstract: This paper analyzes the labor–employer relations during conditions that lead to strike using an evolutionary game and catastrophe theory. During a threat to strike, the employers may accept the whole or only a part of the demands of labors and improve the work conditions or decline the demands, and each selected strategies has its respective costs and benefits. The threat to strike action causes the formation of a game between the strikers and employers that in which, as time goes on, different strategies are evaluated by the players and the effective variables of strike faced gradual and continuous changes, which can lead to a sudden jump of the variables and push the system to very different conditions such as dramatic increase or decrease in the probability of selecting strategies. So the alliance between labors could suffer or reinforce. This discrete sudden change is called catastrophe. In this study after finding evolutionary stable strategies for each player, the catastrophe threshold analyzed by nonlinear evolutionary game and the managerial insight is proposed to employers to prevent the parameters from crossing the border of the catastrophe set that leads to a general strike.

Stability Analysis of Surrounding Rock in Tunnel Crossing Water-Rich Fault Based on Catastrophe Theory

Abstract: The failure of surrounding rock in tunnel crossing water-rich fault is a sudden damage phenomenon with obvious nonlinear and abrupt characteristics, which is characterized by the uncertainty of deformation, the nonlinearity of constitutive relation and the mutation of failure process. Based on the mechanical model of tunnel-fault system, the cusp catastrophe model of the failure of surrounding rock in tunnel crossing water-rich fault is established. Combining with the field monitoring data, the model is employed to judge the stability of surrounding rock. Results show that the deformation curve of tunnel vault has a good correlation to the cusp catastrophe model, and the results for stability analysis is consistent with the actual situation. The influences that the strength of rock mass, groundwater and fault width have on the failure of tunnel are further discussed, which provides scientific guidelines to the construction of tunnel.

Finite-energy accelerating beam dynamics in wavelet-based representations

Abstract: This paper demonstrates that the properties of finite-energy accelerating beams arise from a dynamical self-interference. The authors study examples of accelerating beams derived from catastrophe theory, and they develop a method of analysis based on a combination of spectral techniques to capture the wave packet dynamics at a single-mode level.

Multi-dimensional complex phase transition model by catastrophe theory developing Kolmogorov turbulence theory

Abstract: In this paper, a novel multi-dimensional complex non-equilibrium phase transition model is put forward to describe quantitatively the physical development process of turbulence and develop the Kolm...

Models of Flood-attacks, mathematical catastrophe theory, theory of differential games and security strategies

Abstract: Method of finding of the most optimal security strategies for computer protection from Flood-attacks is given based on differential game theory. Situation of “fallen” server we are describing using mathematical catastrophe theory

A Cusp Catastrophe Model for Alluvial Channel Pattern and Stability

Abstract: The self-adjustment of an alluvial channel is a complicated process with various factors influencing the stability and transformation of channel patterns. A cusp catastrophe model for the alluvial channel regime is established by selecting suitable parameters to quantify the channel pattern and stability. The channel patterns can be identified by such a model in a direct way with a quantified index, which is a 2D projection of the cusp catastrophe surface, and the discriminant function is obtained from the model to distinguish the river state. Predictions based on this model are consistent with the field observations involving about 150 natural rivers of small or medium sizes. This new approach enables us to classify the channel pattern and determine a river stability state, and it paves the way toward a better understanding of the regime of natural rivers to assist decision-making in river management.

Study on the Criterion of Land Subsidence for Gestation Type Soil Cave

Abstract: In order to evaluate the surface stability during the development of underground holes, the process of surface displacement change during the development of the hole is discretized, and the criterion of surface collapse analysis is established according to the catastrophe theory. The validity of the criterion is preliminarily verified by combining the consistency of the cover instability，the simultaneity of instability, the plasticity penetration criterion and the plastic zone cloud map analysis. The above method is used to quantify the determination process of the surface instability so as to obtain the critical hole radius and surface collapse range. The relationship among the critical thickness of the cover layer, internal friction angle, cohesion, elastic modulus and Poisson's ratio is discussed. The results show that the thickness of the cover layer, the internal friction angle and cohesion can be fitted into a linear relationship with good correlation. The thickness of the cover layer is not sensitive to Poisson's ratio and elastic modulus. The criterion of surface instability based on catastrophe theory can provide a useful reference for the study of surface stability above the developed pore. Keywords: catastrophe theory, instability criterion, FLAC3D, land subsidence mechanism

Ray Methods and Special Functions of Wave Catastrophes

Abstract: New asymptotic methods for solving problems of diffraction, focusing and propagation of electromagnetic waves in inhomogeneous anisotropic media are considered: the bicharacteristic method and the wave catastrophe theory. Hamilton-Lukin's bicharacteristic method was used to perform mathematical modeling of ray and caustic structures arising from the propagation of radio waves in the Earth's ionosphere. The results of mathematical modeling of caustic sections of the main, edge and corner wave catastrophes are given. Uniform asymptotic solutions of wave problems leading to special functions of wave catastrophes are considered, and amplitude-phase structures of wave catastrophe special functions are presented.

Deep Neural Network in Cusp Catastrophe Model

Abstract: Catastrophe theory was originally proposed to study dynamical systems that exhibit sudden shifts in behavior arising from small changes in input. These models can generate reasonable explanation behind abrupt jumps in nonlinear dynamic models. Among the different catastrophe models, the Cusp Catastrophe model attracted the most attention due to it's relatively simpler dynamics and rich domain of application. Due to the complex behavior of the response, the parameter space becomes highly non-convex and hence it becomes very hard to optimize to figure out the generating parameters. Instead of solving for these generating parameters, we demonstrated how a Machine learning model can be trained to learn the dynamics of the Cusp catastrophe models, without ever really solving for the generating model parameters. Simulation studies and application on a few famous datasets are used to validate our approach. To our knowledge, this is the first paper of such kind where a neural network based approach has been applied in Cusp Catastrophe model.

Application of Catastrophe Theory to Instabilities Description of Ferroelectric Liquid Crystals in Magnetic Field

Abstract: 1Академия наук Республики Башкортостан, ул. Кирова, д. 15, 450008 Уфа, Россия. E-mail: denis.kondratyev@bk.ru 2Башкирский кооперативный институт (филиал) Российского университета кооперации, ул. Ленина, д. 26, 450000 Уфа, Россия. 3Башкирский государственный медицинский университет, ул. Ленина, д. 3, 450008 Уфа, Россия. 4Восточная экономико-юридическая гуманитарная академия, ул. Мубарякова, д. 3, 450092, Уфа, Россия

Logistic Cusp Catastrophe Regression for Binary Outcome: Method Development and Empirical Testing

Abstract: Cusp catastrophe models are unique to advance life sciences, psychology and behavioral studies. Extensive progresses have been made to utilize this modeling technique for continuous outcome and there is no development for binary data. To fill this gap, this chapter is then aimed to develop a cusp catastrophe modelling method for binary outcome. Building upon our previous research on the nonlinear regression cusp (RegCusp) catastrophe model for continuous outcome, we propose a logistic cusp catastrophe regression (LogisticCusp). LogisticCusp is based on the principles of logistic regression for binary outcome variable y (yes/no) being expressed as a latent binary variable Y through a logit link. This latent regression provides a mathematical connection between an observed outcome variable as a binomially distributed random variable and the deterministic cusp catastrophe at its equilibrium. By connecting the two, Y in the LogisticCusp is considered as one of the true roots of the deterministic cusp catastrophe model determined using the Maxwell or Delay conventions. We validate the method using a 5-step Monte-Carlo simulation with two predictors and three parameters for both bifurcation and asymmetry control variables. We further tested the method with binge drinking behavior in youth with data from the Monitoring the Future Study. Results from 5000 Monte-Carlo simulations indicate that the parameter estimates obtained through LogisticCusp are unbiased and efficient using maximum likelihood estimation with quasi-Newton numerical search algorithm. Results from empirical testing with real data are consistent with those estimated using other methods. LogisticCusp adds a new tool for researchers to examine many issues in psychology, life sciences, and behavioral studies, particularly, issues in medicine and public health with the powerful cusp catastrophe modeling for binary outcome.

Evaluation of theoretical strength of porous materials according to catastrophe theory

Abstract: With the rapid development of modern science, in particular, applied mechanics, the catastrophe theory proved to be quite effective in the analysis of classical results and the development of modern ones. This theory has developed significantly in the study of a number of issues in the theory of elastic stability, which studies the response of elastic bodies and structures to existing mechanical loads. Catastrophe theory predictions have important technical applications for estimating the critical forces that initiate the loss of stability of elastic bodies and engineering structures. The main basics of the research are analysed in this paper; based on the catastrophe theory, the problems are set; the main types of catastrophes’ functions are described; and the simplest of them, in particular the fold catastrophe, is applied. Based on the set analytical relations for the calculations of effective electrical conductivities and elastic modules by the pore concentration of the electrically conductive material, the estimation of the element strength of the composite sample is simulated in the form of a rod.

Development of Forecasting Theory and Methods for Developing Forecasts in Electroenergetics

Abstract: The article discusses several fundamental problems in forecasting problems that form the scientific novelty of the research. One of the problems is formulated as the need to determine the key parameters that form the basis of the forecasting model and allow determining the state of the subject area. The next key problem in the literature is formulated as the «curse of dimensionality». Occurs when the researcher tries to take into account the maximum number of indicators and criteria for evaluating the subject area in the model, and this leads to the fact that the computer model required for its solution approaches the «Turing limit». The third problem is described in the literature as the problem of «supersystem». All elements of the predicted system or process can form higher-level systems that have their own unique properties. This makes it fundamentally impossible to describe a super-system mapping of target functions from the point of view of the systems and processes that make up the super-system. To develop the theory of forecasting and overcome these problems, we propose the use of such sections of modern mathematics as neuromathematics and deep machine learning, chaos theory, catastrophe theory, and the theory of self-organizing systems. It is believed that these methods will increase the depth of the forecast by identifying hidden patterns and relationships among poorly formalized conventional methods of predictive indicators.