Cusp Model for Risk Analysis

Aim To assess simultaneously various risk states of a system. Methods\ Using the catastrophe and fuzzy theory, the energy and uncertainty in a system are set as two control variables and the function of the system is used as the state variable for analysis. Results and Conclusion\ A risk analysis model named the cusp model is presented. Various states regarding the safety of the system such as the accident state, no accident state and miss state can be represented at will on the cusp model.

Application of cusp catastrophe theory to reliability analysis of slopes in open-pit mines

A method of slope reliability analysis was developed by imposing a state equation on the limit equilibrium theory, given the basis of a fixed safety factor technique. Among the many problems of reliability analysis, the most important problem is to find a performance function. We have created a new method of building a limit state equation for planar slip surfaces by applying the mathematical cusp catastrophe theory. This new technique overcomes the defects in the traditional rigid limit equilibrium theory and offers a new way for studying the reliability problem of planar slip surfaces. Consequently, we applied the technique to a case of an open-pit mine and compared our results with that of the traditional approach. From the results we conclude that both methods are essentially consistent, but the reliability index calculated by the traditional model is lower than that from the catastrophic model. The catastrophe model takes into consideration two possible situations of a slope being in the limit equilibrium condition, i.e., it may or may not slip. In the traditional method, however, a slope is definitely considered as slipping when it meets the condition of a limit equilibrium. We conclude that the catastrophe model has more actual and instructive importance compared to the traditional model.

CUSP CATASTROPHE MODEL OF INSTABILITY OF PILLAR IN ASYMMETRIC MINING

A simplified mechanical model of pillar-hang wall was established in asymmetric mining and instability of the system was discussed by means of potential energy principle and cusp catastrophe theory. The necessary-sufficient condition and the jump value of displacement of pillar and the released energy expressions were derived, which established foundation for quantifying of the instability of system. The results show that instability of the system is related to load and its stiffness distribution. The critical load increases with the increasing relative stiffness, and the system is more stable. On the contrary, the instability of system is likely to occur, and the released energy is larger in instability process, and the harm is more tremendous accordingly. Furthermore, an example was calculated, and the estimated results are in good agreement with the practical experience, which provide basis for mining order and arranging stope.

Simulation of maize drought degree in Xi'an City based on cusp catastrophe model

Drought generally has significant impacts on crops.It is essential to quantitatively evaluate the relationship between crop production and drought degree to provide technical support for drought disaster prevention.In this study,a drought degree index that can reflect the changes in precipitation,evapotranspiration,and soil moisture was developed on the basis of crop yield reduction rate.Four drought scenarios were set up to simulate the effects of meteorological drought on drought degree of crops at different growth stages.A cusp catastrophe model was constructed to analyze the mutation characteristics of the drought degree of maize at different growth stages under different meteorological drought conditions.Xi'an City in China was selected as the study area,and summer maize was selected as the research crop.Precipitation and crop yield data from 1951 to 2010 were used as the fundamental data to investigate drought degree mutation of summer maize.The results show that,under the meteorological drought conditions at the emergence-jointing stage,drought degree may change abruptly,and soil moisture content at the sowingemergence,jointing-tasseling,and tasseling-mature stages should be kept higher than 39%.

Research on traffic flow forecasting model based on cusp catastrophe theory

This paper intends to describe the relationship between traffic parameters by using cusp catastrophe theory and to deduce highway capacity and corresponding speed forecasting value through suitable transformation of catastrophe model. The five properties of a catastrophe system are outlined briefly, and then the data collected on freeways of Zhujiang River Delta, Guangdong province, China are examined to ascertain whether they exhibit qualitative properties and attributes of the catastrophe model. The forecasting value of speed and capacity for freeway segments are given based on the catastrophe model. Furthermore, speed-flow curve on freeway is drawn by plotting out congested and uncongested traffic flow and the capacity value for the same freeway segment is also obtained from speed-flow curve to test the feasibility of the application of cusp catastrophe theory in traffic flow analysis. The calculating results of catastrophe model coincide with those of traditional traffic flow models regressed from field observed data, which indicates that the deficiency of traditional analysis of relationship between speed, flow and occupancy in two-dimension can be compensated by analysis of the relationship among speed, flow and occupancy based on catastrophe model in three-dimension. Finally, the prospects and problems of its application in traffic flow research in China are discussed.

A CUSP CATASTROPHE, PRECURSORS PATTERN AND EVOLUTION PROCESS OF ROCKBUST OF COAL PILLAR UNDER A HARD ROCK SUBJECT TO ELASTIC SUPPORT

The rockburst of the poal pillar under a thick hard roof stratum is modelled as the instability failure problem of coal pillars under strata subject to elastic support. The instability mechanism of rockburst is studied by applying cusp catastrophic theory. The effects of the stiffness ratio of the system and loads imposed on the system on the rockburst are explicated.The factors affecting rockbursts are discussed. Based on them, the evolution process, the forewarning regularity arid forewarning sings of rockbursts are studied. It is indicated that the subsidence velocity of roof stratum, which increases quickly and tends to infinity, is the forewarning measurable signs of the rockbursts of coal pillar.

The Study of Chamber Rockburst by the CUSP Model of Catastrophe Theory

By means of CUSP model of catastrophe theory. this paper has studied thephysics process of rockburst occured on circular chamber. The present paper has nolonly described the instability process of rockburst more deeply. but also got the crilicaldepth of plastic softening area of chamber that is valuable in the controlling engineering of rockburst. the chamber displacement jump and energy liberation have been derived. the influence of rock parameters on the rockburst has been discussed .

Catastrophic destabilization of tunnel under rocks slipping in faultage

The model of catastrophic destabilization of tunnel under rock slipping in fault zone based on catastrophic theory and the potential function of fault movement were pre- sented.On the basis of the results above,through Taylor series expansion of the equation of equilibrium surface,its standard form was obtained.Analysis show that catastrophic destabilization of tunnel will occur only when stiffness ratio between elastic sector and strain weakening sector of soft rocks was larger than or equal to 1.On the other hand, sliding behavior and evolution path of fault were directly affected by exogenous process, and it was a major extraneous factor which leads to catastrophic destabilization of tunnel. In the condition of system catastrophe could be generated,if external forces vary from smaller to larger,firstly,fault sticks or creeps,and secondly,when external force equal to or larger than critical value,fault turns to slip suddenly.Inverse,if external forces vary from larger to smaller,fault smoothly slips firstly,when external force equal to or smaller than critical value,and fault turns to stick or creep suddenly.

Instability Analysis of Strike-Slip Fault Based on Cusp Catastrophe Model

The distribution of many active faults in western China is an important reasonfor the frequent earthquakes. With the rapid development of the western region, manymajor projects have been built there and the existence of active faults is bound to have aninfluence on the safety of the engineering structure. Therefore, it is of great significanceto study the mechanism of fault slip instability for evaluating the geological stability ofthe region and for the site selection of major projects. In this paper, cusp catastrophetheory is used to establish a cusp catastrophe model with general softened form ofstrike-slip faults on the basis of strike-slip faults. In this model, the influence of thesoftening property of fault zone on fault instability is considered. Based on this model,the conditions of slip instability of strike-slip faults are derived and further the half-slipdistance, far-field displacement and energy release equation of sliding-slip fault arerevealed. The influences of the system stiffness ratio and the softening property of thefault zone on the half-wave displacement, the far-field displacement and the energyrelease are shown. Which lays a good foundation for further research on activefault-induced earthquake mechanism.

Fundamental and speculative components of the cryptocurrency pricing dynamics

The driving forces behind cryptoassets’price dynamics are often perceived as being dominated by speculative factors and inherent bubble-bust episodes.Fundamental components are believed to have a weak,if any,role in the price-formation process.This study examines five cryptoassets with different backgrounds,namely Bitcoin,Ethereum,Litecoin,XRP,and Dogecoin between 2016 and 2022.It utilizes the cusp catastrophe model to connect the fundamental and speculative drivers with possible price bifurcation characteristics of market collapse events.The findings show that the price and return dynamics of all the studied assets,except for Dogecoin,emerge from complex interactions between fundamental and speculative components,includ-ing episodes of price bifurcations.Bitcoin shows the strongest fundamentals,with on-chain activity and economic factors driving the fundamental part of the dynam-ics.Investor attention and off-chain activity drive the speculative component for all studied assets.Among the fundamental drivers,the analyzed cryptoassets present their coin-specific factors,which can be tracked to their protocol specifics and are economi-cally sound.

DESCRIPTION AND PREDICTION OF CATASTROPHE OF VIBRATION STATE FOR FAULTY ROTORS

A sudden increase of vibration amplitude with no foreboding often results in an abrupt breakdown of a mechanical system.The catastrophe of vibration state of a faulty rotor is a typical nonlinear phenomenon,and very difficult to be described and predicted with linear vibration theory.On the basis of nonlinear vibration and catastrophe theory,fhe eatastrophe of the vibration amplitude of the faulty rotor is described;a way to predict its emergence is developed.

A Nonlinear Coupled Soil Moisture-Vegetation Model

Based on the physical analysis that the soil moisture and vegetation depend mainly on the precipitation and evaporation as well as the growth, decay and consumption of vegetation a nonlinear dynamic coupled system of soil moisture-vegetation is established. Using this model, the stabilities of the steady states of vegetation are analyzed. This paper focuses on the research of the vegetation catastrophe point which represents the transition between aridness and wetness to a great extent. It is shown that the catastrophe point of steady states of vegetation depends mainly on the rainfall P and saturation value υ0, which is selected to balance the growth and decay of vegetation. In addition, when the consumption of vegetation remains constant, the analytic solution of the vegetation equation is obtained.

Catastrophe analysis on pillar instability considered mining effect

The instability of the pillar was discussed based on the potential energy principle and the cusp catastrophe theory, and a simplified mechanical model of the pillar was established considering the mining effect. The necessary-sufficient conditions, the jump value of displacement of pillar and the released energy expressions were deduced. The results show that the instability of the pillar is related to the properties of the rock, the external force and the relative stiffness of the elastic area to the plastic area. The instability of system is like to occur with the enlarging of the softening area or the decreasing of E/λ. The calculation done shows that the estimated results correspond to practical experience.

Application of Catastrophe Theory in 3D Seismic Data Interpretation of Coal Mine

In order to detect fault exactly and quickly, cusp catastrophe theory is used to interpret 3D coal seismic data in this paper. By establishing a cusp model, seismic signal is transformed into standard form of cusp catastrophe and catastrophe parameters, including time-domain catastrophe potential, time-domain catastrophe time, frequency-domain catastrophe potential and frequency- domain degree, are calculated. Catastrophe theory is used in 3D seismic structural interpretation in coal mine. The results show that the position of abnormality of the catastrophe parameter profile or curve is related to the location of fault, and the cusp catastrophe theory is effective to automatically pick up geology information and improve the interpretation precision in 3D seismic data.

A CATASTROPHE ANALYSIS ON THE STABILITY OF THE CRACK GROWTH IN THREE-POINT BENDING SPECIMENS

This paper presents an attempt at the application of catastrophe theory to the stability analysis of J-controlled crack growth in three-point bending specimens. By introducing the solutions of J-integral in the completely yielding state for the ideal plastic material, the critical condition of losing stability for the crack propagation in the specimen is obtained, based on the cusp catastrophe theory. The process of the crack growth from geometrical sense is described.

CATASTROPHE FRACTURE OF THIN-WALL PRESSURE TUBES

Catastrophe theory was used to investigate the fracture behavior of thin-wall cylindrical tubes subjected to internal explosive pressure. Based on the energy theory and catastrophe theory, a cusp catastrophe model for the fracture vas established, and a critical condition associated with the model is given.

Landslide Prediction Based on Wavelet Analysis and Cusp Catastrophe

During the monitoring engineering of landslides, the monitoring data of accumulated displacement are usually affected by the external factors. Therefore, the displacement curve always has step-like character, which brings some difficulties to the accurate prediction of landslides. In order to solve this problem, based on the wavelet analysis and cusp catastrophe, a new kind of analysis method is proposed in this article. First, Fourier transform method can be used to extract the frequency component of the curve of monitoring displacement. Second, the wavelet transform was adopted to inspect the breakpoints of signals, which can be used to analyze the cause of the occurrence of the step-like character in the curve of landslide monitoring. Based on the cusp catastrophe theory, a nonlinear dynamic model was established to conduct the simulation calculation of time forecasting of landslides. According to a case study of landslide, the periodical rainfall and reservoir level fluctuation are the main factors leading to the step-like changes in the curve of monitoring displacement. In addition, the results of simulation calculation are in agreement with the fact of local failure of landslides. This method can provide a new analysis way for the time prediction of landslides.

Stability analysis of the pillars between bedded salt cavern gas storages by cusp catastrophe model

The failure of pillars between bedded salt cavern gas storages can be seen as processes that the deformations of pillars convert from continuous gradual change system to catastrophe state,which are typical nonlinear catastrophe problems.In the paper,the cusp catastrophe model is proposed to obtain the stability factors of pillars.It can overcome the shortages of traditional strength reduction finite element method(SR FEM) and greatly improve the accuracy of stability factors obtained by numerical simulations.The influences of cavern depth,gas pressure,pillar width,and time on the stability factors are studied.Y-1 and Y-2 salt cavern gas storages,located at Jiangsu province of China,were simulated as examples.The stability factors of pillars between Y-1 and Y-2 were evaluated,and the running parameters were recommended to ensure the pillars stability.The results showed that the cusp catastrophe model has high practicability and can precisely predict the stability factors.The stability factors are equidirectional with the increase of gas pressure and pillar width,but reverse to the increase of cavern depth and time.The stability factors of pillars between Y-1 and Y-2 are small for narrow widths,which are influenced greatly by gas pressure,time,pressure difference,and gas production rate.In order to ensure the safety of pillars,the lowest gas pressure,safe running time,max.pressure difference and max.gas production rate of Y-1 and Y-2 were recommended as 7 MPa,5 years,3 MPa,and 0.50 MPa/d,respectively.

Stability of interbed for salt cavern gas storage in solution mining considering cusp displacement catastrophe theory

Cusp displacement catastrophe theory can be introduced to propose a new method about instability failure of the interbed for gas storage cavern in bedded salt in solution mining.We can calculate initial fracture drawing pace of this interbed to obtain 2D and 3D gas storage shapes at this time.Moreover,Stability evaluation of strength reduction finite element method(FEM)based on this catastrophe theory can used to evaluate this interbed stability after initial fracture.A specific example is simulated to obtain the influence of the interbed depth,cavern internal pressure,and cavern building time on stability safety factor(SSF).The results indicate:the value of SSF will be lower with the increase of cavern building time in solution mining and the increase of interbed depth and also this value remains a rise with the increase of cavern internal pressure Especially,we can conclude that the second-fracture of the interbed may take place when this pressure is lower than 6 MPa or after 6 days later of the interbed after initial fracture.According to above analysis,some effective measures,namely elevating the tube up to the top of the interbed,or changing the circulation of in-and-out lines,can be introduced to avoid the negative effects when the secondfracture of the interbed may occur.

THE NONLINEAR CUSP-CATASTROPHE MODEL OF THE SEDIMENT TRANSPORT RATE

In the catastrophe theory of nonlinear science, the intensity of water flow Θ and the coefficient of non uniform sediment m are regarded as two bound variables, and the intensity of bed load transport Φ as the state variable in the motion of non uniform sediment in cusp catastrophe model. Based on the standard equation of the cusp catastrophe theory, the relation equation between the intensity of bed load transport Φ and the intensity of water flow Θ has been derived by used coordinate transform and topology transform. The equation of bed load transport rate was built on the cusp catastrophe theory of nonlinear science. The others are applied to verify this equation, that the results calculated by the cusp catastrophe equation agree well with the other equations. This indicates that the cusp catastrophe equation is reasonable, and the results fully reflect the characteristics of threshold motion and transport of non uniform sediment. The purpose of this paper is to explore the incipient motion and transport laws of non uniform sediment from the viewpoint of nonlinear science.