Catastrophe Theory

Castrigiano, Domenico P. L.; Hayes, Sandra

doi:10.1201/9780429501807

Cited by 25 publications

(60 citation statements)

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“…Elementary catastrophe theory considers a smooth parameterized real valued potential function, which, depends on two or less state variables, and four or less control parameters [8], [12], [14]. The objective of the potential function is to determine the modeling conditions of real world phenomena.…”

Section: Applied Elementary Catastrophe Theorymentioning

confidence: 99%

Catastrophe theory and postmodern general equilibrium: some critical observations

Stiefenhofer¹

2017

ams

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Postmodern general equilibrium theory is the formulation of the Walrasian model in the context of the mathematics of catastrophe theory. In its formulation the model comes equipped with a mapping called the natural projection. The goal of postmodern general equilibrium theory is to extract economic properties from this mapping. However, we show that the natural projection itself is void of any economic content. This observation alone calls for a reexamination of the assumptions and properties of the postmodern general equilibrium model.

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Section: Applied Elementary Catastrophe Theorymentioning

confidence: 99%

Catastrophe theory and postmodern general equilibrium: some critical observations

Stiefenhofer¹

2017

ams

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“…The mathematical tool applicable to the modeling of cutting catastrophe phenomenon is the catastrophe theory [24], which was established in the 1970s. It adopts seven standard catastrophe mathematical models, which are composed of the potential function, the manifold surface equation and the bifurcation set equation (to predict the critical conditions of catastrophe) to describe plenty of catastrophe phenomena when control variables change continuously [25].…”

Section: Introductionmentioning

confidence: 99%

“…In the field of mechanical engineering, the successful application of theoretical modeling methods based on catastrophe theory can be seen in many catastrophe phenomena, such as shear angle catastrophe [27], friction coefficient catastrophe [28], chip flow angle catastrophe phenomenon when cutting with a double-edged tool of 0 edge inclination angle [29] and of arbitrary edge inclination angle [30], etc. In the aspect of experimental modeling research based on experimental data, Luo [31] assumed that the relationship between the actual control variables of the abrupt change of tool wear state and the theoretical control variables of cusp catastrophe [24] was a linear function, the coefficients of the functions were fitted according to the experimental data, and the cusp catastrophe model of tool wear state catastrophe was established. The prediction error of this model for tool wear value was less than 10%.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, the complete data for the critical conditions of CSC are obtained by systematic experimental study on CSC. According to this result, based on the standard bifurcation set equation of the swallowtail catastrophe [24] proposed by the catastrophe theory, this paper assumes that the mapping from the actual control variables of CSC to the theoretical control variables of swallowtail catastrophe is a linear function. The 12 coefficients of the functions are fitted by means of modern optimization technique.…”

Section: Introductionmentioning

confidence: 99%

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Experimental Investigation and Bifurcation Set Equation Modeling of Chip-Splitting Catastrophe

You¹,

Xu²,

Zhu³

et al. 2021

Preprint

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This paper presents the methods of designing experiments for observation of the critical conditions of chip-splitting catastrophe (CSC) based on the dichotomy principle and the methodology of modelling CSC with a bifurcation set equation on the basis of catastrophe theory. 355 groups of experiments are carried out for observing critical conditions of CSC in the symmetrical straight double-edged cutting. It is found that the occurrence of CSC is of obvious regularity. Besides, the specific cutting force is reduced by a maximum of 64.68% after CSC. The bifurcation set equation is established, which can predict the critical conditions of CSC in the symmetrical cutting with a straight double-edged tool with any combinations of cutting edge angles and rake angles. With verification, the predicted values of the critical cutting thickness of CSC based on the established equation are in good agreement with experimental values and yield an average absolute prediction error of 5.34%, which is satisfactorily low. This study lays the groundwork for energy-saving optimization of tool structure and process parameters through reasonable utilization of CSC.

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“…This will be important to show that when a kinetic term is added into the model, the phase diagram still remains qualitatively unchanged, provided that the hopping term is small. Some good references for an introduction to the more general formalism of catastrophe theory are [5] and [13].…”

Section: Chapter 2 Catastrophe Theorymentioning

confidence: 99%

Magnetic-superconducting phase coexistence driven by on-site repulsions in BCS-like models

Alves¹

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This thesis focuses on the study of the thermodynamics of fermionic systems on lattices, from the point of view of the C*-algebraic formulation of quantum mechanics, and on the application of the so-called "catastrophe theory" to the analysis of the phase diagram of an explicit fermionic model. The first chapter is devoted to the introduction of some of the basic and most important properties of C*-algebras, and also to the study of its representations and its set of states. The second chapter is devoted to the development of some important results of catastrophe theory. The results derived in the chapter allows one to analyze the behavior of the minima of members of a family of functions around a degenerate critical point, and they will be used to study the behavior of the thermodynamic pressure for a given fermionic lattice model. The third chapter presents some of the formalism developed in [4], that concerns the existence of the thermodynamics and equilibrium states of lattice fermi systems subject to a suitable set of long-range interactions (i.e., interactions containing mean-field terms). Finally, in the last chapter, using all the formalisms and results already studied in the previous chapters, the thermodynamics of an explicit BCS-like model is analyzed. With the help of catastrophe theory, it is shown that such model exhibits a coexistence of magnetic and superconducting phases for a suitable choice of parameters, and through a perturbative analysis it is also shown that the coexistence still holds when a small kinetic term is added into the model.

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Catastrophe Theory

Cited by 25 publications

References 0 publications

Catastrophe theory and postmodern general equilibrium: some critical observations

Catastrophe theory and postmodern general equilibrium: some critical observations

Experimental Investigation and Bifurcation Set Equation Modeling of Chip-Splitting Catastrophe

Magnetic-superconducting phase coexistence driven by on-site repulsions in BCS-like models

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