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Study on judgment of tunnel face failure in soft surrounding rock and failure control measures
Published in Daniele Peila, Giulia Viggiani, Tarcisio Celestino, Tunnels and Underground Cities: Engineering and Innovation meet Archaeology, Architecture and Art, 2020
J. Du, Z. Mei, H. Gao, B. Zhang, W. Yuan
In 1972, the theory of catastrophe was first proposed in the “Structural Stability and Morphogenesis” published by the French mathematician Thom. The main starting point is the bifurcation theory and the singularity theory, as well as the concept of structural stability. The catastrophe theory mainly explains how the nonlinear system changes from the continuous gradual state to the mutation of the system property, that is, how the continuous change of parameters leads to the discontinuity (Ling. 1987). The cusp catastrophe model is a kind of catastrophe theory, which is proposed by Zeeman (1974). Its potential function V is a two-parameter function (two control variables u and v) whose state variable is x. V=x4+ux2+vx
Study on judgment of tunnel face failure in soft surrounding rock and failure control measures
Published in Daniele Peila, Giulia Viggiani, Tarcisio Celestino, Tunnels and Underground Cities: Engineering and Innovation meet Archaeology, Architecture and Art, 2019
J. Du, Z. Mei, H. Gao, B. Zhang, W. Yuan
In 1972, the theory of catastrophe was first proposed in the “Structural Stability and Morphogenesis” published by the French mathematician Thom. The main starting point is the bifurcation theory and the singularity theory, as well as the concept of structural stability. The catastrophe theory mainly explains how the nonlinear system changes from the continuous gradual state to the mutation of the system property, that is, how the continuous change of parameters leads to the discontinuity (Ling. 1987). The cusp catastrophe model is a kind of catastrophe theory, which is proposed by Zeeman (1974). Its potential function V is a two-parameter function (two control variables u and v) whose state variable is x.
Almost-periodic bifurcations for one-dimensional degenerate vector fields
Published in Dynamical Systems, 2020
Wen Si, Xiaodan Xu, Jianguo Si
The second part of the iteration step is transforming into normal form again. As shown in [1,2] one can apply the machinery of singularity theory to solve normalization problems. The remaining term, can be eliminated by the standard translation which is well-known from the singularity theory. Then system (3.6) becomes where with
Higher-order Landau phenomenological models for perovskite crystals based on the theory of singularities: a new phenomenology of BaTiO3
Published in Phase Transitions, 2018
Ilhwan Kim, Kumok Jang, Ilhun Kim, Lin Li
Already, the singularity theory [23–25] (it is known as the catastrophe theory in physics) became well known as the powerful mathematical tool for construction of the structurally stable phenomenological model and explanation of the complex symmetry changes using it. According to singularity theory [26,23–25], the singularities of the Landau potentials for description of the phase transition in the crystal are determined by the symmetry of the order parameters and the dependence of the phenomenological coefficients on the external parameters such as temperature, pressure and concentration. If the external-dependent variable parameters are given among phenomenological coefficients, then the set of invariants which should surely be included in the potential model is determined, and the potential model in which these invariants are all included is only regarded as the structurally stable phenomenological model. But if the structurally unstable potential model is used, possible qualitative characteristics related to phase transitions of crystalline solid cannot be described entirely. Especially, when one constructs the phenomenological model of the system with the multi-component order parameters, it is very important to choose the simplified structurally stable Landau potential model whose singularity is preserved [27,28,26]. Just, the theory of singularities offers the possibility to consider empirical rules on a rigorous mathematical basis. Therefore, in recent years, the theoretical results in which the theory of singularities was applied in the phenomenological study have been reported [2,3]. Already, this theory was applied to explain the existence of monoclinic and triclinic phases (the lowest-symmetry phases) in the Pb-based perovskite oxide solid solutions, where the structurally stable potential models with three external-dependent coefficients were used [2,3].
Spacelike submanifolds of codimension two in anti-de Sitter space
Published in Applicable Analysis, 2019
On the other hand, singularity theory is proved useful in the description of geometrical properties of submanifolds immersed in different ambient spaces [4–8]. In [5], Izumiya, the first author and Sano investigated the extrinsic differential geometry of hypersurfaces in hyperbolic space by using Legendrian singularity theory. They observed the singularities of lightcone Gauss maps and lightcone Gauss indicatrices, which have special geometrical meanings of spacelike hypersurfaces. Moreover, spacelike submanifold of codimension two in Minkowski space was investigated by Izumiya and Romero Fuster in [6]. They showed a Gauss–Bonnet-type formal in terms of a Gauss–Kronecker curvature with respect to the lightlike normal vectors. We generalize this idea to the anti-de Sitter n-space here. In [9], Chen, Izumiya and the first author study timelike hypersurfaces in anti-de Sitter space. We are motivated to investigate the differential geometry of spacelike submanifolds of other codimension cases. The normal direction of the submanifold cannot be chosen uniquely. However, if we consider the codimension two case, we can determine the lightlike normal frames and define Gauss maps associate with the lightlike normal vectors. In this paper, we study spacelike submanifolds of codimension two in anti-de Sitter n-space. We apply Legendrian singularity theory and obtain new geometric information. It is different from the Minkowski case and the de Sitter case. In our case, it is always possible to choose two lightlike normal directions along the submanifold in the frame of its normal bundle. This is similar to the de Sitter case, but the normalized image is located in the space For the de Sitter case, the normalized image of the lightlike normal is located in the spacelike sphere [7]. The lightcone has its clear background in Physics [10] and the singularities of the lightcone are also studied in [11]. The main results in this paper are Theorems 4.6 and 6.2, which explain the geometric meaning of the contact of submanifolds of codimension two with lightcones and hyperhorospheres from the viewpoint of Legendrian singularity theory.