Structural Stability, the Theory of Catastrophes, and Applications in the Sciences and Singularity Theory and an Introduction to Catastrophe Theory

Physics Today ◽  
1978 ◽  
Vol 31 (1) ◽  
pp. 75-77
Author(s):  
P. Hilton ◽  
Y.‐C. Lu ◽  
Lawrence S. Schulman
Keyword(s):  
Structural Stability ◽  
Catastrophe Theory ◽  
Singularity Theory

The Stability Theorems for Subgroups of and

1994 ◽  
Vol 46 (5) ◽  
pp. 995-1006 ◽  
Author(s):  
Ali Lari-Lavassani ◽  
Yung-Chen Lu
Keyword(s):  
Large Class ◽  
Structural Stability ◽  
Singularity Theory ◽  
Stability Theorems ◽  
The Stability

AbstractIn singularity theory, J. Damon gave elegant versions of the unfolding and determinacy theorems for geometric subgroups of . and . In this work, we propose a unified treatment of the smooth stability of germs and the structural stability of versai unfoldings for a large class of such subgroups.

scholarly journals Structural stability of attractor–repellor endomorphisms with singularities

2011 ◽  
Vol 32 (1) ◽  
pp. 1-33
Author(s):  
PIERRE BERGER
Keyword(s):  
Structural Stability ◽  
Compact Subset ◽  
Disjoint Union ◽  
Singularity Theory ◽  
Compact Manifolds ◽  
Hyperbolic Attractor ◽  
Manifolds With Singularities

AbstractWe prove a theorem on the structural stability of smooth attractor–repellor endomorphisms of compact manifolds, with singularities. By attractor–repellor, we mean that the non-wandering set of the dynamics f is the disjoint union of an expanding compact subset with a hyperbolic attractor on which f acts bijectively. The statement of this result is both infinitesimal and dynamical. To our knowledge, this is the first in this hybrid direction. Our results also generalize Mather’s theorem in singularity theory, which states that infinitesimal stability implies structural stability for composed mappings to the larger category of laminations.

Imperfect bifurcation and Banach space singularity theory

1981 ◽  
Vol 31 (3) ◽  
pp. 294-311
Author(s):  
Leif Arkeryd
Keyword(s):  
Banach Space ◽  
Catastrophe Theory ◽  
Singularity Theory ◽  
Classification Theory ◽  
Universal Unfolding ◽  
Space Setting ◽  
Finite Dimensional ◽  
Space Singularity ◽  
Imperfect Bifurcation

AbstractThis paper generalizes the theory of imperfect bifurcation via singularity theory as developed by M. Golubitsky and D. Schaeffer to a Banach space setting. Like the parameter-free potential catastrophe theory, where similar generalizations have been discussed in the literature, Banach control spaces allow useful uniform control of function parameters through the universal unfolding. Among the results are tests for various germ properties and discussion of their reducibility under a Liapunov—Schmidt type splitting, as well as a generalization of the finite dimensional unfolding and germ classification theory.

scholarly journals Computational Aspects of Classifying Singularities

2000 ◽  
Vol 3 ◽  
pp. 207-228 ◽  
Author(s):  
N. P. Kirk
Keyword(s):  
Catastrophe Theory ◽  
Singularity Theory ◽  
Symbolic Algebra ◽  
Local Singularity ◽  
Computational Aspects ◽  
Maple Package

AbstractA Maple package which performs the symbolic algebra central to problems in local singularity theory is described. This is a generalisation of previous projects, which dealt only with problems in elementary catastrophe theory. Applications to specific problems are described, and a survey given of the powerful techniques from singularity theory that are used by the package. A description of the underlying algorithm is given, and some of the more important computational aspects discussed. The package, user manual and installation instructions are available in the appendices to this article.

Time-resolved high-resolution electron microscopy of structural stability in mgo clusters

1996 ◽  
Vol 54 ◽  
pp. 672-673
Author(s):  
T. Kizuka ◽  
N. Tanaka
Keyword(s):  
Electron Microscopy ◽  
High Resolution ◽  
Magnesium Oxide ◽  
Structural Stability ◽  
High Resolution Electron Microscopy ◽  
Video Tape ◽  
Solid State Physics ◽  
Time Resolved ◽  
Resolution Electron ◽  
Structural And Functional Properties

Structure and stability of atomic clusters have been studied by time-resolved high-resolution electron microscopy (TRHREM). Typical examples are observations of structural fluctuation in gold (Au) clusters supported on silicon oxide films, graphtized carbon films and magnesium oxide (MgO) films. All the observations have been performed on the clusters consisted of single metal element. Structural stability of ceramics clusters, such as metal-oxide, metal-nitride and metal-carbide clusters, has not been observed by TRHREM although the clusters show anomalous structural and functional properties concerning to solid state physics and materials science.In the present study, the behavior of ceramic, magnesium oxide (MgO) clusters is for the first time observed by TRHREM at 1/60 s time resolution and at atomic resolution down to 0.2 nm.MgO and gold were subsequently deposited on sodium chloride (001) substrates. The specimens, single crystalline MgO films on which Au particles were dispersed were separated in distilled water and observed by using a 200-kV high-resolution electron microscope (JEOL, JEM2010) equipped with a high sensitive TV camera and a video tape recorder system.

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