Structurally stable Grassmann transformations
TL;DR: In this paper, the authors characterized the structurally stable Grassmann transformations as the maps which are induced by matrices whose eigenvalues have distinct moduli and showed that the topological classification of these maps is determined by the ordering of the signs of the eigen values of the inducing matrix.
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Abstract: A Grassmann transformation is a diffeomorphism on a Grassmann manifold which is induced by an n x n nonsingular matrix. In this paper the structurally stable Grassmann transformations are characterized to be the maps which are induced by matrices whose eigenvalues have distinct moduli. There is exactly one topological conjugacy class of complex structurally stable Grassmann transformations. For the real case the topological classification is determined by the ordering (relative to modulus) of the signs of the eigenvalues of the inducing matrix.
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Phase portrait of the matrix Riccati equation
TL;DR: The Riccati equation as mentioned in this paper is a quadratic differential equation on the space of real symmetric matrices, which is closely related, via compactification of the phase space, to the differential equations on the Grassmann manifold and the Lagrange-Grassmann manifold.
103
The geometry of matrix eigenvalue methods
Gregory S. Ammar,Clyde F. Martin +1 more
TL;DR: In this paper, the relation between the matrix Riccati equation and the standard matrix eigenvalue methods is explored. And it is demonstrated that the mathematics of the analysis of the two objects is essentially the same; consisting of the flow analysis of flows on the homogeneous spaces of various Lie groups.
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The dynamics of Morse-Smale diffeomorphisms on the torus
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The Dynamics of Eigenvalue Computation
Steve Batterson
- 01 Jan 1993
TL;DR: The theme of this chapter is that techniques from dynamical systems can be applied to the study of certain problems in numerical analysis and this chapter will focus on the particular numerical analysis problem of approximating the eigenvalues of a real matrix.
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Frank Wilson Warner
- 01 Jun 1971
TL;DR: Foundations of Differentiable Manifolds and Lie Groups as discussed by the authors provides a clear, detailed, and careful development of the basic facts on manifold theory and Lie groups, including differentiable manifolds, tensors and differentiable forms.
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A structural stability theorem
TL;DR: In this article, it was shown that certain geometric conditions are sufficient for strutural stability, i.e. qualitative properties of stable diffeomorphisms are unchanged by small C' perturbations.
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Differentiable dynamics;: An introduction to the orbit structure of diffeomorphisms
Zbigniew Nitecki
- 01 Jan 1971
TL;DR: The subject of differentiable dynamical systems in the form recently developed by the group of mathematicians associated with S. Smale and M. Peixoto in the United States and with Ja.
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Structural stability of C1 diffeomorphisms
TL;DR: In this article, it was shown that if f is a C 1 diffeomorphism that satisfies Axiom A and the strong transversality condition then it is structurally stable.
177
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