| Full text | |
| Author(s): |
de Rezende, Ketty A.
;
Lima, Dahisy V. S.
;
Zigart, Murilo A. de Jesus
Total Authors: 3
|
| Document type: | Journal article |
| Source: | SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES; v. 19, n. 1, p. 53-pg., 2025-06-01. |
| Abstract | |
Peixoto's stability theorem stands as a cornerstone in the global dynamical examination of flows on smooth two-manifolds, a significant landmark in Dynamical Systems research. This theorem has served as a blueprint for subsequent global classification theorems within the field. Building upon Peixoto's foundational work, Gutierrez and Sotomayor introduced a compelling generalization and their contribution extends Peixoto's conditions for structural stability of C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C<^>{1}$$\end{document}-vector fields on smooth surfaces to encompass singular two-manifolds M. Furthermore, generalizing this classical theorem to varied and richer topological configurations such as these non-smooth surfaces which feature singular loci comprising cones (C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document}), cross-caps (W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{W}}$$\end{document}), double (D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}}$$\end{document}), and triple points (T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{T}}$$\end{document}) marks a milestone for research in Singular Dynamics. In homage to their contributions, we have named this class of dynamical systems as Gutierrez-Sotomayor flows, GS flows for short. It is our intent, in this article to produce a survey of the state of the art for GS flows which have garnered significant attention in the past years. Our interest is two-fold: firstly present a local and global analysis of GS flows phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi$$\end{document} on singular surfaces M and secondly describe the effects of homotopical deformations on (phi,M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\varphi , M)$$\end{document} which are in correspondence to a spectral sequence of an associated chain complex for phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi$$\end{document}. Herein we address the far reaching results that are obtained by using Spectral Sequence Theory which has yielded several homotopical cancellation theorems within the dynamics. (AU) | |
| FAPESP's process: | 23/03430-8 - Sufficient conditions for the realization of Lyapunov graphs as Gutierrez-Sotomayor flows |
| Grantee: | Murilo André de Jesus Zigart |
| Support Opportunities: | Scholarships in Brazil - Post-Doctoral |
| FAPESP's process: | 22/16455-6 - Algebraic, geometric, and differential topology |
| Grantee: | Daciberg Lima Gonçalves |
| Support Opportunities: | Research Projects - Thematic Grants |
| FAPESP's process: | 23/17645-6 - Morse-Conley Theory, Singular Varieties, and Intersection Homology |
| Grantee: | Dahisy Valadão de Souza Lima |
| Support Opportunities: | Scholarships abroad - Research |
| FAPESP's process: | 18/13481-0 - Geometry of control, dynamical and stochastic systems |
| Grantee: | Marco Antônio Teixeira |
| Support Opportunities: | Research Projects - Thematic Grants |