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Sufficient conditions for the realization of Lyapunov graphs as Gutierrez-Sotomayor flows

Grant number: 23/03430-8
Support Opportunities:Scholarships in Brazil - Post-Doctoral
Effective date (Start): June 01, 2023
Effective date (End): September 30, 2024
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal Investigator:Marco Antônio Teixeira
Grantee:Murilo André de Jesus Zigart
Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
Associated research grant:18/13481-0 - Geometry of control, dynamical and stochastic systems, AP.TEM


With this project, we intend to perform a study about global aspects of flows tangent to singular manifolds with regular points (R), cones (C), cross-caps (W), double crossing points (D) and triple crossing points (T), known as Gutierrez-Sotomayor flows, for short, GS flows. Using Lyapunov graphs as a combinatorial tool and methods from Conley Index Theory, we will approach problems such as the tracking down of the boundaries of isolating blocks which are local realizations of each vertex and its incident edges of an abstract graph as a GS flow on a closed singular manifold. More precisely, we will study a characterization of sufficient conditions that guarantee the realizability of abstract graphs as GS flows, which allows the contruction of several examples and also qualitative studies in general, such as the study of the Euler characteristic and other topological invariants in the singular context. Given the presence of singular orbits associated to the singularities W, D and T of a GS flow, referred to as folds, we have the initial goal of determining tools to identify whether two codimension one singular manifolds present on the boundary of two isolating blocks for GS singularities are homeomorphic, since this analysis becomes more troublesome as the number of folds passing through a block increases arbitrarily. In general, non homeomorphic boundaries are the main obstructing factor which makes the realization of graphs unviable, although in the literature, there are a few cases with a positive outcome, namely: linear graph; graphs with bifurcations for which all the folds are associated to singularities W; and graphs with specific amounts of folds, called minimal graphs. To proceed, we will investigate partial results on graphs with bifurcations for which the folds are associated to singulariteis W and D simultaneously. At last, we will use the achieved results to conduct the study of the equivalent problem in dimension three, where the global realization of abstract graphs as GS flows is a complete open question.

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