Conley theory for Gutierrez-Sotomayor fields

Journal of Singularities
volume 22 (2020), 241-277

Received: 13 March 2019. Received in revised form: 2 June 2020.

DOI: 10.5427/jsing.2020.22q

Add a reference to this article to your citeulike library.

Abstract:

In [6], a characterization and genericity theorem for C^1-structurally stable vector fields tangent to a 2-dimensional compact subset M of R^k are established. Also in [6], new types of structurally stable singularities and periodic orbits are presented. In this work we study the continuous flows associated to these vector fields, which we refer to as the Gutierrez-Sotomayor flows on manifolds M with simple singularities, GS flows, by using Conley Index Theory. The Conley indices of all simple singularities are computed and an Euler characteristic formula is obtained. By considering a stratification of M which decomposes it into a union of its regular and singular strata, certain Euler type formulas which relate the topology of M and the dynamics on the strata are obtained. The existence of a Lyapunov function for GS flows without periodic orbits and singular cycles is established. Using long exact sequence analysis of index pairs we determine necessary and sufficient conditions for a GS flow to be defined on an isolating block. We organize this information combinatorially with the aid of Lyapunov graphs and using a Poincaré-Hopf equality we construct isolating blocks for all simple singularities.

Author(s) information:

H. Montúfar
Faculty of Mathematics
Federal University of Uberlândia
Uberlândia - Brazil
email: montufar@ufu.br

K. A. de Rezende
Institute of Mathematics and Scientific Computing
University of Campinas
Campinas - Brazil
email: ketty@unicamp.br