Gutierrez-Sotomayor flows on singular surfaces - de Rezende, Ketty A.; Grulha Jr., Nivaldo G.; Lima, Dahisy V. de S.; Zigart, Murilo A.J. (Topological Methods in Nonlinear Analysis, 2022)
In this work, we consider the collection of necessary homological conditions previously obtained via Conley index theory for a Lyapunov semi-graphto be associated to a Gutierrez-Sotomayor flow on an isolating block and address their sufficiency. These singular flowsinclude regular $\mathcal{R}$, cone $\mathcal{C}$, Whitney $\mathcal{W}$, double $\mathcal{D}$ and triple $\mathcal{T}$ crossing singularities.Local sufficiency of these conditions are proved in the case of Lyapunov semi-graphs along with a complete characterization ofthe branched $1$-manifolds that make up the boundary of the block. As a consequence, global sufficient conditions are determinedfor Lyapunov graphs labelled with $\mathcal{R}$, $\mathcal{C}$, $\mathcal{W}$,$\mathcal{D}$ and $\mathcal{T}$ and with minimal weightsto be associated to Gutierrez-Sotomayor flows on closed singular $2$-manifolds.By removing the minimality condition, we prove other global realizability resultsby requiring that the Lyapunov graph be labelled with $\mathcal{R}$,$\mathcal{C}$ and $\mathcal{W}$ singularities or that it be linear.