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 \(C^{1}\) -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 ( \({\mathcal{C}}\) ), cross-caps ( \({\mathcal{W}}\) ), double ( \({\mathcal{D}}\) ), and triple points ( \({\mathcal{T}}\) ) 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 \(\varphi\) on singular surfaces M and secondly describe the effects of homotopical deformations on \((\varphi , M)\) which are in correspondence to a spectral sequence of an associated chain complex for \(\varphi\) . 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.
