In this research, a comprehensive theoretical investigation has been conducted on oxygen atom transfer (OAT) reactions from Mimoun complexes to sulfides and sulfoxides. The joint use of the electron localization function (ELF) and Thom's catastrophe theory (CT) provides a powerful tool to analyze the evolution of chemical events along a reaction pathway. The progress of the reaction has been monitored by structural stability domains from ELF topology while the changes between them are controlled by turning points derived from CT which reveal that the reaction mechanism can be separated in several steps: first, a rupture of the peroxo O1-O2 bond, then a rearrangement of lone pairs of the sulfur atom occurs and subsequently the formation of S-O1 bond. The OAT process involving the oxidation of sulfides and sulfoxides is found to be an asynchronous process where O1-O2 bond breaking and S-O1 bond formation processes do not occur simultaneously. Nucleophilic/electrophilic characters of both dimethyl sulfide and dimethyl sulfoxide, respectively, are sufficiently described by our results, which hold the key to unprecedented insight into the mapping of electrons that compose the bonds while the bonds change.