Bifurcation diagrams and caustics of simple quasi border singularities
This paper is devoted to the study of two types of equivalence relations in the space of germs of functions, namely quasi boundary and quasi corner equivalences which were defined in [\textit{V. M. Zakalyukin}, Banach Center Publications 82, 215--225 (2008; Zbl 1151.58024)] and [\textit{F. D. Alharbi} and \textit{V. M. Zakalyukin}, Proc. Steklov Inst. Math. 270, 1--14 (2010) and Trudy Mat. Inst. Steklova 270, 7--20 (2010; Zbl 1225.58017)]. These equivalences are a slight modification of pseudo boundary and pseudo corner equivalences and behave well when the function germs depend on parameters. They play an intermediate role between right equivalence (diffeomorphisms action in the source space) and boundary equivalence (right equivalence preserving a smooth hypersurface). In Section 2 the author gives the main definitions and states the simple singularities and their versal deformations with respect to both equivalence relations. Section 3 explains the correspondence to the classification of Lagrangian projections with a boundary or a corner. Section 4 describes the bifurcation diagrams and the caustics for the simple quasi boundary (corner) singularities. Finally, Section 5 gives an algebraic description of the equivalence classes.
scientific article; zbMATH DE number 6011288