Singularity: A Maple library for local zeros of scalar smooth maps
Majid Gazor, Mahsa Kazemi · arXiv (Cornell University) · 2015
The local zero structure of a smooth map may qualitatively change, when the map is subjected to small perturbations. The changes may include births and/or deaths of zeros. The qualitative properties are defined as the invariances of an appropriate equivalence relation. The occurrence of a qualitative change in the zero structures is called a bifurcation and the map is named a singularity. The local bifurcation analysis of singularities has been extensively studied in singularity theory and many powerful algebraic tools have been developed for their study. However, there does not exist any available symbolic computer-library for this purpose. We suitably generalize some powerful tools from algebraic geometry for correct implementation of the results from singularity theory. We introduce some required criteria along with rigorous proofs for efficient and cognitive computer-implementation. We have accordingly developed a Maple end-user friendly library, named Singularity, for an efficient and complete local bifurcation analysis of real zeros of scalar smooth maps. We have further constructed a built-in help for Singularity and provided a comprehensive user guide. Recent progresses in Maple have been used so that Singularity generates the list of inequivalent persistent bifurcation diagrams for parametric singularities. The approach is independent of the singularity's codimension (a measure of its complexity). The main features of Singularity are briefly illustrated along with a few examples. Our Maple library is not only useful for research goals and engineering applications involving singularities but also for pedagogical purposes.