Symbolic local bifurcation analysis of scalar smooth maps

Abstract

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 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 provide 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 written a comprehensive user-guide for Singularity. The main features of Singularity are briefly illustrated along with a few examples.