Normal Form Analysis of ℤ2-Equivariant Singularities

Abstract

Singular parametric systems usually experience bifurcations when their parameters slightly vary around certain critical values, that is, surprising changes occur in their dynamics. The bifurcation analysis is important due to their applications in real world problems. Here, we provide a brief review of the mathematical concepts in the extension of our developed Maple library, Singularity, for the study of [Formula: see text]-equivariant local bifurcations. We explain how the process of this analysis is involved with algebraic objects and tools from computational algebraic geometry. Our procedures for computing normal forms, universal unfoldings, local transition varieties and persistent bifurcation diagram classifications are presented. Finally, we consider several Chua circuit type systems to demonstrate the applicability of our Maple library. We show how Singularity can be used for local equilibrium bifurcation analysis of such systems and their possible small perturbations. A brief user interface of [Formula: see text]-equivariant extension of Singularity is also presented.