Mode decomposition of global bifurcation diagram with Gröbner bases

Abstract

We propose an algebraic method for the mode decomposition of bifurcation diagram in global parameter space. We investigate the bifurcation behavior of some periodic points of discrete dynamical systems. By using Gröbner bases, we can obtain the algebraic representation of bifurcation diagrams which gives global characteristics of bifurcation diagrams. We also show that the factorization of the algebraic representation gives the mode decomposition of global bifurcation diagrams. Further, we demonstrate that the separation removes singularities on bifurcation points. The essential point of our method is to make use of the ideal generated by the equation of a system.