Gröbner bases and multi-dimensional persistent bifurcation diagram classifications

Abstract

The work presented here is motivated by our ongoing project on symbolic bifurcation analysis of multidimensional germs of singularities, say [EQUATION] where z and λ denote the state variables and a distinguished parameter, respectively. A local solution Z(λ) for f(Z(λ), λ) = 0 is called a bifurcation diagram. Any two bifurcation diagrams Z₁(λ) and Z₂(λ) of f (z, λ) are called qualitatively equivalent when there exists a diffeomorphism (Φ(Z₁, λ₁), Λ(λ₁)) transforming the bifurcation diagram Z₁(λ₁) into Z₂(λ₂) = Φ(Z₁(Λ^-1(lD₂)), Λ^-1(λ₂)). Our proposed bifurcation analysis often corresponds to steady state bifurcations of PDEs, static and dynamical systems and occur in many real life and engineering control problems [5]. This will be integrated into the Singularity library [3, 4] as MultiDimSingularities module. Our library provides systematic tools for symbolic treatment of bifurcation analysis of such systems. The classical approach uses root finding and parametric continuation methods through numerical normal form analysis for differential systems with low codimension (degeneracy). The latter fails on comprehensive analysis for a parametric system and for systems with moderate degeneracies. The goal here is to classify the qualitatively different bifurcation diagrams of an unfolding germ [EQUATION], for f (z, λ), where α = (α₁,...,α_p) are unfolding parameters and F(z, λ, 0) = f(z, λ). This classification is feasible via the concept of persistent bifurcation diagrams. A diagram is called persistent when it is qualitatively self-equivalent under arbitrarily small perturbations. The idea is to find non-persistent sources of bifurcations. Non-persistent sources are divided into three categories: bifurcation B, hysteresis H and double limit point D; see [6, Page 410]. The set [EQUATION] is called a transition local variety, that is a codimension-1 hyperplane in the parameter space. A local variety refers to a neighborhood subset of zeros of a polynomial system. The complement space of this hyperplane consists of a finite number of connected components, say C₁, ..., C_n. For any i and α, β ∈ C_i, the bifurcation diagrams Z(λ, α) and Z(λ, β) are qualitatively equivalent. Hence, a choice of bifurcation diagram from each connected component gives rise to a complete list of persistent bifurcation diagram classifications for F(z, λ, α). The RegularChains library in Maple provides a tool through the command CylindricalAlgebraicDecompose enabling us to generate this list; see [1]. However, there usually exist multiple bifurcation diagrams from each connected component C_i in the generating list. In this abstract paper, we merely focus on the use of Gröbner basis for multi-dimensional persistent bifurcation diagram classifications.