Deciding Hopf Bifurcations by Quantifier Elimination in a Software-component Architecture

Abstract

In this paper we give a semi-algebraic description of Hopf bifurcation fixed points for a given parameterized polynomial vector field. The description is carried out by use of the Hurwitz determinants, and produces a first-order formula which is transformed into a quantifier-free formula by the use of usual-quantifier elimination algorithms. We apply techniques from the theory of sub-resultant sequences and of Gröbner bases to come up with efficient reductions, which lead to quantifier elimination questions that can often be handled by existing quantifier elimination packages.We could implement the algorithms for the conditions on Hopf bifurcations by combining the computer algebra system Maple with packages for quantifier elimination using a Java-based component architecture recently developed by the second author. In addition to some textbook examples we applied our software system to an example discussed in a recent research paper.

Computing Hopf Bifurcations in Chemical Reaction Networks Using Reaction Coordinates

“…A fully algebraic method for the computation of Hopf bifurcation for systems with polynomial vector field has been introduced by El Kahoui and Weber [2] using the powerful technique of quantifier elimination on real closed fields [?]. This technique has already been applied to mass action kinetics of small dimension [3].…”

Algebraic Analysis of Bifurcation and Limit Cycles for Biological Systems

“…Such a condition may be given in terms of the Hurwitz determinants and the constant term of the characteristic polynomial of J according to a simple criterion established by El Kahoui and Weber [7]. The criterion was derived for an arbitrary univariate polynomial A to have one pair of purely imaginary roots and all the other roots with negative real parts by linking the Hurwitz determinants ∆ i of A to the principal subresultant coefficients of A 2 and A 1 , where A 1 (λ 2 ) + λA 2 (λ 2 ) = A(λ), and by investigating the behavior of ∆ i in the case where A has symmetric roots.…”

“…The criterion was derived for an arbitrary univariate polynomial A to have one pair of purely imaginary roots and all the other roots with negative real parts by linking the Hurwitz determinants ∆ i of A to the principal subresultant coefficients of A 2 and A 1 , where A 1 (λ 2 ) + λA 2 (λ 2 ) = A(λ), and by investigating the behavior of ∆ i in the case where A has symmetric roots. Under the condition determined by using El Kahoui and Weber's criterion [7,Theorem 3.6], we can transform system (1.1) to a system of special form and then reduce the transformed system to a two-dimensional system of center-focus type by using the center manifold theorem. We will see how the reduction proceeds from the example in the following subsection.…”

“…Prior to the algebraic analysis of bifurcation investigated by the authors [16], other relevant work has been done by El Kahoui and Weber [7] who applied quantifier elimination [6] to Hopf bifurcation analysis, by Lu and others in [12,13] where limit cycles for a class of biological systems are analyzed by using similar algebraic approaches, and by other researchers in the purely mathematical context of bifurcation and limit cycles (see, e.g., [8,18,19]). Hopf bifurcation analyzed in the mathematical context is usually for systems in the canonical form with the origin as their steady state.…”

Investigating Generic Methods to Solve Hopf Bifurcation Problems in Algebraic Biology

“…Whereas theoretically the problem is known to be decidable [9,2,3] the symbolic investigations carried out for specific parameterized polynomial vector fields arising from larger examples, e.g. the ones investigated in [1,2], have not been fully algorithmic up to now but required a sequence of symbolic computation intervened with ad hoc insights and decisions made by a human, and sometimes of sophisticated coordinate transforms.…”