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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 use of usual quantifier elimination algorithms. We
apply
techniques from the theory of sub-resultant sequences and of Gr¨obner 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.
