Investigating Generic Methods to Solve Hopf Bifurcation Problems in Algebraic Biology

Katsuhisa Horimoto, Georg Regensburger, Markus Rosenkranz, and Hiroshi Yoshida (Editors)
Thomas Sturm and Andreas Weber
In proceedings of Algebraic Biology - Third International Conference (AB 2008), Springer-Verlag, 2008
 

Abstract

Symbolic methods to investigate Hopf bifurcations problems of vector fields arising in the context of algebraic biology have recently obtained renewed attention. However, the symbolic investigations have not been fully algorithmic but required a sequence of symbolic computation intervened with ad hoc insights and decisions made by a human. In this paper we discuss the use of generic methods to reduce questions on the existence of Hopf bifurcations in parameterized polynomial vector fields to quantifier elimination problems over the reals combined with the use of the quantifier elimination over the reals and simplification techniques available in \Redlog. We can reconstruct most of the results given in the literature within some seconds of computation time. As no tedious hand computations are involved we presume that the use of these generic methods will be a useful tool for investigating other examples.

Keywords: Hopf bifurcation, quantifier elimination

Bibtex

@INPROCEEDINGS{SturmWeber2008a,
     author = {Sturm, Thomas and Weber, Andreas},
     editor = {Horimoto, Katsuhisa and Regensburger, Georg and Rosenkranz, Markus and Yoshida, Hiroshi},
      title = {Investigating Generic Methods to Solve Hopf Bifurcation Problems in Algebraic Biology},
  booktitle = {Algebraic Biology - Third International Conference (AB 2008)},
     series = {Lecture Notes in Computer Science},
     volume = {5147},
       year = {2008},
  publisher = {Springer-Verlag},
    address = {Castle of Hagenberg, Austria},
   keywords = {Hopf bifurcation, quantifier elimination},
   abstract = {Symbolic methods to investigate Hopf bifurcations problems of vector fields arising in the context
               of algebraic biology have recently obtained renewed attention. However, the symbolic investigations
               have not been fully algorithmic but required a sequence of symbolic computation intervened with ad
               hoc insights and decisions made by a human. In this paper we discuss the use of generic methods to
               reduce questions on the existence of Hopf bifurcations in parameterized polynomial vector fields to
               quantifier elimination problems over the reals combined with the use of the quantifier elimination
               over the reals and simplification techniques available in \Redlog{}. We can reconstruct most of the
               results given in the literature within some seconds of computation time. As no tedious hand
               computations are involved we presume that the use of these generic methods will be a useful tool for
               investigating other examples.},
       isbn = {978-3-540-85100-4},
        doi = {10.1007/978-3-540-85101-1_15}
}