Algebraic Analysis of Bifurcation and Limit Cycles for Biological Systems

  • Wei Niu
  • Dongming Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5147)


In this paper, we show how to analyze bifurcation and limit cycles for biological systems by using an algebraic approach based on triangular decomposition, Gröbner bases, discriminant varieties, real solution classification, and quantifier elimination by partial CAD. The analysis of bifurcation and limit cycles for a concrete two-dimensional system, the self-assembling micelle system with chemical sinks, is presented in detail. It is proved that this system may have a focus of order 3, from which three limit cycles can be constructed by small perturbation. The applicability of our approach is further illustrated by the construction of limit cycles for a two-dimensional Kolmogorov prey-predator system and a three-dimensional Lotka–Volterra system.


Hopf Bifurcation Center Manifold Algebraic Approach Real Zero Discriminant Variety 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Wei Niu
    • 1
  • Dongming Wang
    • 1
    • 2
  1. 1.Laboratoire d’Informatique de Paris 6Université Pierre et Marie Curie – CNRSParisFrance
  2. 2.LMIB – SKLSDE – School of ScienceBeihang UniversityBeijingChina

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