Limit Cycles and Bifurcations in a Biological Clock Model

  • Bálint Nagy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4818)


A three-variable dynamical system describing the circadian oscillation of two proteins (PER and TIM) in cells is investigated. We studied the saddle-node and Hopf bifurcation curves and distinguished four cases according to their mutual position in a former article. Other bifurcation curves were determined in a simplified, two-variable model by Simon and Volford [6]. Here we show a set of bifurcation curves that divide the parameter plane into regions according to topological equivalence of global phase portraits, namely the global bifurcation diagram, for the three-variable system. We determine the Bautin-bifurcation point, and fold bifurcation of cycles numerically. We also investigate unstable limit cycles and the case when two stable limit cycles exist.


limit cycle bifurcation circadian rhythm model 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Bálint Nagy
    • 1
  1. 1.Department of Mathematical AnalysisCollege of DunaújvárosHungary

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