Detailed study of limit cycles and global bifurcations in a circadian rhythm model

Peter L. Simon, András Volford

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3 Citations (Scopus)


A two variable model describing the circadian fluctuation of two proteins (PER and TIM) in cells is considered. The original model was set up by Leloup and Goldbeter [1998], the present form was developed by Tyson et al. [1999]. Periodic solutions with 24-h period were investigated in those papers. Here the possible phase portraits and bifurcations are studied in detail. The saddle-node and Hopf bifurcation curves are determined in the plane of two parameters by using the parametric representation method [Simon et al., 1999], It is shown that there are four cases according to their mutual position (as the remaining parameters of the system are varied). Using these curves the number and type of the stationary points are determined in all four cases. The global bifurcation diagram, which is a system of bifurcation curves that divide the given parameter plane into regions according to topological equivalence of global phase portraits, is also determined. Finally, the so-called constant period curves are computed numerically. These curves consist of those parameter pairs on the parameter plane for which the system has a limit cycle with a given period. It turns out that in a wide range of parameters the system has limit cycles with period close to 24-h.

Original languageEnglish
Pages (from-to)349-367
Number of pages19
JournalInternational Journal of Bifurcation and Chaos
Issue number2
Publication statusPublished - Feb 2006



  • Parametric representation method
  • Saddle-node and Hopf bifurcation
  • Two-codimensional bifurcations

ASJC Scopus subject areas

  • Modelling and Simulation
  • Engineering (miscellaneous)
  • General
  • Applied Mathematics

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