Limit cycles and bifurcations in a biological clock model

Abstract

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. © 2008 Springer.