LOCAL AND GLOBAL BIFURCATIONS AT INFINITY IN MODELS OF GLYCOLYTIC OSCILLATIONS

We investigate two models of glycolytic oscillations. Each model consi sts of two coupled nonlinear ordinary differential equations. Both mod els are found to have a saddle point at infinity and to exhibit a sadd le-node bifurcation at infinity, giving rise to a second saddle and a stable node at infinity. Depending on model parameters, a stable limit cycle may blow up to infinite period and amplitude and disappear in t he bifurcation, and after the bifurcation, the stable node at infinity then attracts all trajectories. Alternatively, the stable node at inf inity may coexist with either a stable sink (not at infinity) or a sta ble limit cycle. This limit cycle may then disappear in a heteroclinic bifurcation at infinity in which the unstable manifold from one saddl e at infinity joins the stable manifold of the other saddle al infinit y. These results explain prior reports for one of the models concernin g parameter values for which the system does not admit any physical (b ounded) behavior. Analytic results on the scaling of amplitude and per iod close to the bifurcations are obtained and confirmed by numerical computations. Finally, we consider more realistic modified models wher e all solutions are bounded and show that some of the features stemmin g from the bifurcations at infinity are still present.