Local and global bifurcations at infinity in models of glycolytic oscillations |
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Authors: | J Sturis M Brøns |
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Institution: | Scientific Computing, Novo Nordisk, DK-2880 Bagsv?Grd, Denmark, DK Department of Mathematics, Technical University of Denmark, DK-2800 Lyngby, Denmark, DK
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Abstract: | We investigate two models of glycolytic oscillations. Each model consists of two coupled nonlinear ordinary differential
equations. Both models are found to have a saddle point at infinity and to exhibit a saddle-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 the bifurcation, and after the bifurcation, the stable node at infinity
then attracts all trajectories. Alternatively, the stable node at infinity may coexist with either a stable sink (not at infinity)
or a stable limit cycle. This limit cycle may then disappear in a heteroclinic bifurcation at infinity in which the unstable
manifold from one saddle at infinity joins the stable manifold of the other saddle at infinity. These results explain prior
reports for one of the models concerning parameter values for which the system does not admit any physical (bounded) behavior.
Analytic results on the scaling of amplitude and period close to the bifurcations are obtained and confirmed by numerical
computations. Finally, we consider more realistic modified models where all solutions are bounded and show that some of the
features stemming from the bifurcations at infinity are still present.
Received 4 September 1995; received in revised form 18 September 1996 |
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