The Higgins-Selkov oscillator, part 2

Some time ago I wrote about a mathematical model for glycolysis, the Higgins-Selkov oscillator. In fact I now prefer to call it the Selkov oscillator since it is a system of equations written down by Selkov, although it was obtained by modifying a previous model of Higgins. At that time I mentioned some of the difficulties involved in understanding the global properties of solutions of this system. I mentioned that I had checked that this system exhibits a supercritical Hopf bifurcation, thus proving that it has stable periodic solutions for certain values of the parameters. In the hope of getting more insights into this system I decided to make it the theme of a master’s thesis. Pia Brechmann, the student who got this subject, obtained a number of interesting results. After she submitted her thesis she and I decided to carry this a bit further so as to get a picture of the subject which was as comprehensive as possible. We have now written a paper on this subject. I cannot say that we were able to answer all the questions we would have liked to but at least we answered some of them. Here I will mention some of the things which are known and some which are not.

When written in dimensionless form the system contains two parameters (a positive real number) and (an integer which is at least two). I already mentioned a result about a Hopf bifurcation and this was originally only proved for . In the paper we showed that there is a supercritical Hopf bifurcation for any value of . I also previously mentioned doing a Poincaré compactification of the system and blowing up new stationary solutions which appear in this process and are too degenerate to analyse directly. The discussion of using blow-ups in polar coordinates previously given actually concerned the case and does not seem practical for higher values of . It turns out that the technique of quasihomogeneous directional blow-ups, explained in the book ‘Qualitative theory of planar differential systems’ by Dumortier et al. can be used to treat the general case. This type of blow-up has the advantage that the transformations are given by monomials rather than trigonometric functions and that there is a systematic method for choosing good values of the exponents in the monomials.

We discovered a paper by d’Onofrio (J. Math. Chem. 48, 339) on the Selkov oscillator where he obtains interesting results on the uniqueness and stability of periodic solutions. It was suggested by Selkov on the basis of numerical calculations that for large there exist solutions with oscillations which grow arbitrarily large at late times (let us call these unbounded oscillations). We were not able to decide on a rigorous level whether such solutions exist or not. They cannot exist when a periodic solution does. When they exist they have to do with a heteroclinic cycle in the compactification. We showed that when a cycle of this kind exists it is asymptotically stable and in that case solutions with unbounded oscillations exist. However we were not able to decide whether a heteroclinic cycle at infinity ever exists for this system. What we did prove is that for all values of the parameters there exist unbounded solutions which are eventually monotone.

We also proved that when the steady state is stable each bounded solution converges to it and that when there exists a periodic solution it is unique and each bounded solution except the steady state converges to that. I find it remarkable that such an apparently harmless two-dimensional dynamical system is so resistent to a complete rigorous analysis.

Advertisement