I’ll try to fix this, but I’m tempted to check first!

]]>There’s a small mistake in your analysis of the stability of the fixed point of the Higgins-Selkov system. In one of your comments above, you stated that the fixed point is stable if

and unstable if

The correct condition should be that it’s stable if

and unstable if

(Note the reversed inequalities, and the different exponents on the right hand side.) So what you stated in the article about the system having a limit cycle attractor for *larger* values of is backwards as well.

It’s for this reason that the system shown in Mike Martin’s default example (with ) has a limit cycle attractor, i.e. the fixed point is in fact unstable there. While it is “barely” unstable there, the simulations I’ve been doing show that this system needs to be “barely” unstable like that in order to have a limit cycle attractor. I initially varied the parameters by somewhat large increments, and the behavior quickly became unbounded. This is explained pretty well in Rendall’s notes on it (the last link in your article).

By the way, thanks for this article. I’m using it as an example (in fact, a homework exercise) in my nonlinear dynamics class for biologists.

]]>“Escherichia coli Core Metabolism Model in LIM” by Karline Soetaert

https://cran.r-project.org/web/packages/LIM/vignettes/LIMecoli.pdf

]]>The figure (12.3 from MCB, 6th Edition.) is rich in information if you know the code:

In the case of NAD, the “little plus sign” is located on a quaternary nitrogen – four bonds instead of the usual three for nitrogen (cf the -NH2). So, the two O-s have no bearing on the matter, other than making the overall charge -1. You can’t write a quaternary nitrogen as ‘neutral’. On reduction, the hydrogen can in principle come in from either side of the nicotinamide ring, and reduction to NADH *may* be stereospecific (I haven’t checked). Ideally one would draw the NAD/NADH molecule without the ‘extended’ anhydride bridge (-P-O-P-). Using Rasmol or some other free 3D-modelling tool would provide a better idea of the 3D-shape. It’s worth mentioning that NADH is fluorescent and allows reactions to be followed.

I found this set of teaching slides on “Glycolytic Oscillations” of interest – the insulin oscillation is mentioned as well as the Sel’kov and later models (with references). You also get some experimental chart data!!

Apologies if the comment appears in the wrong place, it can be confusing …

]]>Having only one parameter restricts equilibria bifurcations to three generic possibilities: Andronov-Hopf, saddle-node (roughly, two equilibria collide and disappear) and pitchfork bifurcations (like, stable equilibrium becomes unstable and spawns two stable equilibria; it happens when system has symmetry and thus it’s less generic). In nice (and, possibly, simple) cases bifurcation set is a finite union of hypersurfaces in parameter space; other bifurcation points belong to subsets of codimension greater than 1, but they still belong to this union of hypersurfaces. So this illustrates why it is more likely that we hit Andronov-Hopf/saddle-node/pitchfork bifurcation than any other. ]]>

Theorem.Given a differentiable real dynamical system defined on an open subset of the plane, then every non-empty compact ω-limit set of an orbit, which contains only finitely many fixed points, is either:a) a fixed point,

b) a periodic orbit, or

c) a connected set composed of a finite number of fixed points together with homoclinic and heteroclinic orbits connecting these.

There must be some theorem explaining what catastrophes can happen in this situation if we smoothly vary the system in a way that depends on a single parameter… at least in the generic cases. But I don’t know that theorem.

Clearly Hopf bifurcations are the most famous of these catastrophes.

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