In this case, the Hopf bifurcation is a singular Hopf, in which case the Hopf bifurcation vanishes in the singular limit (i.e. the relevant eigenvalues for the Hopf bifurcation are λ = ±iω with ω = O , and one is not tempted to deduce the criticality of the Hopf bifurcation in the full problem from the dynamics of the layer problem (or the reduced problem).
We may want to know if one can relate the criticality of the Hopf bifurcation obtained in the layer problem to the criticality of the Hopf bifurcation in the full problem.
In recent study [38], Guckenheimer and Osinga investigate two slow-fast systems in which the criticality of a Hopf bifurcation in the full system does not match the criticality of the corresponding Hopf bifurcation in the layer problem.
Thus, we see that, in general, it is not possible to predict the criticality of a Hopf bifurcation in a slow-fast system with two or more fast variables in the limit ε → 0 simply by observing the criticality of the associated Hopf bifurcation in the layer problem.
They show that in each case there is a nearby torus bifurcation in the slow-fast system, and that the family of periodic orbits in the full system is O close to the family of periodic orbits in the layer problem, regardless of the criticality of the Hopf bifurcation.
We then go on to show that there can be problems with the use of GSPT in analysing models with Hopf bifurcations, and in particular show that the criticality of a Hopf bifurcation in a full slow-fast system may not match the criticality of the corresponding Hopf bifurcation in the associated layer problem.
We believe that the above-mentioned approach was critical to the integrity and criticality of the study.
Knowledge of the dynamics in the layer problem is therefore insufficient to predict the criticality of the Hopf bifurcation in the full system.
Criticality of the Hopf bifurcation in the full system is then as in the reduced problem, the slow subsystem.g.
This means that the layer problem cannot be used to make predictions about the criticality of the Hopf bifurcation in the full system.
