This post continues the discussion in the previous post on this subject. It turns out that the (exact) normal form is a system whose dynamics are not easy to analyse. The bifurcation diagram can be found as Figure 8.8 in the book of Kuznetsov. The case presented there is the supercritical one. The diagram can easily be converted into that describing the subcritical case, the most important thing being that the time direction is reversed so that the types of stability are interchanged. The bifurcation point is the origin in the plane of the parameters . The plane can be written as the union of four open regions separated by bifurcation curves. These regions are distinguished by the types of dynamical behaviour which occur for the given values of the parameters. is simplest, having a trivial dynamics, without steady states or periodic solutions. The boundary of is a smooth curve passing through the origin. If the origin is deleted the remainder consists of two connected components and . The curve is the boundary between and . There a fold bifurcation takes place, producing a saddle point and a sink. The region is that between and the negative part of axis, which we call . On that axis a Hopf bifurcation takes place. The sink turns into a source and a stable periodic solution is born. This is a supercritical Hopf bifurcation. For the other sign of the parameter there is an unstable periodic solution and the Hopf bifurcation is subcritical. This is the reason that we have also adopted the terms sub- and supercritical in the case of a BT bifurcation. (Note that the sign of can be computed from the parameters and .) On the other side of we come to . In that region there is a saddle point and the periodic solution already mentioned. It is the unique periodic solution for given values of the parameters. The other boundary of is a curve where a non-local bifurcation takes place. On the curve there is an orbit which is homoclinic to the saddle. On the other side of is where the homoclinic loop has broken and the qualitative behaviour is similar to that in except that there is a source instead of a sink. From we can pass through and return to by means of a fold bifurcation. It is perhaps worth mentioning what happens on the upper half of the axis. There is no bifurcation there but the two eigenvalues of the linearization are equal and opposite (a neutral saddle). This type of situation may be dangerous when looking for Hopf bifurcations because it shares certain algebraic properties with a Hopf point without being one. For this reason it could make sense to call it a ‘false Hopf point’, which is my internal name for it.

The proofs of the statements about the homoclinic orbit and the periodic solution are hard and I will just make a few comments on that. There is a subtle rescaling which, in particular, ensures that the two steady states are a fixed distance apart, independent of the parameters. Doing this transformation brings the system into the form of a perturbation of a Hamiltonian system. In the latter case the homoclinic orbit is obtained as a level surface of the Hamiltonian. This perturbation can be analysed by a method due to Pontryagin. Use is made of elliptic functions. These things are sketched in some detail by Kuznetsov in an appendix. He also mentions an alternative method for proving the uniqueness of the periodic orbit due to Dumortier and Rousseau. Concerning the fact that the approximate normal form can be transformed to the exact normal form Kuznetsov only gives a very brief sketch of the proof. Combining what is known about the qualitative behaviour of the solutions in normal form and the fact that any system satisfying the Bogdanov-Takens conditions can be reduced to this normal form provides a method for proving the existence and stability of periodic solutions and the existence of a homoclinic loop in given dynamical systems. The genericity conditions can be easier to check than for a Hopf bifurcation since they are conditions on the second derivatives rather than on the third derivatives. How can it be that it is easier to analyse a more complicated bifurcation than a simpler one? The point is that the hardest work is hidden in the proofs of the theorems about the normal form and does not have to be repeated when analysing a concrete system. The use of a BT bifurcation to help prove the existence of a Hopf bifurcation is connected to the idea of an organizing centre. The idea is to obtain insight into a dynamical system by looking for points with extremely special properties. In a given system a point like this of a given type may not exist but when it exists it may be easier to find than a point with a lower degree of speciality, even if the latter occurs more commonly. For instance we can look for a BT point instead of looking for a Hopf point. This type of strategy may be especially useful in systems where there are many parameters over which there is a lot of control. It gives a way of focussing the search in the parameter space. It is the opposite of the situation where you feel that you are looking for something you expect to be plentiful but do not know where to start.

January 26, 2021 at 7:31 am |

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