Talk:Global bifurcations

The entry is nicely structured and informative. However, the following issues require attention:

1. The terms "singular point" and "fixed point" are used at the moment where the usual "equilibrium point" would sound better. There is also a misprint to correct: The second section is about honhyperbolic cycles, not "singular points".

2. Figure 5 apparently shows the bifurcation of a STABLE limit cycle (b) from the UNSTABLE homoclinic loop (a), which is impossible. The homoclinic loop in Fig. 5(a) should be drawn as stable from inside.

3. Although complete details seem inappropriate in such a text, it should be mentioned that "a countable set of Smale horseshoes" occurs near a homoclinic loop of a hyperbolic saddle with one real and two complex eigenvalues ONLY if the complex eigenvalues are closer to the imaginary axis than the real one (if the real eigenvalue is positive - as in the figure - than the saddle value must be positive). If the real eigenvalue is closer (the saddle value is then negative), than only one cycle appears. All this is explained in the very good entry "Shilnikov bifurcation" of Scholarpedia, that has to be referred to.

4. I also miss several references to textbooks, including

   * Shilnikov L.P., Shilnikov A., Turaev D. and Chua, L. [1998] Methods of Qualitative Theory in Nonlinear Dynamics. Part I. World Scientific. 

   * Shilnikov L.P., Shilnikov A., Turaev D. and Chua, L. [2001] Methods of Qualitative Theory in Nonlinear Dynamics. Part II.World Scientific. 

   * Yu.A. Kuznetsov [2004] Elements of Applied Bifurcation Theory, Springer, 3rd edition.

Reviewer B:

The manuscript is a good example showing how it is possible to present in clear form an essence of such huge material as Global Bifurcations. I just have one suggestion. I believe that an encyclopedical article should have "definicia" but in the article it is absent. I suggest to use the author book and before the word "Contents" to add the following:

The formal definition [AAIS] can be done as follows:

Definition. The bifurcation is local if the support of bifurcation is a singular point or a periodic orbit, otherwise the bifurcation is global. A finite subset of phase space is said to be the support of bifurcation if there exists an arbitrary small neighborhood of this subset and the neighborhood of the bifurcation value of paramenters (depending on it) such that, outside this neighborhood of this subset, the systems are topologically equivalent.

The mostly studied global bifurcations are those for which the support of bifurcation consists of finitely many homoclinic or heteroclinic trajectories.

END OF INSERTION

Small remarks:

p.1, 4th line from above: nonwondering <-> nonwandering. p.2, The title: Nonhyperbolic singular points <-> nonhyperbolic cycles. references: ABS 212. <-> Trans. of the Moscow Math. Soc. 44(1983)153-216.

Further reading: I believe it should be added the books of Shil'nikov, Shil'nikov, Turaev, Chua and the book of Kuznietsov (and maybe some others).

Reviewer A:

from the second reviewer: In my view the paper has added a little to nothing to the existing articles on global bifurcations including Shilnikov saddle-node, homoclinic saddle-focus, blue sky bifurcations etc, which are less laconically presented and more impressively visualized. So, i do not really see a need for one covering global bifurcations, rather then a collection of internal pointers to the previously submitted texts.

I concur with saying earlier. Besides, the article does not cite the original work by L Shilnikov who is named the founding father of global/homoclinic bifurcation theory. This was pointed out by other reviews as well

• Shilnikov L.P., Shilnikov A., Turaev D. and Chua, L. [1998] Methods of Qualitative Theory in Nonlinear Dynamics. Part I. World Scientific.

• Shilnikov L.P., Shilnikov A., Turaev D. and Chua, L. [2001] Methods of Qualitative Theory in Nonlinear Dynamics. Part II.World Scientific.

Original papers by Shilnikov and Turaev on the Blue Sky Catastrophe are missing