Talk:Heteroclinic cycles

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    Heteroclinic cycles article: Review

    The article reviews and defines heteroclinic cycles in dynamical systems. The author shows how one may find a heteroclinic cycle and presents examples of dynamical systems exhibiting heteroclinic cycles.

    This article is very well written, short and ideal for Scholarpedia. The details appear to be correct. I have the following suggestions which may help:

    1. In the definition, I feel the `homoclinic cycle' should mentioned as a specific case of the heteroclinic cycle where the sequences of connections returns to the beginning. Homoclinic cycles are commonly cited in the literature. In the sentence beginning 'As time evolves...', the author talks about spending increasingly longer periods of time near each solution. Does the author mean solution or `equilibrium and/or periodic orbit'?

    RESPONSE: Homoclinic cycles are now mentioned in the article.

    2. In the section on Finding Heteroclinic cycles, the author generalises the paper of Melbourne, Choosat and Golubitsky (1989) to consider O(N) equivariant dynamical systems. Are readers expected to know what the orthogonal symmetry group is? The author deals with finding a specific type of heteroclinic cycle i.e., where the connections exist in 2D space. In this specific setting the Poincare-Bendixson theorem can be used to prove the existence of the connecting trajectories. More complicated heteroclinic cycles exist and all that is generally needed to be known is that the equilibria in Fix(\sum_j) is a saddle and the equilibria in Fix(\sum_{j+1}) is a sink in the Fix(T_j) (See Krupa and Melbourne (1995)) though the connections cannot in general be proved. I feel this should be mentioned as it would relate to cycles in PDEs. I also feel it could be made a bit more clear that the equilibrium in Fix(\sum_j) must be a saddle in Fix(T_j) whereas the equilibrium in Fix(\sum_{j+1}) must be a sink in Fix(T_j).

    RESPONSE: Done, text has been modified accordingly.

    3. Robustness, Asymptotic stability and nearly asymptotic stability are interesting questions about heteroclinic cycles and it might be worth mentioning by citing the Krupa and Melbourne (1995) paper.

    RESPONSE: Done.

    4. In the examples section, i feel the elementary (and possibly first) example of a heteroclinic cycle described by Guckenhiemer and Holmes Math. Proc. Camb. Phil. Soc 103:189-92 (1988) would be a good idea to go through. Here you can write out the ODE simply and describe the fixed point subspaces etc. I feel this would be very instructive to people who are not experts in area but may come across heteroclinic cycles. Also one could draw a picture of the phase space and show the fixed point subspaces. I also feel the ODEs used in Figures 2 and 3 should be written down and the parameters stated.

    RESPONSE: A new section describing the Guckenheimer-Holmes cycle has been added, as well as phase-space pictures and the ODEs, including parameters.

    5. I feel it would be worth mentioning the heteroclinic cycle in the Kuramoto-Sivashinsky PDE as an example of the heteroclinic cycle found in the O(2)-equivariant system by citing Holmes, Lumley and Berkooz `Turbulence, Coherent Structures, Dynamical systems and Symmetry' (1996) Cambridge University Press.

    RESPONSE: A section describing the heteroclinic cycle in the Kuramoto-Sivashinsky equation has been added. Phase-space pictures with the corresponding spatio-temporal patterns of the dynamics associated with the heteroclinic cycle have also been added.

    Review 2

    This article provides a clear concise summary of what heteroclinic cycles are.

    Unfortunately the literature cited gives a misleading view of the history of the subject. For example, the 'Guckenheimer-Holmes' cycle was first written down, and its behaviour described, by Busse and Heikes in 1980 (Science 208, 173-175).

    RESPONSE: Thank you for the clarification. The correction is now done in the first paragrph that describes the 'Guckenheimer-Holmes' cycle.

    Another example is the heteroclinic cycle in the 1:2 resonance problem. This was first identified by Jones and Proctor in 1987 (Phys Lett A 121, 224-228), followed up by a longer paper in J Fluid Mech.

    RESPONSE: A sentence has been added to reference this other example. The author is not sure about the exact reference for the 'longer paper in J. Fluid Mech'. So perhaps this reviewer would like to edit the article to include the exact reference. Peer review/editing is allowed for this article and would be welcome. Thank you again for the comments.

    Thanks for including those references so quickly. The longer paper is JFM 188, 301-335 (1988) but the article doesn't really need both. I was thinking that the Jones-Proctor citation could go in the final section of the article (Cycle in a PDE with circular symmetry)? Also I wonder if that cycle should be regarded as homoclinic since I think the fixed points are on the same group orbit in that case, like the Guckenheimer-Holmes-Busse-Heikes example. Also I fixed a few typos.
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