Talk:Homoclinic orbits of time-reversible systems

 : Report on the manuscript Homoclinic time reversible system

Dynamics of homoclinic orbits in vector fields forms an active field of research, see A.J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in Handbook of Dynamical Systems, vol. 3, 379-524 North-Holland, Amsterdam, 2010.

The present article deals with homoclinic orbits in reversible systems. But, from the beginning on, the authors limit their scope of consideration to

- reversible systems in in \({\mathbb R}^{2n}\) - the dimension of the state space is even,

- symmetric homoclinic orbits.

In the course of this they survey the behavior of symmetric homoclinic orbits and the dynamics nearby those orbits. Altogether this article conveys a brief glimpse of homoclinic dynamics in reversible systems. This is also reflected in the rather concise list of references.

Amendments and complements

I suggest to rename the article in Homoclinic orbits of time reversible systems.

I suggest the following rearrangements: The subparagraphs Robust existence of symmetric homoclinics and An example should become separate paragraphs. New succession: Formal definitions, An example, Robust existence of symmetric homoclinics, ... .

The given definition of a symmetric solution is very special\[x(t)=R x(-t)\] implies that \(x(0)\in {\rm Fix}\,(R)\). More general it is true that the corresponding orbit is an \(R\)-image of itself -- this implies that \(x(\tau)\in {\rm Fix}\,(R)\) for some \(\tau\). Possibly a brief description of the impact of reversibility would be in order (cf. Vanderbauwhede and Fiedler 1992).

A hyperbolic symmetric equilibrium implies that the dimension of the state space has to be even, and that \(\dim{\rm Fix}\,(R)=n\), cf. paragraph Robust existence of symmetric homoclinics. At the end of this paragraph some references would be appreciated.

At the end of the paragraph Dynamics near homoclinic orbits also J. Härterich, Cascades of reversible homoclinic orbits to a saddle-focus equilibrium, Physica D 112, No. 1-2, 187-200 (1998) should be cited.

Mention dynamics in more general reversible systems, e.g. J.S.W. Lamb, M.-A. Teixeira and K.N. Webster, Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in \({\mathbb R}^3\), J. Differ. Equations 219, No. 1, 78-115 (2005).

Mention dynamics related to nonsymmetric homoclinic orbits (in reversible systems), e.g. A. J. Homburg and J. Knobloch, Multiple homoclinic orbits in conservative and reversible systems, Trans. Am. Math. Soc. 358, No. 4, 1715-1740 (2006).

Second Review of this scholarpedia contribution

This is a well written summary. My main concern is the frequent use of the word "typical", which does not always seem to correct. Two examples:

- "Typically such systems of ordinary-differential equations (ODEs) arise as steady state or travelling-wave reductions". Time reversible systems are much more general than this, though I agree that homoclinic orbits are of interest primarily in the context of travelling-wave equations. Perhaps replace "Typically" by "Often"?

- "In the typical situation, we have dim Fix(R)=n, and the equilibrium at is hyperbolic." This is not correct as nonhyperbolic reversible equilibria are structurally stable. The suggestion mentioned by the first reviewer remedies this.

Author Champneys : corrections to article

We have made all the changes suggested by the reviewers. Thank you for these positive suggestions.