Talk:Piecewise smooth dynamical systems
I have considered the article on 'Piecewise smooth dynamical systems' submitted by A.R. Champneys and M. di Bernado. I must admit that I am not very impressed by the pedagogical value of this contribution. I think the authors forfeit a good chance to provide insights into the intriguing dynamics that one can observe in piecewise smooth systems and into the scientific and technical areas where such insight is useful. Instead they just present an unsystematic account of a number of largely unexplained phenomena. I suggest that the authors reduce the number of example and instead explain the qualitative aspects of the observed phenomena in sufficient detail for the reader to actually understand why the phenomena appear as they do. The authors can then use the remaining part of the paper to explain under which conditions these phenomena arise. The figures are not of a quality one expects today, and the figure captions do not provide adequate explanations.
Author Champneys :
This reviewer is of course entitled to their opinion on the pedagogical value of the article. However, the author makes no specific criticsms. He claims that our account is "unsystematic" and involves "largely unexplained phenomena". We would of course take issue with this. Is the problem that we have not supplied sufficient references? The explanations may ALL be found in the literature. Perhaps this reviewer is not aware of the literature. Or perhaps the reviewer has a complete mathematically rigorous approach to problems such that unless the theorem has been proved in all generality then one should not state a result. However, on looking at other pages on scholarpedia, we see that such an article would be out of line with others.
Of course we would welcome the opportunity to imporve our article, and therefore we would be most grateful if this reviewer could provide a list of more specific points that we could deal with.
The comment about the quality of the figures is also hard to understand as they have all appeared in the scientific literature.
Reviewer B (second round)
The article has improved significantly during revision. However, there are still a number of details that require attention, before the article can be accepted:
The abbreviations PWS and PWL are used repeatedly in the text without ever being defined.
The figures are fairly poor in quality compared to many other illustrations one can find on the net.
The term ‘system evolution operator’ may be mathematically correct. However, it is hardly appropriate for the level of explanation provided in the article.
It would be useful to indicate x, F(x,mu), S1, and S2 on Fig. 1.
The article refers to the fact that the bifurcations known from smooth systems also occur in piecewise-smooth systems, but fails to mention that they often take a very different form.
Why are DIBs restricted to interactions of fixed points, equilibria, and limit cycles with switching manifolds. What about invariant tori and chaotic sets.
The text describing Fig. 2 should include a direct references to the different panels, i.e. references to Fig. 2a, Fig. 2b, etc. should be given in the text. Moreover, the dynamics on the right hand side of Fig. 1a should be identified to the reader.
The parameter omega used in Fig. 3 appears not to be defined. The information sigma=0 in the figure caption is not meaningful. Instead the coefficient of restitution r should be specified. The reference [3] given in the caption is not correct. Perhaps the proper reference is [5].
Why does the bifurcation in Fig. 4 occur at mu=-.315, and how is mu defined in this example?
The caption to Fig. 5 should have a reference to the book by Feigen.
On Fig. 6, the non-visible parts of the trajectories could be indicated by dotted curves.
The interesting aspects of Fig. 8 remain uncommented: Do we actually observe coexisting solutions, and what is the precise route to chaos?
I think the usual term for periodically driven systems is ‘stroboscopic map’ without reference to Poincare.
It is not necessary to use Monte Carlo techniques to obtain the bifurcation diagram in Figs. 9 and 10.
The authors ought to provide an explanation to the bifurcation diagrams in Fig. 9 in the form, for instance, of a set of figures with iterations on invariable sets of the map at different parameter values.
Likewise, the authors ought to explain the main differences between Fig. 10 and the bifurcation diagram we are used to see for smooth one-dimensional maps.
In connection with Fig. 11, two different specifications are given for the matrix N1. One of these is for N2. Anyway, these specifications are of little general interest, and it would be worth to try to explain in words what the conditions are for a direct transition to chaos in the considered three-dimensional flow system.
The text refers to the different references by numbers, but such numbers are not given in the list of references. Please, refer by author name (year).
Author di Bernardo :
We thank the anonymous reviewer for his comments. We revised the manuscript accordingly to take into account the comments he/she raised. We also reconverted the figures in .png format and modified them where necessary in order to improve their quality (there is an inevitable loss of quality in the conversion). We now believe the article to be ready for pubblication.
