Talk:Unstable periodic orbits

Review 1

Second report:

I checked the paper and found the revisions satisfactory. There are two corrections I would like to be implemented: Since (3) is an approximation to (2) for large Lyapunov exponents, it would be best to say so. The current wording leaves the impression that there are two equally good arguments that arrive at different expressions. For instance, replace 'In particular,....' with:

This shows the intimate relation between the invariant density of chaotic sets and periodic orbits. For strongly unstable systems, the determinants can be expanded and only the unstable directions contribute. Then (2) becomes approximately... (and then display eq (3)).

After the sixth equation in the section on cycle expansion, the word 'interoperate' should probably be 'interpret'

And there are a couple of typos.

First report:

Overall, the description of the theory and applications of periodic orbits are very good and should give the reader an idea of what is involved. The examples are well chosen, and should also help applications in the Neurosciences. The comments I have are about a few points where a little more mathematical rigor could be presented.

In the first paragraph, when the exponential growth of periodic orbits is mentioned, I would add a reference to

Stephen Smale, 1967. Differentiable dynamical systems. Bulletin of the American Mathematical Society, 73: 747–817

He already uses zeta functions to calculate entropies.

The second paragraph starts with the statement: A chaotic system can be modelled by a hierarchy of UPOS,... I do not like the verb 'model' in this context. I would rather say that the system can be represented by its periodic orbits. To me, the model is the choice of the set of equations or maps, which capture a subset of the features of the system. Once the mathematical equations are chosen, the use of periodic orbit theory is the application of a mathematical technique with very few degrees of freedom (in contrast, there are quite a few choices for the model itself). Secondly, any system (but pathological ones) can be represented by their periodic orbits. That the orbits are, in addition, hierarchically organized, is an additional feature. Thus I would say: A chaotic system can be represented and approximated by its periodic orbits. The periodic orbits are typically hierarchially organized, and an increasing accuracy can be achieved by ..

In the section on the natural measure, the descriptions of the weights is qualitatively ok, but mathematically incomplete: formally, the probability to remain in a set \(S\) is given by \[ \mu(S)=\lim_{n\rightarrow\infty} \int_S dx \delta(x-f^{(n)}(x)) \] and this integrates to \[ \mu(S)=\sum_{Fix(n)} \lim_{n\rightarrow\infty} \frac{1}{|det(1-Df^{(n)}(x))|}\,. \] The origin of this is

L.P. Kadanoff and C. Tang, Escape from strange repellors, PNAS 81, 1276--1279 (1984),

and the continuous situation is discussed in

Cvitanovi´c, P. and Eckhardt, B. 1991. Periodic orbit expansions for classical smooth flows. Journal of Physics A, 24: L237–-L241

Denoting the eigenvalues of the derivative matrix \(Df^{(n)}\) by \(\Lambda(x,n)\), this becomes to leading order equal to the expression (1) given in the notes, but only to leading order. The difference is that the full zeta function calculated from this expression (which becomes a product of zeta functions of the type given later) is analytic, whereas the individual ones are not (as demonstrated in

G. Russberg and B. Eckhardt, Resummation of classical and semiclassical periodic orbit expressions, Phys. Rev. E 47, (1993) 1578--1588.)

I would generally recommend to use captial \(\Lambda\) for the eigenvalues of the derivative matrix of a map, and small \(\lambda\) for the Lyapunov exponents. If, in the case of a continuous system, \(Df\) are the eigenvalues of the monodromy matrix for an orbit of period \(T\), then \(\Lambda = \exp(\lambda T)\).

In the section on applications, it would seem natural to first describe how periodic orbits can be extracted from experimental data, and the to describe how they can be used for prediction and control etc.

As far as extracting them is concerned, a reference to Auerbach et al might be appropriate, as they also obtained their orbits from time series.

An application of periodic orbit theory I like a lot is the paper

Flepp, L., Holzner, T., Brun, E., Finardi, M. \& Badii, R. 1991. Model identification by periodic-orbit analysis for NMR-Laser chaos. Physical Review Letters, 67: 2244–2247.

They have a very good idea which equations describe their system, and then use a comparison between the periodic orbits extracted from experiment and from the model to determine the values of certain coupling constants and to identify the significance of a nonlinear term.

When it comes to applications of periodic orbit theory to neural systems, I think the work of Frank Moss and his group deserves mention. There are, for instance, the articles

Xing Pei and Frank Moss Nature 379, 618 - 621 (15 February 1996) Characterization of low-dimensional dynamics in the crayfish caudal photoreceptor

and

Pierson, David; Moss, Frank Detecting Periodic Unstable Points in Noisy Chaotic and Limit Cycle Attractors with Applications to Biology Physical Review Letters, Volume 75, Issue 11, September 11, 1995, pp.2124-2127

which should be mentioned.

reply to review #1

I appreciate greatly on all suggestions and comments from Reviewer #1. They are very helpful. I have incorporated all the suggestions in the revision.

Absolutely, you are correct about the missing sum over PO in Eq. 2. I have corrected it and put in the other changes also. Thanks again for all the suggestions in making this a better article.

Review 2

<I appreciate the extensive comments by the reviewer. I have tried to address them fully. Please see my inline notes below and the changes made in the revisions.>

Sorry for the delay in writing this review - it took me some time to get used to wikipedia stylearticles in general and this article in particular. Unfortunately, I am not very happy with the entry as it stands. Having said that, I think it is well written and scientifically sound - however,there are some more general issues I would like to address. These are in particular:

(i) what is the purpose of the article in the first place

(ii) what is the intended target readership.

The article starts with a very brief description of what UPO's actually are and then launches straight into fairly specialised subject areas such as periodic orbit formula for the calculation of thermodynamical quantities, zeta functions etc. and ends with detection of UPO's in data. That is hardly what I and probably the large majority of people (> 90%) will expect when clicking on a link 'unstable periodic orbits' and they will thus probably not get very much out of it. I try to imagine here what a typical scientist (which may well be a biologist/chemist/engineer/experimental physicist) wants and needs to know about UPOs - I doubt whether the 'proof' of eqn (2) is the most useful starting point. In fact - coming back to point (i) - this article should actually be called 'Periodic orbit theories' or 'Periodic orbit formalisms' or something like that as it deals with a specific applications using the set of all UPO's - which is only a very small part of what could be associated with an unstable orbit.

<The issue is that there is already a fairly comprehensive description on periodic orbits (in general) and the stability of a periodic orbit is further explained in the same entry. I decided to put my emphasis on the PO formalism in this article instead. My viewpoint is that the PO formalism is one of the more appealing aspects in the experimental application of UPOs. For experimentalists interested in extracting UPOs, they would like to know the various dynamical properties of the chaotic system that they can estimate from the UPOs and the different ways that they can affect the systems through the UPOs. The periodic orbit formalism is the theoretical under-pinning for these arguments and usage.>

<Actually, I was orginally going to include a section on the basic properties of an UPO in this article. However, I noticed that the Scholarpedia article on "periodic orbit" is actually quite comprehensive in describing the basic properties of an UPO already. I will go back to that article and to see what I can add here and to put links to there if needed.>

I suggest to add a chapter in the beginning containing more elementary properties of UPOs. Now, admittedly, both the key words 'unstable' and 'periodic orbit' exist already as entry in scholarpedia; however, what makes unstable orbits so special and important for chaos is not explained there. Such a section could contain

basic properties of a particular unstable PO:

(such as linear stability, eigenvalues of Jakobian J larger 1 (or 0 for flow), hyperbolic orbit, non-hyperbolic (unstable) orbits (beyond linear stability), simple phase portraits, exponentially diverging trajectories, stable and unstable manifolds - eigenvectors of J in tangent spaces)

<Actually, most of these local properties are described in the Scholarpedia article on periodic orbits. I will go through it in details and to see what I can add additionally here.>

global properties:

unstable periodic orbits in ergodic component (stochastic layer) of dynamical system may form dense set, unstable periodic orbits in Smale horse shoe, connection between unstable orbits and homoclinic/heteroclinic intersections, UPOs and symbolic dynamics/Markov partition etc, UPOs near intermittent points, centre manifolds, KAM and Poincare-Birkhoff

<I will include some of the global properties at the beginning of the article.>

This could than be followed by an application section such as given by the author.

These are only suggestions of course, not all may be necessary and others could be included. Still, the author should have in mind that many of the periodic orbit formulae introduced rely heavily on assumptions which are specific to global properties of the system (completely chaotic), are in fact rare in practise and have nothing to do with UPOs per se. On the other hand, UPOs may be important in systems which are not completely chaotic such as the experimental systems mentioned at the end of the article - these aspects are neglected so far. I think these facts need to be stressed when starting on the periodic orbit formalism - any form of stable regions, tangencies etc can render these PO formulae useless or may make an application very complicate (rewriting the symbolic dynamics, pruning, critical parameter exists for which natural measure becomes singular (intermittency)).

<I will put a cautionary statement at the beginning of the UPO formalism section as suggested. I agreed that for systems without a strong expanding direction (such as systems with mixed regions and tengencies), UPO formalism might fail. However, I believe that in most experimental applications with strong chaoticity, a good approximated characterization of the system can still be achieved from a knowledge of the set of the shorter fundamental orbits. In particular, for some engineering applications, optimal performance can be achieved by considering only the lowest orbits (Hunt 1996).>

Regarding the actual article:

please explain in more detail in which sense orbits form a 'skeleton' and why they are hierarchically organised (some keywords such as dense set, symbolic dynamics, homo/heteroclinic points have been mentioned earlier). <I will address this point in the added section on the basic properties of UPOs.>

I doubt whether the landscape picture is very informative for somebody who is not an expert in dynamical systems already - please give more information (such as what is the z-axis) and explain how the picture was produced. What are the straight lines connecting some of the peaks?

<Actually, I think that the landscape picture is a good schematic diagram illustrating how the motion of a chaoic trajectory will be controlled by the UPOs. The image that I want to convey is the motion of a ball rolling around in a landscape punctuated with many peaks (UPOs). Obviously, UPOs are mostly saddles and not repellers but one can get the sense that the motion of the ball will be strongly affected by the sharpness of the peak (the stability of the UPOs) and the landscape is "densely" populated by these "speed bumps". I will put a short description here to explain this visual picture. The straight lines in the diagram are just visual aids in identifying the corresponding cyclic pieces of the period-2 cycles in the diagram.>

Eq (1) - I assume there is a summation over periodic orbits missing? <yes. there should be a sum over all POs here. It is fixed now.>

Eq (2) - it should be noted that \Lambda_u is the largest expanding eigenvalue <The system under consideration is a simple 2D hyperbolic map. \Lambda_u is the only expanding eigenvalue. I will make a note about the higher dimensional case, when \Lambda_u will be replaced by the largest expanding eigenvalue.>

Lyapunov exponents:

I am not sure about the formula - first of all, the symbol h_{u,s} is normally reserved for entropies. Secondly, for higher-dimensional systems, there is a spectrum of Lyapunov exponents. Is the formula only for the largest ( h_u ) and smallest (h_s) exponent. In both cases, \Lambda_{u,s} would be largest expanding/contracting eigenvalue. Please define $u$ and $s$.

<I have change the "h" symbol back to \lambda. As stated at the beginning of the section, these results are valid for the 2D hyperbolic example chosen and "u" and "s" are the labels for the unstable and stable directions of the 2D map. A defintion for these two subscripts were given at the beginning of the section. I have added a reminder statement about this point here also.>

Cycle expansion:

A word of caution should be including, saying that the zeta function as well as the cycle expansion may behave badly if the symbolic dynamics is not complete - which is usually the case. Keywords - pruning, branch cuts, meromorphic function. HH Rugh gave a proof for 1d expanding maps giving the conditions under which the zeta function or spectral determinant is analytic/meromorphic. It may be in order to cite Cvitanovic et al's web-book here (or earlier) - maybe with direct link?

<I have included a caution statement in the cycle expansion section and I agree that it is a good idea to include a direct link to Cvitanovic's web-book here.>

Some spelling errors:

Sec. 'Predicting, Tracking ...'

line 9: short terms --> short term <done>

last page: ...notable results for modeling phyiscal --> modelling physical <done>

reply by referee 2

I am very happy about the changes and I think this article has greatly benefitted from the extension made. This is a brilliant article, now gently leading the unsuspected reader all the way into zeta functions and cycle expansion!