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In <2q3gj1$n6f@network.ucsd.edu> [sci.nonlinear], Matt Kennel (mbk@inls1.ucsd.edu) wrote:
>Jim Heagy (jheagy@zoltar.nrl.navy.mil) wrote:
>: In article <2poha4$fqa@portal.gmu.edu>, herwin@mason1.gmu.edu (HARRY R.
>: ERWIN) wrote:

>: > Does anyone have any references to papers addressing the analysis of
>: > data from systems with dissipative chaos?
>: > 
>: There is a method of topolocial data analysis that concerns the
>: connectivity and intertwining properties of unstable periodid orbits in a
>: chaotic attractor. As far as I know, this is the only method of chaotic
>: data analysis that can be used to definitively REJECT proposed models for
>: the data.  See the references

>: G. B. Mindlin, H. G. Solari, et al, "Topological Analysis of Chaotic Time
>: Series ...", J. Nonlin. Sci. 2, 147 (1991).

>: G. B. Mindlin, X.-J. Hou, et al, "Classification of Strange Attractors by
>: Integers", Phys. Rev. Lett. 64, 2350 (1990).

>: N. B. Tufillaro, H. G. Solari and R. Gilmore, "Relative Rotation Rates:
>: Fingerprints for Strange Attractors", Phys. Rev. A 41, 5717, (1990).

There has been also a few experimental studies:

@ARTICLE{tufillaro91a,
   author = {N. B. Tufillaro and R. Holzner and L. Flepp and E. Brun, M. Finardi
   and R. Badii},
   title = {Template analysis for a chaotic NMR laser},
   journal = {Phys. Rev. A},
   year = 1991,
   volume = 44,
   pages = {R4786},
}
@ARTICLE{papoff92a,
   author = {F. Papoff and A. Fioretti and E. Arimondo and G. B. Mindlin and
   H. G. Solari and R. Gilmore},
   title = {Structure of chaos in the laser with a saturable absorber},
   journal = {Phys. Rev. Lett.},
   year = 1992,
   volume = 68,
   pages = {1128},
}
@ARTICLE{lefranc93a,
   author = {M. Lefranc and P. Glorieux},
   title = {Topological analysis of chaotic signals from a ${\rm CO}_2$ laser},
   journal = {Int. J. Bifurcation and Chaos},
   year = 1993,
   volume = 3,
   number = {3},
   pages = 643,
}

>Does any of this work for dimensions > 3 ? 
 >I thought heard/read that all knots untie in 4 dimensions, preventing
>this interesting topological analysis from working.

No. As you state all closed curves in 4-d space are isotopic. So topological
analysis in its present form can be applied only to systems which can
be embedded in a 3-d space. Note that strictly speaking the knot invariants
you compute from the unstable periodic orbits depend on the embedding chosen
(all 3d knots are themselves embeddings of the trivial knot in 3d space, i.e.
embeddings of S^1 in S^3). However, it could be that templates determined with
different embeddings differ only by a global torsion, but that has not yet been
addressed, I think.


>If there were some way to make this work in arbitrary dimensions, it could
>be quite powerful.

Some speculations:
In 3d space, we can use classical knot theory but in general knot theory
studies embeddings of S^n in S^{n+2}. So in 4d space we could study intertwining
of surfaces, for examples invariant manifolds of periodic orbits. The template
could perhaps also be generalized. In a 3d space, it is a 2d branched
manifold which can be roughly be viewed as a compression of the phase space
along the stable (fractal) direction of the attractor, i.e. the two dimensions
which survive in the template are the ones with positive and zero 
lyapunov exponents: the unstable direction and the direction of the flow.
In a 4d chaotic systems with two positive lyapunov exponents
there could be a template-like thing which would be a branched 3d manifold. This
thing could be anlyzed with knot theory in S^4.
Of course extracting unstable periodic orbits from experimental time series
is one thing, determining their invariant manifolds is another ans seems
at first glance much more difficult.

--
________________________________________________________________________
| Marc Lefranc, Laboratoire de Spectroscopie Hertzienne (URA CNRS 249) |
| Bat P5, UFR de Physique                                              |
| Universite des Sciences et Technologies de Lille                     |
| F-59655 Villeneuve d'Ascq CEDEX (FRANCE)                             |
| e-mail: lefranc@lsh.univ-lille1.fr                                   |
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