The Topology of Chaos: Alice in Stretch and Squeezeland

Here is the (under construction) home page of "The Topology of Chaos: Alice in Stretch and Squeezeland", a book about topological analysis written by Robert Gilmore Nonlinear dynamics research group at the Physics department of Drexel University, Philadelphia and Marc Lefranc, Laboratoire de Physique des Lasers, Atomes, Molécules, Université des Sciences et Technologies de Lille, France and published by Wiley. Topological analysis is about extracting from chaotic data the topological signatures that determine the stretching and squeezing mechanisms which act on flows in phase space and are responsible for generating chaotic behavior. In this book, we provide a detailed description of the fundamental concepts and tools of topological analysis. For three-dimensional systems, the methodology is well established and relies on sophisticated mathematical tools such as knot theory and templates (i.e., branched manifolds such as the one shown on the cover, see here for a color version with explanations). This illustrated by a few case studies of experimental systems such as the Belousov-Zhabotinskii reaction or various lasers. The last chapters discuss how topological analysis could be extended to handle higher-dimensional systems, and how it can be viewed as a key part of a general program for dynamical systems theory. Problems for which topological analysis has proved invaluable are: classification of strange attractors (show me how you are stretched and squeezed, I will tell you who you are), understanding of bifurcation sequences (orbits organized in a complex way cannot appear in an arbitrary order), extraction of symbolic dynamical information and construction of symbolic codings (itineraries on templates translate to braid types and vice versa). As such, it has become a fundamental tool of nonlinear dynamics. Here is the table of contents.

Errare humanum est

More generally, any suggestion or criticism is welcome: do not hesitate to share with us your reactions and opinions about this work, positive or negative. We hope this will help us to improve subsequent editions.

Links

Topological analysis

• The Nonlinear Dynamics archive of Nick "Dr. Chaos" Tufillaro has a lot of interesting things on nonlinear dynamics, including topological analysis, which he knows quite a few things about. In particular you can find sample sections of his book on chaotic dynamics. Here you can find the review he wrote about our book in the American Journal of Physics (subscription required).
• The research team on "Nonlinear dynamics, complex systems and econophysics" at the University of Buenos Aires is home to Gabriel Mindlin and Hernan Solari.
• Christope Letellier is in the "topological analysis and modelization of dynamical systems" group at CORIA in Rouen, France.
• Mario Natiello is at the Mathematics centre of Lund University, in Sweden.

Tools

• The libtop package provides programs to determine the simplest template compatible with a given set of topological invariants.

References

• This book grew out of a review paper by R. Gilmore :

  "Topological analysis of chaotic dynamical systems" , Rev. Mod. Phys. 70 , 1455-1530 (1998).

• Templates as a tool to study knots in dynamical systems were introduced by J. S. Birman and R. F. Williams:

  "Knotted periodic orbits in dynamical systems I: Lorenz's equations" , Topology 22 , 47-82 (1983).

• G. B. Mindlin, X.-J. Hou, H. G. Solari, R. Gilmore, and N. B. Tufillaro proposed to use templates to characterize and classify chaotic systems:

  "Classification of strange attractors by integers" , Phys. Rev. Lett. 64 , 2350-2353 (1990).

• And soon after,G. B. Mindlin, H. G. Solari, M. A. Natiello, R. Gilmore, and X.-J. Hou showed the experimental feasibility of this approach by analyzing the Belousov-Zhabotinskii reaction:

  "Topological analysis of chaotic time series data from Belousov-Zhabotinski reaction" , J. Nonlinear Sci. 1 , 147-173 (1991).

• Experimental systems from various fields were characterized with topological analysis, from lasers and chemical reactions to a chaotically vibrating string (N. B. Tufillaro, P. Wyckoff, R. Brown, T. Schreiber and T. Molteno):

  "Topological time series analysis of a string experiment and its synchronized model" , Phys. Rev. E 51 , 164-174 (1995).

• As emphasized by G. B. Mindlin, R. Lopez-Ruiz, H. G. Solari, and R. Gilmore, the linking of periodic orbits determines which bifurcation sequences can occur and which cannot:

  "Horseshoe implications" , Phys. Rev. E 48 , 4297-4304 (1993).

• R. Gilmore and J. W. L. McCallum showed that different topological organizations can be observed in different parts of parameter space and that transitions from one structure to another occur in a systematic way:

  "Structure in the bifurcation diagram of the Duffing oscillator" , Phys. Rev. E 51 , 935-956 (1995).

• In early experiments, only standard Smale's horseshoes had been observed. G. Boulant, S. Bielawski, D. Derozier, and M. Lefranc described a regime that looked like a regular horseshoe, but whose topological invariants revealed that it was in fact on the other side of the mirror:

  "Experimental observation of a chaotic attractor with a reverse horsehoe topological structure" , Phys. Rev. E 55 , R3801-3804 (1997).

• R.G. and J. W. L. McCallum had predicted that from one chaotic tongue in parameter space to the next, the global torsion of the template increases or decreases by 1 . G. Boulant, M. Lefranc, S. Bielawski, and D. Derozier observed that this was indeed the case in their fiber laser:

  "Horseshoe templates with global torsion in a driven laser" , Phys. Rev. E 55 , 5082-5091 (1997).

• Bursting oscillations have received a great deal of attention lately because of their importance in biological systems. The study of a model of a neuron with subtreshold oscillations by R. Gilmore, X. Pei, and F. Moss shows that there is a lot of topology behind these oscillations:

  "Topological analysis of chaos in neural spike train bursts" , Chaos 9 , 812-817 (1999).

• Topological invariants of orbits contain a lot of information about their symbolic dynamics. M. Lefranc, P. Glorieux, F. Papoff, F. Molesti, and E. Arimondo proposed to use this information to construct symbolic encodings for chaotic attractors, and tested this approach with a modulated laser:

  "Combining topological analysis and symbolic dynamics to describe a strange attractor and its crises" , Phys. Rev. Lett. 73 , 1364-1367 (1994).

• There is a natural coding for template orbits induced by the branched structure of the template. Building on the reference above, J. Plumecoq and M. Lefranc showed how to lift this coding from the template to the attractor in a systematic way:

  "From template analysis to generating partitions I: Periodic orbits, knots and symbolic encodings" , Physica D 144 , 231-258 (2000).

• In this paper, J. Plumecoq and M. Lefranc tested the validity of partitions constructed using topological analysis as described in part I. In particular, they showed that these partitions are fully compatible with those obtained by computing lines of homoclinic tangencies:

  "From template analysis to generating partitions II: Characterization of the symbolic encodings" , Physica D 144 , 259-278 (2000).