## The Chaos Program

Prof. Gilmore and his colleagues have been studying chaos in physical systems for many years.  He is particularly interested in establishing an understanding
of the structure of chaos.  After all, what would be the point of analyzing, for example, a bunch of  chemicals if the Mendelyeev Periodic Table of the Chemical Elements did not exist?
Prof. Gilmore's group has created a hierarchy of discrete classifications for chaotic dynamical systems --- and their strange attractors --- in three dimensional spaces.  There are four levels in this hierarchy.  The first two levels have been described in Refs. 1 and 2.

From the smallest to the largest, these levels are:

1.  Basis sets of orbits.  These are sets of orbits in a strange attractor whose presence forces all other (unstable) periodic orbits that exist in the attractor.  Up to any finite period, a finite basis set can be constructed algorithmically [1,2,3,4].

2.  Branched manifolds.  These are two-dimensional structures that are manifolds almost everywhere.  They are not manifolds
because they possess two types of singularities: splitting points and branch lines.  The splitting points describe the stretching process that is ultimately responsible for the sensitivity to initial conditions exhibited by chaotic systems.  The branch lines describe
the return process that is ultimately responsible for maintaining bounded motion in the attractor.  Branched manifolds organize all the unstable periodic orbits in a strange attractor in a very specific way, which is teased out by computing the gauss linking numbers of pairs of orbits.  A more refined topological invariant, the relative rotation rates, offer even more information.  We have created an algorithm for determining the branched manifold underlying physical experiments and applied it to determine the mechanism underlying some chemical data.  Other laboratories throughout the world have adopted this procedure for analyzing chaotic data [1,2,5-9].

3.  Bounding tori.  In the same way that branched manifolds organize periodic orbits, bounding tori organize branched manifolds. Every branched manifold can be embedded in a minimal'' three dimensional manifold, called an inertial manifold.  Its boundary is a torus. The flow generating the strange attractor, restricted to the torus surface, provides a canonical form that can be/has been used to classify every strange attractor studied in three dimensions [10-13].

4.  Embeddings of bounding tori.  The bounding tori described above are as seen from the inside'' (the classification is intrinsic).
In analyses of data, we see the bounding torus as it is embedded in 3-space.  Any bounding torus can be embedded in the surrounding space in a discrete number of ways.  This enumeration is currently being worked out [14].

Prof. Gilmore and his colleagues have created the analog of Fourier analysis for nonlinear dynamical systems in three dimensions. Strange attractors, or their caricature, branched manifolds/bounding tori, are build up Lego-style from two basic building blocks, one containing splitting singularities, the other joining singularities, in a way that is systematic yet with sufficient degrees of freedom to allow an even richer variety of behavior in physical systems than has yet been seen.

A.  R. Gilmore, Topological analysis of chaotic dynamical systems, Revs. Mod. Phys. 70, 1455-1530 (1998).

B.  R. Gilmore and M. Lefranc, The Topology of Chaos: Alice in Stretch and Squeezeland, NY: Wiley, 2002.

C.  R. Gilmore and C. Letellier, The Symmetry of Chaos: Alice in Mirrorland, NY: Oxford University Press, 2007.

D.  R. Gilmore, Presentation at the Physics and Topology Workshop, Drexel University, September 8-9, 2008.

1.  H. G. Solari, E. Eschenazi, R. Gilmore and J. Tredicce, Influence of Coexisting Attractors on the Dynamics of a Laser System, Optics Commun. {\bf 64},49-53 (1987).

2.  R. Gilmore, Catastrophe Theory, Nuclear Physics { B2} (Proc. Suppl.), 191-200 (1987).

3.  J. R. Tredicce, R. Gilmore, H. G. Solari and E. Eschenazi, Winding Numbers and Collisions Between Attractors in a Laser System in: {\sl Fundamentals in Quantum Optics II}, F. Ehlotsky (Ed), Lecture Notes in Physics {\bf \#282}, Berlin: Springer-Verlag, 1987, pp. 273-275.

4a.  H. G. Solari and R. Gilmore, Relative rotation rates for driven dynamical systems, Phys. Rev.A37, 3096-3109 (1988).

4b.  H. G. Solari and R. Gilmore, Relative rotation rates for driven dynamical systems, Phys. Rev.A37, 3096-3109 (1988).

5.  E. Eschenazi, H. G. Solari and R. Gilmore, Basins of Attraction in Driven Dynamical Systems, Phys. Rev. {A39}, 2609-2627 (1989).

6a.  N. B. Tufillaro, H. G. Solari, and R. Gilmore,  Relative rotation rates: Fingerprints for strange attractors, Phys. Rev. A41, 5717-5720 (1990).

6b.  N. B. Tufillaro, H. G. Solari, and R. Gilmore,  Relative rotation rates: Fingerprints for strange attractors, Phys. Rev. A41, 5717-5720 (1990).

7a.  G. B. Mindlin, X.-J. Hou, H. G. Solari, R. Gilmore, and N. B. Tufillaro, Classification of strange attractors
by integers, Phys. Rev. Lett. 64, 2350-2353 (1990).

7b.  G. B. Mindlin, X.-J. Hou, H. G. Solari, R. Gilmore, and N. B. Tufillaro, Classification of strange attractors
by integers, Phys. Rev. Lett. 64, 2350-2353 (1990).

8.  G. B. Mindlin, H. G. Solari, M. A. Natiello, R. Gilmore, and X.-J. Hou, Topological analysis of chaotic time series data from
the Belousov-Zhabotinskii reaction, J. Nonlinear Science 1, 147-173 (1991).

8a.  G. B. Mindlin, H. G. Solari, M. A. Natiello, R. Gilmore, and X.-J. Hou, Topological analysis of chaotic time series data from
the Belousov-Zhabotinskii reaction, J. Nonlinear Science 1, 147-173 (1991).

9.  F. A. Papoff, A. Fioretti, E. Arimondo, G. B. Mindlin, H. G. Solari, and R. Gilmore, Structure of chaos in the laser with
saturable absorber, Phys. Rev. Lett. 68, 1128-1131 (1992).

10.   G. B. Mindlin and R. Gilmore, Topological Analysis and Synthesis of Chaotic Time Series, Physica {D58}, 229-242 (1992).

11.   R. Gilmore, Summary of the Second Workshop on Measures of Complexity and Chaos, Int. J. Bifurcation and Chaos { 3}, 491-524 (1993).

12a.  G. B. Mindlin, R.  Lopez-Ruiz, H. G. Solari, and R. Gilmore, Horseshoe implications, , Phys. Rev. E48, 4297-4304 (1993).

12b.  G. B. Mindlin, R.  Lopez-Ruiz, H. G. Solari, and R. Gilmore, Horseshoe implications, , Phys. Rev. E48, 4297-4304 (1993).

13.   J. L. W. McCallum and R. Gilmore, A Geometric Model of the Driven Duffing Oscillator, Int. J. Bifurcation and Chaos { 3}, 685-691 (1993).

14.   P. T. Boyd, G. B. Mindlin, R. Gilmore, and H. G. Solari, Topological Analysis of Chaotic Orbits: Hyperion Revisited, Ap. J. {\sl 431}, 425-431 (1994).

15.   R. Gilmore and J. W. L. McCallum, Superstructure in the Bifurcation Diagram of the Duffing Oscillator, Phys. Rev. {\bf E51}, 935-956 (1995).

16.   R. Gilmore, in: { Chaos and the Changing Nature of Science and Medicine: An Introduction}, D. E. Herbert (Ed)NY: AIP CP 376 (1996): pp. 35-53.

17.   E. Rold\'an, G. J. de Valc\'arel, R. Vilaseca, R. Corbal\'an, V. J. Mart\'inez, and R. Gilmore, The Dynamics of Optically Pumped Molecular Lasers. On its Relation with the Lorenz-Haken Model, Quantum and Semiclassical Optics {\bf 9}, R1-R35, (1997).

18.   R. Gilmore, Topological analysis of chaotic time series, {\sl Applications of Soft Computing, Vol 3165, Proceedings of SPIE}, (B. Bosacchi, J. C. Bezdek, and D. B. Vogel, Eds.), SPIE: Bellingham, WA, 1997: pp. 243-257.

19.  R. Gilmore, R. Vilaseca, R. Corbal\'an, and E. Rold\'an, Topological Analysis of chaos in the optically pumped laser, Phys. Rev. {\bf E55}, 2479-2487 (1997).

20.   R. Gilmore, X. Pei, and F. Moss, Topological analysis of chaos in neural spike train bursts, Chaos {\bf 9}(3), 812-817 (1999).

21.   R. Gilmore and X. Pei, The topology and organization of unstable periodic orbits in Hodgkin-Huxley models of receptors with subthreshold oscillations,{\sl Handbook of Biological Physics, Vol. 4, Neuro-informatics, Neural Modeling}, (F. Moss and S. Gielen, Eds.), Amsterdam, North Holland, 2001, pp. 155-203.

22.  C. Letellier and R. Gilmore, Covering dynamical systems: Two-fold covers, Phys. Rev. E63, 016206 (2000).

23.   C. Letellier, P. Werny, J.-M. Malasoma, and R. Gilmore, Multichannel intermittencies induced by symmetries, Physical Review {\bf E66} , 036220 (2002).

24.  C. Letellier and R. Gilmore, Dressed symbolic dynamics, Phys. Rev. E67, 036205 (2003).

25.  T. D. Tsankov and R. Gilmore, Strange attractors are classified by bounding tori, Phys. Rev. Lett. 91(13), 134104 (2003).

26.  T. D. Tsankov and R. Gilmore, Topological aspects of the structure of chaotic attractors in R^3, Phys. Rev. E69, 056206 (2004).

27.  T. D. Tsankov, A. Nishtala, and R. Gilmore, Embeddings of a strange attractor into R^3, Phys. Rev. E69, 056215 (2004).

87.  G. Byrne, R. Gilmore, and C. Letellier, Distinguishing between folding and tearing mechanisms in strange attractors, Phys. Rev. E70, 056214 (2004).

29.  C. Letellier, T. D. Tsankov, G. Byrne, and R. Gilmore, Large scale structural reorganization of strange attractors, Phys. Rev. E72, 026212 (2005).

30.  C. Letellier, L. A. Aguirre, J. Maquet, and R. Gilmore, Evidence for low-dimensional chaos in sunspot cycles, Astronomy and Astrophysics 449, 379-387 (2006).

31.  C. Letellier and R. Gilmore, Symmetry groups for 3D dynamical systems, Journal of Physics A40, 5597-5620 (2007).

32.  R. Gilmore, Two-parameter families of strange attractors, Chaos 17, 013104 (2007).

33. N. Romanazzi, M. Lefranc, and R. Gilmore, Embeddings of low-dimensional strange attractors: Topological invariants and degrees of freedom, Physical Review E75, 066214 (2007).

34. C. Letellier, R. Gilmore, and Timothy Jones, Peeling bifurcations of toroidal chaotic attractors, Physical Review E76, 066204 (2007).

35. R. Gilmore, C. Letellier, and N. Romanazzi, Global topology from an embedding, J. Phys. A: Math. Theor.40, 13,291 - 13,297 (2007) .

35a. R. Gilmore, C. Letellier, and N. Romanazzi, Global topology from an embedding, J. Phys. A: Math. Theor.40, (2007) (in press) (different format).

36. C. Letellier, V. Messager, and R. Gilmore, From quasi-periodicity to toroidal chaos: analogy between the Curry-Yorke map and the van der Pol system, Physical Review E (accepted).

37. I. Moroz, C. Letellier, and R. Gilmore, When are projections also embeddings?, Phys. Rev. E75, 046201 (2007).

38. C. Letellier and R. Gilmore, Poincar\'e sections for a new three-dimensional toroidal attractor, Journal of Physics A: Mathematics and Physics (in press).

39. C. Letellier, I. Moroz, and R. Gilmore, A topological test for embeddings, (work in progress).

40. N. Romanazzi, V. Messager, C. Letellier, M. Lefranc, and R. Gilmore, A useful canonical form for low dimensional attractors, (work in progress).

41.  J. Katriel and R. Gilmore, Entropy of bounding tori, (to be submitted).

42.  C. Letellier, I. M. Moroz, and R. Gilmore, A topological test for embeddings, (in process).

43.  R. Gilmore, C. Letellier, and M. Lefranc, Chaos - Topology, Scholarpedia 3(7):4592 (2007).

44.  C. Letellier, I. M. Moroz, and R. Gilmore, A comparison of tests for embeddings, Phys. Rev. E78(2), 026203 (2008)

45.  Daniel J. Cross and R. Gilmore, Representation theory for strange attractors, Phys. Rev. E80(1), 056207 (2009)

46.  R. Gilmore, Jean Marc Ginoux, Timothy Jones, C. Letellier, and U. S. Freitas, Connecting curves for dynamical systems, Journal of Physics A (in press)

47.  Daniel J. Cross, Ryan Michaluk, and R. Gilmore, Biological algorithm for data reconstruction, Physical Review E 81, 036217 (2010)

48.  Daniel J. Cross and R. Gilmore, Differential embedding of the Lorenz attractor, Physical Review E 81, 066220 (2010)

49.  Daniel J. Cross and R. Gilmore, Equivariant differential embeddings, Journal of Mathematical Physics 51, 092706 (2010)

50.  Daniel J. Cross and R. Gilmore, Complete set of representations for dissipative chaotic three-dimensional dynamical systems, Physical Review E 82, 056211 (2010)

51.  R. Gilmore, How topology came to chaos, World Scientific Review (in press)

52.  Keith Gilmore and R. Gilmore, Introduction of the sphere map with application to spin-torque nano-oscillators, World Scientific Review (in press)

53.  R. Gilmore, Atomic Coherent States and Sphere Maps, Journal of Physics A (submitted)

54.  C. Letellier and Robert Gilmore, The Universal Template is a subtemplate of the double-scroll template, Journal of Physics A: Math. Theor. 46, 065102 (2013).

55.  Daniel J. Cross and R. Gilmore, Dressed return maps distinguish chaotic mechanisms, Physical Review E 87, 012919 (2013).

Last Revision: January 30, 2013