From template analysis to generating partitions. I: Periodic orbits, knots and symbolic encodings. (English) Zbl 0978.37006
The distribution of unstable periodic orbits reflects essential topological properties of any chaotic attractor in a 3-dimensional flow. It can be described by a template or knot holder under suitable conditions, as was done in now classical work by J. S. Birman and R. F. Williams [Knotted periodic orbits in dynamical systems. I: Topology 22, 47-82 (1983; Zbl 0507.58038); II: Contemp. Math. 20, 1-60 (1983; Zbl 0526.58043)]. The authors develop a method to encode the knot theoretic information into symbolic dynamics. This involves Delaunay triangulations derived from partitions of a Poincaré plane. The encoding is done by finite approximations taking into account orbits of higher and higher periods. Its consistency is discussed, and its validity is checked in the context of numerical simulations concerning a chaotic attractor in a particular laser model. – For Part II see the review Zbl 0098.37008 below.
Reviewer: Dieter Erle (Dortmund)
MSC:
| 37B10 | Symbolic dynamics |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 37C27 | Periodic orbits of vector fields and flows |
Keywords:
unstable periodic orbits; chaotic attractor; template; knot holder; Delaunay triangulations; encodingReferences:
| [1] | J. Guckenheimer, P.J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, Berlin, 1983.; J. Guckenheimer, P.J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, Berlin, 1983. · Zbl 0515.34001 |
| [2] | A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambdrige, 1995.; A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambdrige, 1995. · Zbl 0878.58020 |
| [3] | Hao Bai-Lin, Elementary Symbolic Dynamics and Chaos in Dissipative Systems, World Scientific, Singapore, 1989.; Hao Bai-Lin, Elementary Symbolic Dynamics and Chaos in Dissipative Systems, World Scientific, Singapore, 1989. · Zbl 0724.58001 |
| [4] | R. Badii, A. Politi, Complexity: Hierarchical Structures and Scaling in Physics, Cambridge Nonlinear Science Series, Vol. 6, Cambridge University Press, Cambridge, 1997.; R. Badii, A. Politi, Complexity: Hierarchical Structures and Scaling in Physics, Cambridge Nonlinear Science Series, Vol. 6, Cambridge University Press, Cambridge, 1997. · Zbl 1042.82500 |
| [5] | Hayes, S.; Grebogi, C.; Ott, E., Communicating with chaos, Phys. Rev. Lett., 70, 3031-3034 (1993) |
| [6] | Hayes, S.; Grebogi, C.; Ott, E.; Mark, A., Experimental control of chaos for communication, Phys. Rev. Lett., 73, 1781-1784 (1994) |
| [7] | Grassberger, P.; Kantz, H., Generating partitions for the dissipative Hénon map, Phys. Lett. A, 113, 235-238 (1985) |
| [8] | Lefranc, M.; Glorieux, P.; Papoff, F.; Molesti, F.; Arimondo, E., Combining topological analysis and symbolic dynamics to describe a strange attractor and its crises, Phys. Rev. Lett., 73, 1364-1367 (1994) |
| [9] | Gilmore, R., Topological analysis of chaotic dynamical systems, Rev. Mod. Phys., 70, 1455-1530 (1998) · Zbl 1205.37002 |
| [10] | Flepp, L.; Holzner, R.; Brun, E.; Finardi, M.; Badii, R., Model identification by periodic-orbit analysis for NMR-laser chaos, Phys. Rev. Lett., 67, 2244-2247 (1991) |
| [11] | Finardi, M.; Flepp, L.; Parisi, J.; Holzner, R.; Badii, R.; Brun, E., Topological and metric analysis of heteroclinic crisis in laser chaos, Phys. Rev. Lett., 68, 2989-2991 (1992) |
| [12] | Badii, R.; Brun, E.; Finardi, M.; Flepp, L.; Holzner, R.; Parisi, J.; Reyl, C.; Simonet, J., Progress in the analysis of experimental chaos through periodic orbits, Rev. Mod. Phys., 66, 1389-1415 (1994) |
| [13] | Davidchack, R. L.; Lai, Y.-C.; Bollt, E. M.; Dhamala, M., Estimating generating partitions of chaotic systems by unstable periodic orbits, Phys. Rev. E, 61, 1353-1356 (2000) |
| [14] | Sterling, D.; Dulling, H. R.; Meiss, J., Homoclinic bifurcations in the Hénon map, Physica D, 134, 153-184 (1999) · Zbl 0987.37040 |
| [15] | N.B. Tufillaro, T.A. Abbott, J.P. Reilly, An Experimental Approach to Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, MA, 1992.; N.B. Tufillaro, T.A. Abbott, J.P. Reilly, An Experimental Approach to Nonlinear Dynamics and Chaos, Addison-Wesley, Reading, MA, 1992. · Zbl 0762.58001 |
| [16] | H.G. Solari, M.A. Natiello, G.B. Mindlin, Nonlinear Dynamics: A Two-way Trip from Physics to Math, IOP Publishers, London, 1996.; H.G. Solari, M.A. Natiello, G.B. Mindlin, Nonlinear Dynamics: A Two-way Trip from Physics to Math, IOP Publishers, London, 1996. · Zbl 0867.58059 |
| [17] | Mindlin, G. B.; Hou, X.-J.; Solari, H. G.; Gilmore, R.; Tufillaro, N. B., Characterization of strange attractors by integers, Phys. Rev. Lett., 64, 2350-2353 (1990) · Zbl 1050.37509 |
| [18] | Mindlin, G. B.; Solari, H. G.; Natiello, M. A.; Gilmore, R.; Hou, X.-J., Topological analysis of chaotic time series data from Belousov-Zhabotinski reaction, J. Nonlinear Sci., 1, 147-173 (1991) · Zbl 0797.58057 |
| [19] | Birman, J. S.; Williams, R. F., Knotted periodic orbits in dynamical systems I: Lorenz’s equations, Topology, 22, 47-82 (1983) · Zbl 0507.58038 |
| [20] | R.W. Ghrist, P.J. Holmes, M.C. Sullivan, Knots and Links in Three-dimensional Flows, Lecture Notes in Mathematics, Vol. 1654, Springer, Berlin, 1997.; R.W. Ghrist, P.J. Holmes, M.C. Sullivan, Knots and Links in Three-dimensional Flows, Lecture Notes in Mathematics, Vol. 1654, Springer, Berlin, 1997. · Zbl 0869.58044 |
| [21] | J. Plumecoq, M. Lefranc, From template analysis to generating partitions. II: Characterization of the symbolic encodings, Physica D 144 (2000) 259-278.; J. Plumecoq, M. Lefranc, From template analysis to generating partitions. II: Characterization of the symbolic encodings, Physica D 144 (2000) 259-278. · Zbl 0978.37007 |
| [22] | Abarbanel, H. D.; Brown, R.; Sidorowich, J. J.; Tsimring, L. S., The analysis of observed chaotic data in physical systems, Rev. Mod. Phys., 65, 1331-1392 (1993) |
| [23] | Holmes, P. J., Knotted periodic orbits in suspensions of Smale’s horseshoe: period multiplying and cabled knots, Physica D, 21, 7-41 (1986) · Zbl 0623.58014 |
| [24] | P.J. Holmes, in: T. Bedford, J. Swift (Eds.), New Directions in Dynamical Systems, Cambridge University Press, Cambridge, 1988, pp. 150-191.; P.J. Holmes, in: T. Bedford, J. Swift (Eds.), New Directions in Dynamical Systems, Cambridge University Press, Cambridge, 1988, pp. 150-191. · Zbl 0635.00009 |
| [25] | Grassberger, P.; Kantz, H.; Moenig, U., On the symbolic dynamics of the Hénon map, J. Phys. A, 22, 5217-5230 (1989) · Zbl 0722.58016 |
| [26] | Cvitanović, P.; Gunaratne, G. H.; Procaccia, I., Topological and metric properties of Hénon-type strange attractors, Phys. Rev. A, 38, 1503-1520 (1988) |
| [27] | D’Alessandro, G.; Grassberger, P.; Isola, S.; Politi, A., On the topology of the Hénon map, J. Phys. A, 23, 5285-5294 (1990) · Zbl 0716.58010 |
| [28] | Giovannini, F.; Politi, A., Homoclinic tangencies, generating partitions and curvature of invariant manifolds, J. Phys. A, 24, 1837-1848 (1991) · Zbl 0733.58033 |
| [29] | Jaeger, L.; Kantz, H., Structure of generating partitions for two-dimensional maps, J. Phys. A, 30, L567-L576 (1997) · Zbl 0922.58050 |
| [30] | Wu, Z.-B., Symbolic dynamics analysis of chaotic time-series with a driven frequency, Phys. Rev. E, 53, 1446-1452 (1996) |
| [31] | Jaeger, L.; Kantz, H., Homoclinic tangencies and non-normal Jacobians — effects of noise in nonhyperbolic chaotic systems, Physica D, 105, 79-96 (1997) · Zbl 0933.37021 |
| [32] | R. Gilmore, Catastrophe Theory for Scientists and Engineers, Wiley, New York, 1981 [reprinted by Dover, New York, 1993].; R. Gilmore, Catastrophe Theory for Scientists and Engineers, Wiley, New York, 1981 [reprinted by Dover, New York, 1993]. · Zbl 0497.58001 |
| [33] | Solari, H. G.; Gilmore, R., Relative rotation rates for driven dynamical systems, Phys. Rev. A, 37, 3096-3109 (1988) |
| [34] | Auerbach, D.; Cvitanović, P.; Eckmann, J.-P.; Gunaratne, G.; Procaccia, I., Exploring chaotic motion through periodic orbits, Phys. Rev. Lett., 58, 2387-2389 (1987) |
| [35] | Lathrop, D.; Kostelich, E. J., Characterization of a strange attractor by periodic orbits, Phys. Rev. A, 40, 4028-4031 (1989) |
| [36] | Ott, E.; Grebogi, C.; Yorke, J., Controlling chaos, Phys. Rev. Lett., 64, 1196-1199 (1990) · Zbl 0964.37501 |
| [37] | L.H. Kaufmann, Knots and Physics, World Scientific, Singapore, 1991.; L.H. Kaufmann, Knots and Physics, World Scientific, Singapore, 1991. · Zbl 0733.57004 |
| [38] | Natiello, M. A.; Solari, H. G., Remarks on braid theory and the characterisation of periodic orbits, J. Knot Theory Ramifications, 3, 511 (1994) · Zbl 0824.57004 |
| [39] | Solari, H. G.; Natiello, M. A.; Vázquez, M., Braids on the Poincaré section: a laser example, Phys. Rev. E, 54, 3185-3195 (1996) |
| [40] | Hall, T., The creation of horseshoes, Nonlinearity, 7, 861-924 (1994) · Zbl 0806.58015 |
| [41] | Birman, J.; Williams, R., Knotted periodic orbits in dynamical systems II: knot holders for fibered knots, Cont. Math., 20, 1-60 (1983) · Zbl 0526.58043 |
| [42] | Holmes, P. J.; Williams, R. F., Knotted periodic orbits in suspensions of Smale’s horseshoe: torus knots and bifurcation sequences, Arch. Rational Mech. Anal., 90, 15-194 (1985) · Zbl 0593.58027 |
| [43] | Holmes, P. J., Knotted periodic orbits in suspensions of annulus maps, Proc. R. Soc. London A, 411, 351-378 (1987) · Zbl 0642.58039 |
| [44] | Holmes, P. J., Knotted period orbits in suspensions of Smale’s horseshoe: extended families and bifurcation sequences, Physica D, 40, 42-64 (1989) · Zbl 0825.58035 |
| [45] | Tufillaro, N. B.; Holzner, R.; Flepp, L.; Brun, R.; Finardi, M.; Badii, R., Template analysis for a chaotic NMR laser, Phys. Rev. A, 44, R4786-R4788 (1991) |
| [46] | Papoff, F.; Fioretti, A.; Arimondo, E.; Mindlin, G. B.; Solari, H. G.; Gilmore, R., Structure of chaos in the laser with a saturable absorber, Phys. Rev. Lett., 68, 1128-1131 (1992) |
| [47] | Fioretti, A.; Molesti, F.; Zambon, B.; Arimondo, E.; Papoff, F., Topological analysis of laser with saturable absorber in experiments and models, Int. J. Bifurc. Chaos, Appl. Sci. Eng., 3, 559-564 (1993) · Zbl 0875.78008 |
| [48] | Lefranc, M.; Glorieux, P., Topological analysis of chaotic signals from a \(CO_2\) laser with modulated losses, Int. J. Bifurc. Chaos, Appl. Sci. Eng., 3, 643-649 (1993) · Zbl 0875.78007 |
| [49] | Braun, T.; Correira, R. R.B.; Altmann, N., Topological model of homoclinic chaos in a glow discharge, Phys. Rev. E, 51, 4165-4168 (1995) |
| [50] | Letellier, C.; Sceller, L. L.; Dutertre, P.; Gouesbet, G.; Fei, Z.; Hudson, J. L., Topological characterization and global vector field reconstruction of an experimental electrochemical system, J. Phys. Chem., 99, 7016-7027 (1995) |
| [51] | Tufillaro, N. B.; Wyckoff, P.; Brown, R.; Schreiber, T.; Molteno, T., Topological time series analysis of a string experiment and its synchronized model, Phys. Rev. E, 51, 164-174 (1995) |
| [52] | Letellier, C.; Gouesbet, G.; Rulkov, N. F., Topological analysis of chaos in equivariant electronic circuits, Int. J. Bifurc. Chaos, Appl. Sci. Eng., 6, 2531-2555 (1996) · Zbl 1298.94157 |
| [53] | Boulant, G.; Lefranc, M.; Bielawski, S.; Derozier, D., Horseshoe templates with global torsion in a driven laser, Phys. Rev. E, 55, 5082-5091 (1997) |
| [54] | Boulant, G.; Bielawski, S.; Derozier, D.; Lefranc, M., Experimental observation of a chaotic attractor with a reverse horseshoe topological structure, Phys. Rev. E, 55, R3801-R3804 (1997) |
| [55] | McCallum, J. W.L.; Gilmore, R., A geometric model for the Duffing oscillator, Int. J. Bifurc. Chaos, Appl. Sci. Eng., 3, 685-691 (1993) · Zbl 0900.70331 |
| [56] | Gilmore, R.; McCallum, J. W.L., Structure in the bifurcation diagram of the Duffing oscillator, Phys. Rev. E, 51, 935-956 (1995) |
| [57] | Letellier, C.; Dutertre, P.; Gouesbet, G., Characterization of the Lorenz system, taking into account the equivariance of the vector field, Phys. Rev. E, 49, 3492-3495 (1994) |
| [58] | Letellier, C.; Dutertre, P.; Maheu, B., Unstable periodic orbits and templates of the Rössler system: toward a systematic topological characterization, Chaos, 5, 271-282 (1995) |
| [59] | Tufillaro, N. B., Braid analysis of a bouncing ball, Phys. Rev. E, 50, 4509-4522 (1994) |
| [60] | Letellier, C.; Gouesbet, G.; Soufi, F.; Buchler, J. R.; Kolláth, Z., Chaos in variable stars: topological analysis of W Vir model pulsations, Chaos, 6, 466-476 (1996) · Zbl 1055.85500 |
| [61] | Boulant, G.; Lefranc, M.; Bielawski, S.; Derozier, D., A non-horseshoe template in a chaotic laser model, Int. J. Birfurc. Chaos, Appl. Sci. Eng., 8, 965-975 (1998) |
| [62] | Gilmore, R.; Vilaseca, R.; Corbalan, R.; Roldan, E., Topological analysis of chaos in the optically pumped laser, Phys. Rev. E, 55, 2479-2487 (1997) |
| [63] | Roldan, E.; de Valcárcel, G. J.; Vilaseca, R.; Martínez, V. J.; Gilmore, R., The dynamics of optically pumped molecular lasers. On its relation with the Lorenz-Haken laser model, Quant. Semiclassical Opt., 9, 1-35 (1997) |
| [64] | Gilmore, R.; Pei, X.; Moss, F., Topological analysis of chaos in neural spike train bursts, Chaos, 9, 812-8817 (1999) · Zbl 1070.92505 |
| [65] | Melvin, P.; Tufillaro, N. B., Templates and framed braids, Phys. Rev. A, 44, 3419-3422 (1991) |
| [66] | M. Lefranc, unpublished.; M. Lefranc, unpublished. |
| [67] | Hansen, K., Remarks on the symbolic dynamics for the Hénon map, Phys. Lett. A, 165, 100-104 (1992) |
| [68] | Giovannini, F.; Politi, A., Generating partitions in Hénon-type maps, Phys. Lett. A, 161, 332-336 (1992) · Zbl 0979.37502 |
| [69] | Hénon, M., A two-dimensional mapping with a strange attractor, Commun. Math. Phys., 50, 69-77 (1976) · Zbl 0576.58018 |
| [70] | Mindlin, G. B.; Lopez-Ruiz, R.; Solari, H. G.; Gilmore, R., Horseshoe implications, Phys. Rev. E, 48, 4297-4304 (1993) |
| [71] | Hansen, K. T.; Cvitanović, P., Bifurcation structures in maps of Hénon type, Nonlinearity, 11, 1233-1261 (1998) · Zbl 0965.37040 |
| [72] | Hall, T., Weak universality in two-dimensional transitions to chaos, Phys. Rev. Lett., 71, 58-61 (1993) |
| [73] | Tredicce, J. R.; Arecchi, F. T.; Puccioni, G. P.; Poggi, A.; Gadomski, W., Dynamics behavior and onset of low-dimensional chaos in a modulated homogeneously broadened single-mode laser: experiments and theory, Phys. Rev. A, 3, 2073-2081 (1986) |
| [74] | Dangoisse, D.; Glorieux, P.; Hennequin, D., Chaos in a \(CO_2\) laser with modulated parameters: experiments and numerical simulations, Phys. Rev. A, 36, 4775-4791 (1987) |
| [75] | Aurenhammer, F., Voronoi diagrams — a survey of a fundamental geometric data structure, ACM Comput. Surveys, 23, 345-405 (1991) |
| [76] | F.P. Preparata, M.I. Shamos, Computational Geometry: An Introduction, Springer, Berlin, 1985.; F.P. Preparata, M.I. Shamos, Computational Geometry: An Introduction, Springer, Berlin, 1985. · Zbl 0575.68059 |
| [77] | A. Okabe, B. Boots, K. Sugihara, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, Wiley, Chichester, UK, 1992.; A. Okabe, B. Boots, K. Sugihara, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, Wiley, Chichester, UK, 1992. · Zbl 0877.52010 |
| [78] | Watson, D. F., Computing the \(n\)-dimensional Delaunay tesselation with application to Voronoi polytopes, Comput. J., 24, 167-172 (1981) |
| [79] | J.-D. Boissonat, M. Teillaud, in: Proceedings of the Second Annual ACM Symposium on Computing Geometry, ACM, New York, 1986, pp. 260-268.; J.-D. Boissonat, M. Teillaud, in: Proceedings of the Second Annual ACM Symposium on Computing Geometry, ACM, New York, 1986, pp. 260-268. |
| [80] | Watson, D. F., Natural neighboring sorting, Austral. Comput. J., 17, 189-193 (1985) |
| [81] | R. Gilmore, in: B. Bosacchi, J.C. Bezdek, D.B. Fogel (Eds.), Applications of Soft Computing, Proceedings of SPIE, Vol. 3165, SPIE, Bellingham, 1997, pp. 243-257.; R. Gilmore, in: B. Bosacchi, J.C. Bezdek, D.B. Fogel (Eds.), Applications of Soft Computing, Proceedings of SPIE, Vol. 3165, SPIE, Bellingham, 1997, pp. 243-257. |
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