Topological analysis of chaotic dynamical systems

Reviews of Modern Physics

Volumn 70, Issue 4 PART II, 1998, Pages 1455-1529

  Gilmore, Robert ^a  

^a DREXEL UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 22444455689     PISSN: 00346861     EISSN: None     Source Type: Journal    
DOI: 10.1103/revmodphys.70.1455     Document Type: Article

References (186)

1. Abarbanel, H. D. I., 1996, Analysis of Observed Chaotic Data (Springer, New York).
2. Abarbanel, H. D. I., R. Brown, J. J. Sidorowich, and L. S. Tsimring, 1993, Rev. Mod. Phys. 65, 1331.
3. Abraham, N. B., D. Dangoise, P. Glorieux, and P. Mandel, 1985, J. Opt. Soc. Am. B 2, 23.
4. Abraham, N. B., P. Mandel, and L. M. Narducci, 1988, in Progress in Optics XXV, edited by E. Wolf (North-Holland, Amsterdam), p. 1.
5. Abraham, R. H., and C. D. Shaw, 1984, Dynamics, The Geometry of Behavior, Part III (Ariel Press, Santa Cruz, CA).
6. Adams, C. C., 1994, The Knot Book (Freeman, New York).
7. Alligood, K. T., 1985, Trans. Am. Math. Soc. 292, 713.
8. Alligood, K. T., T. D. Sauer, and J. A. Yorke, 1997, Chaos: An Introduction to Dynamical Systems (New York, Springer).
9. Alseda, L., J. Llibre, and M. Misiurewicz, 1993, Combinatorial Dynamics and Entropy in Dimension One (World Scientific, Singapore).
10. Alvarez, R., 1996, Ph.D. thesis (Drexel University).
11. Arecchi, F. T., and R. G. Harrison, 1993, Selected Papers on Optical Chaos (SPIE Milestone Series, Vol. MS 75).
12. Arecchi, F. T., R. Meucci, G. P. Puccioni, and J. R. Tredicce, 1982, Phys. Rev. Lett. 49, 1217.
13. Argoul, F., A. Arneodo, P. Richetti, J. C. Roux, and H. L. Swinney, 1987, Acc. Chem. Res. 86, 119.
14. Arnol'd, V. I., 1968, Russian Math. Surveys 23, 1.
15. Arnol'd, V. I., 1973, Ordinary Differential Equations (MIT, Cambridge, MA).
16. Arnol'd, V. I., 1975, Russian Math. Surveys 30, 1.
17. Arnol'd, V. I., 1986, Catastrophe Theory, second edition (Springer, New York).
18. Atiyah, M., 1990, The Geometry and Physics of Knots (Cambridge University, Cambridge, England).
19. Auerbach, D., P. Cvitanovic, J.-P. Eckmann, G. Gunaratne, and I. Procaccia, 1987, Phys. Rev. Lett. 58, 2387.
20. Beiersdorfer, P., J. M. Wersinger, and Y. Treve, 1983, Phys. Lett. A 96, 269.
21. Bestvina, M., and M. Handel, 1992, Ann. Math. 135, 1.
22. Birman, J. S., 1975, Braids, Links, and Mapping Class Groups (Princeton University, Princeton, NJ).
23. Birman, J. S., and R. F. Williams, 1983a, Topology 22, 47.
24. Birman, J. S., and R. F. Williams, 1983b, Contemp. Math. 20, 1.
25. Biswas, D. J., R. G. Harrison, C. O. Weiss, W. Klische, D. Dangoise, P. Glorieux, and N. M. Lawandy, 1986, in Instabilities and Chaos in Quantum Optics, edited by F. T. Arecchi and R. G. Harrison (Springer, Berlin), p. 109.
26. Block, L., and W. A. Coppel, 1992, Dynamics in One Dimension, Lecture Notes in Mathematics No. 1513 (Springer, New York).
27. Block, L., J. Guckenheimer, M. Misiurewicz, and L.-S. Yang, 1980, Global Theory of Dynamical Systems, edited by Z. Nitecki and C. Robinson, Lecture Notes in Mathematics No. 819 (Springer, New York), p. 18.
28. Bocko, M. F., D. H. Douglas, and H. H. Fruchty, 1984, Phys. Lett. A 104, 388.
29. Boulant, G., S. Bielawski, D. Derozier, and M. Lefranc, 1997a, Phys. Rev. E 55, R3801.
30. Boulant, G., M. Lefranc, S. Bielawski, and D. Derozier, 1997b, Phys. Rev. E 55, 5208.
31. Boulant, G., M. Lefranc, S. Bielawski, and D. Derozier, 1997c, Int. J. Bifurcation Chaos Appl. Sci. Eng. (in press).
32. Broomhead, D. S., and G. P. King, 1986, Physica D 20, 217.
33. Brown, R., N. Rulkov, and E. R. Tracy, 1994, Phys. Rev. E 49, 3784.
34. Chauve, M. P., and P. LeGal, 1992, Physica D 58, 407.
35. Coffman, K., W. D. McCormick, Z. Nosziczius, R. H. Simoyi, and H. L. Swinney, 1987, J. Chem. Phys. 86, 119.
36. Collet, P., and J.-P. Eckmann, 1980, Iterated Maps on the Interval as Dynamical Systems (Birkhäuser, Boston).
37. Creveling, H. F., J. F. De Paz, J. Y. Baladi, and R. J. Schoenhals, 1975, J. Fluid Mech. 67, 65.
38. Cvitanovic, P., 1984, Universality in Chaos (Adam Hilger, Ltd., Bristol).
39. Cvitanovic, P., G. H. Gunaratne, and I. Procaccia, 1988, Phys. Rev. A 38, 1503.
40. Dangoisse, D., A. Bekkale, F. Papoff, and P. Glorieux, 1988, Europhys. Lett. 6, 335.
41. DeMelo, W., and S. Van Strien, 1993, One Dimensional Dynamics (Springer, New York).
42. Devaney, R., and Z. Nitecki, 1979, Commun. Math. Phys. 67, 137.
43. Draper, N. R., and H. Smith, 1966, Applied Regression Analysis (Wiley, New York).
44. Duffing, G., 1918, Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz (Vieweg, Braunschweig).
45. Dupertuis, M. A., R. R. E. Salomaa, and M. R. Siegrist, 1986, Opt. Commun. 57, 410.
46. Eckmann, J.-P., and D. Ruelle, 1985, Rev. Mod. Phys. 57, 617.
47. Eckmann, J.-P., and D. Ruelle, 1987, Europhys. Lett. 4, 973.
48. Efron, B., 1979, SIAM (Soc. Ind. Appl. Math.) Rev. 21, 460.
49. Englisch, V., and W. Lauterborn, 1991, Phys. Rev. A 44, 916.
50. Eschenazi, E., H. G. Solari, and R. Gilmore, 1989, Phys. Rev. A 39, 2609.
51. Feigenbaum, M. J., 1978, J. Stat. Phys. 19, 25.
52. Feigenbaum, M. J., 1980, Los Alamos Sci. 1, 4.
53. Franks, J., and R. F. Williams, 1985, Trans. Am. Math. Soc. 20, 241.
54. Fujisaka, H., and T. Yamada, 1983, Prog. Theor. Phys. 69, 32.
55. Gaito, S. T., and G. P. King, 1990, in Measures of Complexity and Chaos, edited by N. B. Abraham, A. M. Albano, A. P. Passamante, and P. E. Rapp, Vol. 208 of NATO ARW Series B (Plenum, New York), p. 235.
56. Ghrist, R. W., P. J. Holmes, and M. C. Sullivan, 1996, Knots and Links in Three-Dimensional Flows (unpublished).
57. Gilmore, C., 1993a, Int. J. Bifurcation Chaos Appl. Sci. Eng. 3, 583.
58. Gilmore, C., 1993b, J. Econ. Behav. Organ. 22, 209.
59. Gilmore, R., 1974, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, New York).
60. Gilmore, R., 1981, Catastrophe Theory for Scientists and Engineers (Wiley, New York).
61. Gilmore, R., 1996, A program for dynamical systems theory, Drexel University, preprint (unpublished).
62. Gilmore, R., and J. W. L. McCallum, 1995, Phys. Rev. E 51, 935.
63. Gilmore, R., R. Vilaseca, R. Corbalan, and E. Roldân, 1997, Phys. Rev. E 55, 2479.
64. Gioggia, R. S., and N. B. Abraham, 1983, Phys. Rev. Lett. 51, 650.
65. Golubitsky, M., and V. Guillemin, 1973, Stable Mappings and Their Singularities (Springer, New York).
66. Gorman, M., P. J. Widmann, and K. A. Robins, 1984, Phys. Rev. Lett. 52, 2241.
67. Gorman, M., P. J. Widmann, and K. A. Robins, 1986, Physica D 19, 255.
68. Grassberger, P., and I. Procaccia, 1983, Phys. Rev. A 28, 2591.
69. Grebogi, C., E. Ott, and J. A. Yorke, 1983, Physica D 7, 181.
70. Guckenheimer, J., and P. J. Holmes, 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York).
71. Gunaratne, G. H., P. S. Linsay, and M. J. Vinson, 1989, Phys. Rev. Lett. 63, 1.
72. Haken, H., 1975, Phys. Lett. A 53, 77.
73. Hall, T., 1991, Ph.D. Thesis (Cambridge University).
74. Hall, T., 1993, Phys. Rev. Lett. 71, 58.
75. Hall, T., 1994a, Nonlinearity 7, 367.
76. Hall, T., 1994b, Nonlinearity 7, 861.
77. Halsey, T. C., M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, 1986, Phys. Rev. A 33, 1141.
78. Hammel, S., 1990, Phys. Lett. A 148, 421.
79. Hao, B.-L., 1984, Chaos (World Scientific, Singapore).
80. Harrison, R. G., I. A. Al-Saidi, and D. J. Biswas, 1985, IEEE J. Quantum Electron. 21, 1491.
81. Harrison, R. G., and D. J. Biswas, 1985, Phys. Rev. Lett. 55, 63.
82. Hayashi, C., 1964, Nonlinear Oscillations in Physical Systems (McGraw Hill, New York).
83. Hayashi, C., 1975, Selected Papers on Nonlinear Oscillations (Nippon Printing and Publishing, Osaka).
84. Hennequin, D., F. de Tomais, B. Zambon, and E. Arimondo, 1988, Phys. Rev. A 37, 2243.
85. Holmes, P. J., 1986, Physica D 21, 7.
86. Holmes, P. J., 1988, in New Directions in Dynamical Systems, edited by T. Bedford and J. Swift (Cambridge University, Cambridge, England), p. 150.
87. Holmes, P. J., and R. F. Williams, 1985, Arch. Ration. Mech. Anal. 90, 115.
88. Hong, F. L., M. Tachikawa, T. Oda, and T. Shimizu, 1989, J. Opt. Soc. Am. B 6, 1378.
89. Jackson, E. A., 1990, Perspectives in Nonlinear Dynamics (Cambridge University, Cambridge, England).
90. Kaplan, J. L., and J. A. Yorke, 1979, in Functional Difference Equations and the Approximation of Fixed Points, edited by H. O. Pietgen and H. O. Walther, Springer Lecture Notes in Mathematics No. 730 (Springer, Berlin), p. 204.
91. Kaufmann, L. H., 1987, On Knots (Princeton University, Princeton, NJ).
92. Khanin, Ya. I., 1995, Principles of Laser Dynamics (North-Holland, Amsterdam).
93. King, G. P., and S. T. Gaito, 1992, Phys. Rev. A 46, 3092.
94. Klinker, T., T. W. Meyer-Ilse, and W. Lauterborn, 1984, Phys. Lett. A 101, 371.
95. Knop, W., and W. Lauterborn, 1990, J. Chem. Phys. 93, 3950.
96. Kocarev, L., Z. Tasev, and D. Dimovski, 1994, Phys. Lett. A 190, 399.
97. Kurz, T., and W. Lauterborn, 1987, Phys. Rev. A 37, 1029.
98. Lathrop, D. P., and E. J. Kostelich, 1989, Phys. Rev. A 40, 4028.
99. Lauterborn, W., and R. Steinhoff, 1988, J. Opt. Soc. Am. B 5, 1097.
100. Lawandy, N. M., and G. A. Koepf, 1980, IEEE J. Quantum Electron. 16, 701.
101. Lefebvre, M., D. Dangoise, and P. Glorieux, 1984, Phys. Rev. A 29, 758.
102. Lefranc, M., and P. Glorieux, 1993, Int. J. Bifurcation Chaos Appl. Sci. Eng. 3, 643.
103. Lefranc, M., D. Hennequin, and D. Dangoise, 1991, J. Opt. Soc. Am. B 8, 239.
104. Letellier, C., and G. Gouesbet, 1995, Phys. Rev. E 52, 4754.
105. Lorenz, E. N., 1956, Statistical Forecasting Program (Department of Meteorology, MIT, Cambridge, MA), Technical Report No. 1.
106. Lorenz, E. N., 1963, J. Atmos. Sci. 20, 130.
107. Lorenz, E. N., 1980, Ann. (N.Y.) Acad. Sci. 357, 282.
108. Los, J., 1993, Proc. London Math. Soc. (3) 66, 1.
109. Mancho, A. M., A. A. Duarte, and G. B. Mindlin, 1996, Phys. Lett. A 221, 181.
110. Marié, R., 1981, in Dynamical Systems and Turbulence, Warwick, 1980, edited by D. Rand and L.-S. Young, Lecture Notes in Mathematics No. 898 (Springer, Berlin), p. 230.
111. Matorin, I. I., A. S. Pikovskii, and Ya. I. Khanin, 1984, Sov. J. Quantum Electron. 14, 1401.
112. Melvin, P., and N. B. Tufillaro, 1991, Phys. Rev. A 44, 3419.
113. Metropolis, M., M. L. Stein, and P. R. Stein, 1973, J. Comb. Theory, Ser. A 15, 25.
114. Mickens, R. E., 1981, Nonlinear Oscillations (Cambridge University, Cambridge, England).
115. Midavaine, T., D. Dangoisse, and P. Glorieux, 1986, Phys. Rev. Lett. 55, 1989.
116. Milnor, J., and W. Thurston, 1987, in Dynamical Systems, edited by J. C. Alexander, Lecture Notes in Mathematics No. 1342 (Springer, New York), p. 465.
117. Mindlin, G. B., and R. Gilmore, 1992, Physica D 58, 229.
118. Mindlin, G. B., X.-J. Hou, H. G. Solari, R. Gilmore, and N. B. Tufillaro, 1990, Phys. Rev. Lett. 64, 2350.
119. Mindlin, G. B., R. Lopez-Ruiz, H. G. Solari, and R. Gilmore, 1993, Phys. Rev. E 48, 4297.
120. Mindlin, G. B., and H. G. Solari, 1995, Phys. Rev. E 54, 1497.
121. Mindlin, G. B., H. G. Solari, M. A. Natiello, R. Gilmore, and X.-J. Hou, 1991, J. Nonlinear Sci. 1, 147.
122. Minorsky, N., 1962, Nonlinear Oscillations (Van Nostrand, Princeton, NJ).
123. Narducci, L. M., and N. B. Abraham, 1988, Laser Physics and Laser Instabilities (World Scientific, Singapore).
124. Nayfeh, A. H., 1973, Perturbation Methods (Wiley, New York).
125. Newhouse, S. E., 1974, Topology 13, 9.
126. Oppenheim, A. V., and R. W. Schafer, 1989, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, NJ).
127. Oseledec, V. I., 1968, Trans. Mosc. Math. Soc. 19, 197.
128. Ott, E., 1993, Chaos in Dynamical Systems (Cambridge University, Cambridge, England).
129. Packard, N. H., J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, 1980, Phys. Rev. Lett. 45, 712.
130. Papoff, F., A. Fioretti, E. Arimondo, G. B. Mindlin, H. G. Solari, and R. Gilmore, 1992, Phys. Rev. Lett. 68, 1128.
131. Parlitz, U., V. Englisch, C. Scheffczyk, and W. Lauterborn, 1990, J. Acoust. Soc. Am. 88, 1061.
132. Parlitz, U., and W. Lauterborn, 1985, Phys. Lett. 107A, 351.
133. Pecora, L. M., and T. L. Carroll, 1990, Phys. Rev. Lett. 64, 821.
134. Poincaré, H., 1892, Les Méthodes Nouvelle de la Mécanique Céleste (Gauthier Villars, Paris).
135. Poston, T., and I. N. Stewart, 1978, Catastrophe Theory and Its Applications (Pitman, London).
136. Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, 1986, Numerical Recipes (Cambridge University, Cambridge, England).
137. Puccioni, G. P., A. Poggi, W. Gadomski, J. R. Tredicce, and F. T. Arecchi, 1985, Phys. Rev. Lett. 55, 339.
138. Richetti, P., P. De Keeper, J. C. Roux, and H. L. Swinney, 1987, J. Stat. Phys. 48, 977.
139. Roldán, E., G. J. de Valcárcel, R. Vilaseca, R. Corbálan, V. J. Martinez, and R. Gilmore, 1997, Quantum Semiclassic. Opt. 9, R1.
140. Rolfsen, D., 1976, Knots and Links (Publish or Perish, Berkeley, CA).
141. Roux, J.-C., R. H. Simoyi, and H. L. Swinney, 1983, Physica D 8, 257.
142. Rössler, O. E., 1976a, Phys. Lett. A 57, 397.
143. Rössler, O. E., 1976b, Z. Naturforsch. A 31a, 1664.
144. Satija, I. I., A. R. Bishop, and K. Fesser, 1985, Phys. Lett. A 112, 183.
145. Sauer, T., 1992, Physica A 58, 193.
146. Sauer, T., J. A. Yorke, and M. Casdagli, 1991, J. Stat. Phys. 65, 579.
147. Scheffczyk, C., U. Parlitz, T. Kurz, W. Knop, and W. Lauterborn, 1991, Phys. Rev. A 43, 6495.
148. Schwartz, I. B., and H. L. Smith, 1983, J. Math. Biol. 18, 233.
149. Scott, S. K., 1991, Chemical Chaos (Oxford University, London).
150. Shaw, R., 1981, Z. Naturforsch. A 36a, 80.
151. Shil'nikov, A. L., 1993, Physica D 62, 338.
152. Shimizu, T., and N. Morioka, 1980, Phys. Lett. A 76, 201.
153. Shinriki, M., M. Yamamoto, and S. Mori, 1981, Proc. IEEE 69, 394.
154. Šilnikov, L. P., 1965, Sov. Math. Dokl. 66, 163.
155. Singer, J., Y.-Z. Wang, and H. Bau, 1991, Phys. Rev. Lett. 66, 1123.
156. Smale, S., 1967, Bull. Am. Math. Soc. 73, 747.
157. Solari, H. G., E. Eschenazi, R. Gilmore, and J. R. Tredicce, 1987, Opt. Commun. 64, 49.
158. Solari, H. G., and R. Gilmore, 1988, Phys. Rev. A 37, 3096.
159. Solari, H. G., M. A. Natiello, and G. B. Mindlin, 1996, Nonlinear Physics and Its Mathematical Tools (IOP Publishing Ltd., Bristol).
160. Solari, H. G., M. A. Natiello, and M. Vazquez, 1996, Phys. Rev. E 54, 3185.
161. Sparrow, C., 1982, The Lorenz Equations (Springer, New York).
162. Takens, F., 1981, in Dynamical Systems and Turbulence, Warwick, 1980, edited by D. Rand and L.-S. Young, Lecture Notes in Mathematics No. 898 (Springer, Berlin), p. 366.
163. Thom. R., 1975, Structural Stability and Morphogeneiss (Benjamin-Addison Wesley, New York).
164. Thompson, J. M. T., and H. B. Stewart, 1986, Nonlinear Dynamics and Chaos (Wiley, New York).
165. Tredicce, J. R., N. B. Abraham, G. P. Puccioni, and F. T. Arecchi, 1985, Opt. Commun. 55, 131.
166. Tredicce, J. R., F. T. Arecchi, G. L. Lippi, and G. P. Puccioni, 1985, J. Opt. Soc. Am. B 2, 173.
167. Tredicce, J. R., F. T. Arecchi, G. P. Puccioni, A. Poggi, and W. Gadomski, 1986, Phys. Rev. A 34, 2073.
168. Tufillaro, N. B., 1994, Phys. Rev. E 50, 4509.
169. Tufillaro, N. B., T. Abbott, and J. Reilly, 1992, An Experimental Approach to Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, MA).
170. Tufillaro, N. B., R. Holzner, L. Flepp, E. Brun, M. Finardi, and R. Badii, 1991, Phys. Rev. A 44, 4786.
171. Tufillaro, N. B., H. G. Solari, and R. Gilmore, 1990, Phys. Rev. A 41, 5717.
172. Tufillaro, N. B., P. Wyckkoff, R. Brown, T. Schreiber, and T. Molteno, 1995, Phys. Rev. E 51, 164.
173. Uezu, T., 1983, Phys. Lett. A 93, 161.
174. Uezu, T., and Y. Aizawa, 1982, Prog. Theor. Phys. 68, 1907.
175. van Buskirk, R., and C. Jeffries, 1985, Phys. Rev. A 31, 3332.
176. van der Pol, B., 1927, Philos. Mag. (7) 3, 65.
177. Waldner, F., R. Badii, D. R. Barberis, G. Broggi, W. Floeder, P. F. Meier, R. Stoop, M. Warden, and H. Yamazaki, 1986, J. Magn. Magn. Mater. 54-57, 1135.
178. Wedding, B., A. Gasch, and D. Jaeger, 1984, Phys. Lett. A 105, 105.
179. Weiss, C. O., 1985, J. Opt. Soc. Am. B 2, 137.
180. Weiss, C. O., and J. Brock, 1986, Phys. Rev. Lett. 50, 413.
181. Weiss, C. O., and W. Klisch, 1984, Opt. Commun. 57, 2804.
182. Weiss, C. O., and R. Vilaseca, 1991, Dynamics of Lasers (VCH, New York).
183. Whitney, H., 1936, Ann. Math. 37, 645.
184. Wolf, A., J. B. Swift, H. L. Swinney, and J. A. Vastano, 1985, Physica D 16, 285.
185. Zambon, B., 1991, Phys. Rev. A 44, R688.
186. Zeeman, E. C., 1977, Catastrophe Theory: Selected Papers 1972-1977 (Addison-Wesley, Reading, MA).