- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Reviews of Modern Physics

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Leaking chaotic systems

Eduardo G. Altmann, Jefferson S. E. Portela, and Tamás Tél

Eduardo G. Altmann

  • Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany

Jefferson S. E. Portela

  • Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany and Fraunhofer Institute for Industrial Mathematics ITWM, 67663 Kaiserslautern, Germany

Tamás Tél

  • Institute for Theoretical Physics-HAS Research Group, Eötvös University, Pázmány Péter sétány 1/A, Budapest, H-1117, Hungary

PDF

Rev. Mod. Phys. 85, 869 – Published 29 May, 2013

DOI: https://doi.org/10.1103/RevModPhys.85.869

Abstract

There are numerous physical situations in which a hole or leak is introduced in an otherwise closed chaotic system. The leak can have a natural origin, it can mimic measurement devices, and it can also be used to reveal dynamical properties of the closed system. A unified treatment of leaking systems is provided and applications to different physical problems, in both the classical and quantum pictures, are reviewed. The treatment is based on the transient chaos theory of open systems, which is essential because real leaks have finite size and therefore estimations based on the closed system differ essentially from observations. The field of applications reviewed is very broad, ranging from planetary astronomy and hydrodynamical flows to plasma physics and quantum fidelity. The theory is expanded and adapted to the case of partial leaks (partial absorption and/or transmission) with applications to room acoustics and optical microcavities in mind. Simulations in the limaçon family of billiards illustrate the main text. Regarding billiard dynamics, it is emphasized that a correct discrete-time representation can be given only in terms of the so-called true-time maps, while traditional Poincaré maps lead to erroneous results. Perron-Frobenius-type operators are generalized so that they describe true-time maps with partial leaks.

View figure in article

Article Text

References (276)

  1. Abdullaev, S. S., T. Eich, and K. H. Finken, 2001, Phys. Plasmas 8, 2739.
  2. Afraimovich, V., and L. A. Bunimovich, 2010, Nonlinearity 23, 643.
  3. Aguirre, J., and M. A. F. Sanjuán, 2003, Phys. Rev. E 67, 056201.
  4. Aguirre, J., R. L. Viana, and M. A. F. Sanjuán, 2009, Rev. Mod. Phys. 81, 333.
  5. Akaishi, A., and A. Shudo, 2009, Phys. Rev. E 80, 066211.
  6. Alt, H., H. D. Gräf, H. L. Harney, R. Hofferbert, H. Lengeler, A. Richter, P. Schardt, and H. A. Weidenmüller, 1995, Phys. Rev. Lett. 74, 62.
  7. Alt, H., H.-D. Gräf, H. Harney, R. Hofferbert, H. Rehfeld, A. Richter, and P. Schardt, 1996, Phys. Rev. E 53, 2217.
  8. Altmann, E., and T. Tél, 2008, Phys. Rev. Lett. 100, 174101.
  9. Altmann, E., and T. Tél, 2009, Phys. Rev. E 79, 016204.
  10. Altmann, E. G., 2009, Phys. Rev. A 79, 013830.
  11. Altmann, E. G., E. C. da Silva, and I. L. Caldas, 2004, Chaos 14, 975.
  12. Altmann, E. G., and A. Endler, 2010, Phys. Rev. Lett. 105, 255102.
  13. Altmann, E. G., T. Friedrich, A. E. Motter, H. Kantz, and A. Richter, 2008, Phys. Rev. E 77, 016205.
  14. Altmann, E. G., and H. Kantz, 2007, Europhys. Lett. 78, 10008.
  15. Altmann, E. G., J. C. Leitao, and J. V. Lopes, 2012, Chaos 22, 026114.
  16. Altmann, E. G., G. Del Magno, and M. Hentschel, 2008, Europhys. Lett. 84, 10008.
  17. Altmann, E. G., A. E. Motter, and H. Kantz, 2005, Chaos 15, 033105.
  18. Altmann, E. G., A. E. Motter, and H. Kantz, 2006, Phys. Rev. E 73, 026207.
  19. Altmann, E. G., J. S. E. Portela, and T. Tél, 2013, unpublished.
  20. Aref, H., 1984, J. Fluid Mech. 143, 1.
  21. Aref, H., and M. E. Naschie, 1995, Chaos Applied to Fluid Mixing (Pergamon, New York), 1st ed.
  22. Ares, N., and D. A. Wisniacki, 2009, Phys. Rev. E 80, 046216.
  23. Armstead, D. N., B. R. Hunt, and E. Ott, 2004, Physica (Amsterdam) 193D, 96.
  24. Arnoldi, J.-F., F. Faure, and T. S. Weich, 2013, arXiv:1302.3087.
  25. Artuso, R., E. Aurell, and P. Cvitanovic, 1990, Nonlinearity 3, 361.
  26. Artuso, R., L. Cavallasca, and G. Cristadoro, 2008, Phys. Rev. E 77, 046206.
  27. Artuso, R., and C. Manchein, 2009, Phys. Rev. E 80, 036210.
  28. Artuso, R., and A. Prampolini, 1998, Phys. Lett. A 246, 407.
  29. Bäcker, A., and H. Dullin, 1997, J. Phys. A 30, 1991.
  30. Bahsoun, W., and S. Vaienti, 2012, arXiv:1206.3654v1.
  31. Bakhtin, Y., and L. Bunimovich, 2011, J. Stat. Phys. 143, 943.
  32. Barthélemy, J., O. Legrand, and F. Mortessagne, 2005, Europhys. Lett. 70, 162.
  33. Bauer, W., and G. Bertsch, 1990, Phys. Rev. Lett. 65, 2213.
  34. Berry, M., 1981, Eur. J. Phys. 2, 91.
  35. Berry, M., N. Balazs, M. Tabor, and A. Voros, 1979, Ann. Phys. (N.Y.) 122, 26.
  36. Berry, M. V., 1987, Proc. R. Soc. A 413, 183.
  37. Berry, M. V., 2009, J. Phys. A 42, 165208.
  38. Berry, M. V., 2010, in New Directions in Linear Acoustics and Vibration, edited by M. Wright, and R. Weaver (Cambridge University Press, Cambridge, England), pp. vii–ix.
  39. Bittner, S., E. Bogomolny, B. Dietz, M. Miski-Oglu, P. Oria Iriarte, A. Richter, and F. Schäfer, 2010, Phys. Rev. E 81, 066215.
  40. Bittner, S., B. Dietz, R. Dubertrand, J. Isensee, M. Miski-Oglu, and A. Richter, 2012, Phys. Rev. E 85, 056203.
  41. Bleher, S., C. Grebogi, E. Ott, and R. Brown, 1988, Phys. Rev. A 38, 930.
  42. Bodai, T., E. G. Altmann, and A. Endler, 2013, Phys. Rev. E 87, 042902.
  43. Bogomolny, E., R. Dubertrand, and C. Schmit, 2008, Phys. Rev. E 78, 056202.
  44. Buljan, H., and V. Paar, 2001, Phys. Rev. E 63, 066205.
  45. Bunimovich, L., 2012, Contemp. Math. 567, 11 236.
  46. Bunimovich, L., and A. Yurchenko, 2011, Isr. J. Math. 182, 229.
  47. Bunimovich, L. A., 2008, Nonlinearity 21, T13.
  48. Bunimovich, L. A., and C. P. Dettmann, 2005, Phys. Rev. Lett. 94, 100201.
  49. Bunimovich, L. A., and C. P. Dettmann, 2007, Europhys. Lett. 80, 40 001.
  50. Bunimovich, L. A., and B. Z. Webb, 2012, arXiv:1211.4617.
  51. Buric, N., A. Rampioni, G. Turchetti, and S. Vaienti, 2003, J. Phys. A 36, L209.
  52. Casati, G., G. Maspero, and D. L. Shepelyansky, 1997, Phys. Rev. E 56, R6233.
  53. Casati, G., G. Maspero, and D. L. Shepelyanski, 1999a, Physica (Amsterdam) 131D, 311.
  54. Casati, G., G. Maspero, and D. L. Shepelyansky, 1999b, Phys. Rev. Lett. 82, 524.
  55. Chappell, D. J., and G. Tanner, 2013, J. Comput. Phys. 234, 487.
  56. Chappell, D. J., G. Tanner, N. Sondergaard, and D. Loechel, 2013, arXiv:1303.4249.
  57. Chernov, N., and R. Markarian, 2006, Chaotic Billiards, Mathematical Surveys and Monographs (American Mathematical Society, Providence), Vol. 112.
  58. Chirikov, B. V., and D. L. Shepelyansky, 1984, Physica (Amsterdam) 13D, 395.
  59. Chirikov, B. V., and D. L. Shepelyansky, 1999, Phys. Rev. Lett. 82, 528.
  60. Chirikov, B. V., and D. L. Shepelyansky, 2002, Phys. Rev. Lett. 89, 239402.
  61. Christiansen, F., and P. Grassberger, 1993, Phys. Lett. A 181, 47.
  62. Cioffi, F., F. Gallerano, and E. Napoli, 2005, J. Hydraul. Res. 43, 290.
  63. Claus, I., and P. Gaspard, 2001, Phys. Rev. E 63, 036227.
  64. Collet, S., Pierre Martínez, and V. Maume-Deschamps, 2000, Nonlinearity 13, 1263.
  65. Cristadoro, G., G. Knight, and M. Degli Esposti, 2012, arXiv:1212.0673v1.
  66. Cristadoro, G., and R. Ketzmerick, 2008, Phys. Rev. Lett. 100, 184101.
  67. Custódio, M. S., and M. W. Beims, 2011, Phys. Rev. E 83, 056201.
  68. Cvitanović, P., R. Artuso, P. Dahlquist, R. Mainieri, G. Tanner, G. Vattay, N. Whelan, and A. Wirzba, 2004, “Chaos: Classical and Quantum,” http://chaosbook.org.
  69. da Silva, E. C., I. L. Caldas, R. L. Viana, and M. A. F. Sanjuán, 2002, Phys. Plasmas 9, 4917.
  70. Dembowski, C., B. Dietz, T. Friedrich, H.-D. Gräf, A. Heine, C. Mejía-Monasterio, M. Miski-Oglu, A. Richter, and T. Seligman, 2004, Phys. Rev. Lett. 93, 134102.
  71. Demers, M., and P. Wright, 2011, arXiv:1112.4812.
  72. Demers, M., P. Wright, and L.-S. Young, 2010, Commun. Math. Phys. 294, 353.
  73. Demers, M. F., and L.-S. Young, 2006, Nonlinearity 19, 377.
  74. de Moura, A. P. S., and P. S. Letelier, 1999, Phys. Lett. A 256, 362.
  75. Dennis, M. R., 2010, in New Directions in Linear Acoustics and Vibration, edited by M. Wright, and R. Weaver (Cambridge University Press, Cambridge, England), pp. 65–84.
  76. Dettmann, C. P., 2011, in Frontiers in the Study of Chaotic Dynamical Systems with Open Problems, edited by Z. Elhadj, and J. C. Sprott (World Scientific, Singapore), pp. 195–218.
  77. Dettmann, C. P., 2013, Nonlinearity 26, 307.
  78. Dettmann, C. P., and O. Georgiou, 2009, Physica (Amsterdam) 238D, 2395.
  79. Dettmann, C. P., and O. Georgiou, 2011a, J. Phys. A 44, 195102.
  80. Dettmann, C. P., and O. Georgiou, 2011b, Phys. Rev. E 83, 036212.
  81. Dettmann, C. P., and O. Georgiou, 2012, Chaos 22, 026113.
  82. Dettmann, C. P., and T. B. Howard, 2009, Physica (Amsterdam) 238D, 2404.
  83. Dettmann, C. P., and E. D. Leonel, 2012, Physica (Amsterdam) 241D, 403.
  84. Dettmann, C. P., G. V. Morozov, M. Sieber, and H. Waalkens, 2009, Phys. Rev. A 80, 063813.
  85. Dietz, B., T. Friedrich, M. Miski-Oglu, A. Richter, T. H. Seligman, and K. Zapfe, 2006, Phys. Rev. E 74, 056207.
  86. Dietz, B., A. Heine, A. Richter, O. Bohigas, and P. Leboeuf, 2006, Phys. Rev. E 73, 035201.
  87. Ding, M., T. Bountis, and E. Ott, 1990, Phys. Lett. A 151, 395.
  88. Dorfman, J. R., 1999, An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge University Press, Cambridge,England), 1st ed.
  89. Doron, E., and U. Smilansky, 1992a, Chaos 2, 117.
  90. Doron, E., and U. Smilansky, 1992b, Phys. Rev. Lett. 68, 1255.
  91. Dullin, H. R., and A. Baecker, 2001, Nonlinearity 14, 1673.
  92. Dumont, R., and P. Brumer, 1992, Chem. Phys. Lett. 188, 565.
  93. Eberspächer, A., J. Main, and G. Wunner, 2010, Phys. Rev. E 82, 046201.
  94. Eckhardt, B., 2003, J. Phys. A 36, 371.
  95. Engelhardt, W., and W. Feneberg, 1978, J. Nucl. Mater. 76–77, 518.
  96. Ermann, L., G. Carlo, and M. Saraceno, 2009, Phys. Rev. Lett. 103, 054102.
  97. Ermann, L., G. G. Carlo, J. M. Pedrosa, and M. Saraceno, 2012, Phys. Rev. E 85, 066204.
  98. Ermann, L., and D. L. Shepelyansky, 2010, Eur. Phys. J. B 75, 299.
  99. Evans, T. E., R. A. Moyer, and P. Monat, 2002, Phys. Plasmas 9, 4957.
  100. Ezra, G. S., H. Waalkens, and S. Wiggins, 2009, J. Chem. Phys. 130, 164118.
  101. Faisst, H., and B. Eckhardt, 2003, Phys. Rev. E 68, 026215.
  102. Falconer, K., 1985, The Geometry of Fractal Sets, Falconer-book (Cambridge University Press, Cambridge, England), 1st ed.
  103. Faure, F., N. Roy, and J. Sjoestrand, 2008, Open Mat. J. 1, 35 [http://www.benthamscience.com/open/tomatj/articles/V001/35TOMATJ.htm].
  104. Fendrik, A. J., A. M. F. Rivas, and M. J. Sánchez, 1994, Phys. Rev. E 50, 1948.
  105. Fendrik, A. J., and M. J. Sánchez, 1995, Phys. Rev. E 51, 2996.
  106. Fendrik, A. J., and D. A. Wisniacki, 1997, Phys. Rev. E 55, 6507.
  107. Ferguson, A., and M. Pollicott, 2012, Ergod. Theory Dyn. Syst. 32, 961.
  108. Friedman, N., A. Kaplan, D. Carasso, and N. Davidson, 2001, Phys. Rev. Lett. 86, 1518.
  109. Froyland, G., and K. Pradberg, 2009, Physica (Amsterdam) 238D, 1507.
  110. Froyland, G., and O. Stancevic, 2010, Discrete Contin. Dyn. Syst. Series B 14, 457.
  111. Fyodorov, Y. V., D. V. Savin, and H.-J. Sommers, 2005, J. Phys. A 38, 10731.
  112. García-Mata, I., and D. A. Wisniacki, 2011, J. Phys. A 44, 315101.
  113. Gaspard, P., 1996, Phys. Rev. E 53, 4379.
  114. Gaspard, P., 1998, Chaos, Scattering and Statistical Mechanics, Cambridge Nonlinear Science Series (Cambridge University Press, Cambridge, England), Vol. 9.
  115. Gaspard, P., and J. R. Dorfman, 1995, Phys. Rev. E 52, 3525.
  116. Geisel, T., A. Zacherl, and G. Radons, 1988, Z. Phys. B 71, 117.
  117. Georgiou, O., C. P. Dettmann, and E. G. Altmann, 2012, Chaos 22, 043115.
  118. Gmachl, C., F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, and D. L. Sivco, 1998, Science 280, 1556.
  119. Gorin, T., T. Prosen, T. H. Seligman, and M. Znidaric, 2006, Phys. Rep. 435, 33.
  120. Goussev, A., and K. Richter, 2007, Phys. Rev. E 75, 015201(R).
  121. Goussev, A., D. Waltner, K. Richter, and R. Jalabert, 2008, New J. Phys. 10, 093010.
  122. Grete, P., and M. Markus, 2007, Phys. Rev. E 75, 036207.
  123. Gutiérrez, M., and A. Goussev, 2009, Phys. Rev. E 79, 046211.
  124. Gutiérrez, M., D. Waltner, J. Kuipers, and K. Richter, 2009, Phys. Rev. E 79, 046212.
  125. Harayama, T., and S. Shinohara, 2011, Laser Photonics Rev. 5, 247.
  126. Haydn, N., Y. Lacroix, and S. Vaienti, 2005, Ann. Probab. 33, 2043.
  127. Höhmann, R., U. Kuhl, and H.-J. Stöckmann, 2008, Phys. Rev. Lett. 100, 124101.
  128. Hu, H., A. Rampioni, L. Rossi, G. Turchetti, and S. Vaienti 2004, Chaos 14, 160.
  129. Jacobs, J., E. Ott, and B. Hunt, 1998, Phys. Rev. E 57, 6577.
  130. Jacquod, P., and C. Petitjean, 2009, Adv. Phys. 58, 67.
  131. Joyce, W. B., 1975, J. Acoust. Soc. Am. 58, 643.
  132. Joyce, W. B., 1978, J. Acoust. Soc. Am. 64, 1429.
  133. Jung, C., T. Tél, and E. Ziemniak, 1993, Chaos 3, 555.
  134. Kac, M., 1959, Probability and Related Topics in Physical Sciences (Interscience, New York).
  135. Kantz, H., and P. Grassberger, 1985, Physica (Amsterdam) 17D, 75.
  136. Kantz, H., and P. Grassberger, 1987, Phys. Lett. A 123, 437.
  137. Kaplan, A., N. Friedman, M. Andersen, and N. Davidson, 2001, Phys. Rev. Lett. 87, 274101.
  138. Karney, C. F. F., 1983, Physica (Amsterdam) 8D, 360.
  139. Károlyi, G., M. Pattanyús-Ábrahám, T. Krámer, J. Józsa, and T. Tél, 2010, Engineering Comp. Mech. 163, 251.
  140. Katok, A., and B. Hasselblatt, 1995, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, England).
  141. Kaufmann, Z., and H. Lustfeld, 2001, Phys. Rev. E 64, 055206.
  142. Keating, J., M. Novaes, S. Prado, and M. Sieber, 2006, Phys. Rev. Lett. 97, 150406.
  143. Keller, G., and C. Liverani, 2009, J. Stat. Phys. 135, 519.
  144. Ketzmerick, R., L. Hufnagel, F. Steinbach, and M. Weiss, 2000, Phys. Rev. Lett. 85, 1214.
  145. Klages, R., 2007, Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics, Advanced Series in Nonlinear Dynamics (World Scientific, Singapore).
  146. Knight, G., O. Georgiou, C. Dettmann, and R. Klages, 2012, Chaos 22, 023132.
  147. Köber, B., U. Kuhl, H.-J. Stöckmann, A. Goussev, and K. Richter, 2011, Phys. Rev. E 83, 016214.
  148. Kokshenev, V., and M. Nemes, 2000, Physica (Amsterdam) 275A, 70.
  149. Kokshenev, V., and E. Vicentini, 2003, Phys. Rev. E 68, 016221.
  150. Kopp, M., and H. Schomerus, 2010, Phys. Rev. E 81, 026208.
  151. Körber, M. J., M. Michler, A. Bäcker, and R. Ketzmerick, 2013, arXiv:1304.2662.
  152. Kovács, Z., and T. Tél, 1992, Phys. Rev. A 45, 2270.
  153. Kuhl, U., H.-J. Stöckmann, and R. Weaver, 2005, J. Phys. A 38, 10433.
  154. Lai, Y.-C., and T. Tél, 2011, Transient Chaos: Complex Dynamics in Finite Time Scales, Applied Mathematical Sciences (Springer, New York), Vol. 173.
  155. Lai, Y.-C., K. Życzkowski, and C. Grebogi, 1999, Phys. Rev. E 59, 5261.
  156. Laskar, J., and M. Gastineau, 2009, Nature (London) 459, 817.
  157. Lau, Y.-T., J. M. Finn, and E. Ott, 1991, Phys. Rev. Lett. 66, 978.
  158. Lebental, M., J. S. Lauret, J. Zyss, C. Schmit, and E. Bogomolny, 2007, Phys. Rev. A 75, 033806.
  159. Lee, S.-Y., S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, 2004, Phys. Rev. Lett. 93, 164102.
  160. Legrand, O., and D. Sornette, 1990a, Physica (Amsterdam) 44D, 229.
  161. Legrand, O., and D. Sornette, 1990b, Europhys. Lett. 11, 583.
  162. Legrand, O., and D. Sornette, 1991a, Phys. Rev. Lett. 66, 2172.
  163. Legrand, O., and D. Sornette, 1991b, in Large Scale Structures in Nonlinear Physics, edited by J.-D. Fournier, and P.-L. Sulem, Lecture Notes in Physics (Springer, Berlin/Heidelberg), Vol. 392, pp. 267–274.
  164. Leonel, E. D., M. W. Beims, and L. A. Bunimovich, 2012, Chaos 22, 026101.
  165. Leonel, E. D., and C. P. Dettmann, 2012, arXiv:1203.6235v1.
  166. Lippolis, D., J.-W. Ryu, S.-Y. Lee, and S. W. Kim, 2012, Phys. Rev. E 86, 066213.
  167. Liu, Z., and Y.-C. Lai, 2002, Phys. Rev. E 65, 046204.
  168. Lu, W. T., S. Sridhar, and M. Zworski, 2003, Phys. Rev. Lett. 91, 154101.
  169. Matyas, L., and I. F. Barna, 2011, Chaos, Solitons and Fractals 44, 1111.
  170. Matyas, L., and P. Gaspard, 2005, Phys. Rev. E 71, 036147.
  171. Matyas, L., and R. R. Klages, 2004, Physica (Amsterdam) 187D, 66.
  172. Meiss, J. D., 1992, Rev. Mod. Phys. 64, 795.
  173. Meiss, J. D., 1997, Chaos 7, 139.
  174. Meiss, J. D., and E. Ott, 1985, Phys. Rev. Lett. 55, 2741.
  175. Meiss, J. D., and E. Ott, 1986, Physica (Amsterdam) 20D, 387.
  176. Mekis, A., J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, 1995, Phys. Rev. Lett. 75, 2682.
  177. Miyaguchi, T., 2007, Phys. Rev. E 75, 066215.
  178. Morrison, P. J., 2000, Phys. Plasmas 7, 2279.
  179. Mortessagne, F., O. Legrand, and D. Sornette, 1992, Europhys. Lett. 20, 287.
  180. Mortessagne, F., O. Legrand, and D. Sornette, 1993, Chaos 3, 529.
  181. Motter, A., 2001, Phys. Lett. A 285, 127.
  182. Nagler, J., 2002, Das eingeschränkte Drei-Körper-Problem im Crashtest, Ph.D. thesis, Bremen University.
  183. Nagler, J., 2004, Phys. Rev. E 69, 066218.
  184. Nagler, J., 2005, Phys. Rev. E 71, 026227.
  185. Nagler, J., M. Krieger, M. Linke, J. Schönke, and J. Wiersig, 2007, Phys. Rev. E 75, 046204.
  186. Nándori, P., and D. Szász, 2012, Chaos 22, 026115.
  187. Neufeld, Z., P. H. Haynes, and G. Picard, 2000, Phys. Fluids 12, 2506.
  188. Neufeld, Z., and E. Hernández-Garcia, 2009, Chemical and Biological Processes in Fluid Fows: A Dynamical Systems Approach (Imperial College Press, Cambridge), 1st ed.
  189. Ngan, K., and R. Pierrehumbert, 2000, Phys. Fluids 12, 822.
  190. Nöckel, J. U., and A. D. Stone, 1997, Nature (London) 385, 45.
  191. Nonnenmacher, S., 2011, Nonlinearity 24, R123.
  192. Nonnenmacher, S., and E. Schenk, 2008, Phys. Rev. E 78, 045202R.
  193. Nonnenmacher, S., and M. Zworski, 2005, J. Phys. A 38, 10 683.
  194. Novaes, M., 2012, arXiv:1211.7248v1.
  195. Novaes, M., J. M. Pedrosa, D. Wisniacki, G. G. Carlo, and J. P. Keating, 2009, Phys. Rev. E 80, 035202.
  196. Ott, E., 1993, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, England).
  197. Ottino, J. M., 1989, The Kinematics of Mixing: Stretching, Chaos and Transport (Cambridge University Press, Cambridge, England), 1st ed.
  198. Paar, V., and H. Buljan, 2000, Phys. Rev. E 62, 4869.
  199. Paar, V., and N. Pavin, 1997, Phys. Rev. E 55, 4112.
  200. Pedrosa, J. M., G. G. Carlo, D. A. Wisniacki, and L. Ermann, 2009, Phys. Rev. E 79, 016215.
  201. Pedrosa, J. M., D. Wisniacki, G. G. Carlo, and M. Novaes, 2012, Phys. Rev. E 85, 036203.
  202. Peres, A., 1984, Phys. Rev. A 30, 1610.
  203. Pianigiani, G., and J. A. Yorke, 1979, Trans. Am. Math. Soc. 252, 351.
  204. Pierrehumbert, R., H. Brogniez, and R. Roca, 2007, in The Global Circulation of the Atmosphere, edited by T. Schneider, and A. Sobel (Princeton University Press, Princeton, NJ).
  205. Pierrehumbert, R. T., 1994, Chaos Solitons Fractals 4, 1091.
  206. Pikovsky, A., and O. Popovich, 2003, Europhys. Lett. 61, 625.
  207. Pikovsky, A. S., 1992, J. Phys. A 25, L477.
  208. Poli, C., B. Dietz, O. Legrand, F. Mortessagne, and A. Richter, 2009, Phys. Rev. E 80, 035204.
  209. Poli, C., G. A. Luna-Acosta, and H.-J. Stöckmann, 2012, Phys. Rev. Lett. 108, 174101.
  210. Portela, J. S., I. L. Caldas, and R. L. Viana, 2008, Eur. Phys. J. Special Topics 165, 195.
  211. Portela, J. S. E., I. L. Caldas, R. L. Viana, and M. A. F. Sanjuán, 2007, Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 4067.
  212. Potzuweit, A., T. Weich, S. Barkhofen, U. Kuhl, H.-J. Stöckmann, and M. Zworski, 2012, Phys. Rev. E 86, 066205.
  213. Puhlmann, M., H. Schanz, T. Kottos, and T. Geisel, 2005, Europhys. Lett. 69, 313.
  214. Ramilowski, J. A., S. D. Prado, F. Borondo, and D. Farrelly, 2009, Phys. Rev. E 80, 055201(R)
  215. Redding, B., L. Ge, Q. Song, J. Wiersig, G. S. Solomon, and H. Cao, 2012, Phys. Rev. Lett. 108, 253902.
  216. Ree, S., and L. Reichl, 2002, Phys. Rev. E 65, 055205.
  217. Richter, A., 1999, in The IMA Volumes in Mathematics and its Applictaions (Springer-Verlag, New York), Vol. 109, p. 479.
  218. Robnik, M., 1983, J. Phys. A 16, 3971.
  219. Rotter, I., 2009, J. Phys. A 42, 153001.
  220. Ryu, J.-W., S.-Y. Lee, C.-M. Kim, and Y.-J. Park, 2006, Phys. Rev. E 73, 036207.
  221. Sadurní, E., and T. H. Seligman, 2008, J. Phys. A 41, 102002.
  222. Saito, N., H. Hirooka, J. Ford, F. Vivaldi, and G. Walker, 1982, Physica (Amsterdam) 5D, 273.
  223. Sanjuán, M. A. F., T. Horita, and K. Aihara, 2003, Chaos 13, 17.
  224. Savin, D. V., O. Legrand, and F. Mortessagne, 2006, Europhys. Lett. 76, 774.
  225. Savin, D. V., and V. V. Sokolov, 1997, Phys. Rev. E 56, R4911.
  226. Schelin, A., I. Caldas, R. Viana, and S. Benkadda, 2011, Phys. Lett. A 376, 24.
  227. Schenck, E., 2009, Ann. Henri Poincaré 10, 711.
  228. Schmitz, O., 2012, Nucl. Fusion 52, 054001.
  229. Schneider, J., V. Fernández, and E. Hernández-Garcia, 2005, J. Mar. Syst. 57, 111.
  230. Schneider, J., J. Schmalzl, and T. Tél, 2007, Chaos 17, 033115.
  231. Schneider, J., and T. Tél, 2003, Ocean Dynamics 53, 64.
  232. Schneider, J., T. Tél, and Z. Neufeld, 2002, Phys. Rev. E 66, 066218.
  233. Schomerus, H., and M. Hentschel, 2006, Phys. Rev. Lett. 96, 243903.
  234. Schomerus, H., and J. Tworzydło, 2004, Phys. Rev. Lett. 93, 154102.
  235. Schwefel, H. G. L., N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, 2004, J. Opt. Soc. Am. B 21, 923.
  236. Shen, Y., M. Miyake, S. Takamura, T. Kuroda, and T. Okuda, 1989, J. Nucl. Mater. 168, 295.
  237. Shepelyansky, D. L., 2008, Phys. Rev. E 77, 015202(R).
  238. Shinohara, S., T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, and E. E. Narimanov, 2010, Phys. Rev. Lett. 104, 163902.
  239. Shinohara, S., T. Harayama, T. Fukushima, M. Hentschel, S. Sunada, and E. Narimanov, 2011, Phys. Rev. A 83, 053837.
  240. Shinohara, S., M. Hentschel, J. Wiersig, T. Sasaki, and T. Harayama, 2009, Phys. Rev. A 80, 031801.
  241. Sieber, M., and K. Richter, 2001, Phys. Scr. 2001 T90, 128.
  242. Skipetrov, S. E., and B. A. van Tiggelen, 2006, Phys. Rev. Lett. 96, 043902.
  243. Song, Q., W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, 2009, Phys. Rev. A 80, 041807.
  244. Song, Q., L. Ge, B. Redding, and H. Cao, 2012, Phys. Rev. Lett. 108, 243902.
  245. Song, Q. H., L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, W. Fang, and G. S. Solomon, 2010, Phys. Rev. Lett. 105, 103902.
  246. Spina, M. A., I. Garcia-Mata, and M. Saraceno, 2010, J. Phys. A 43, 392003.
  247. Stockmann, H.-J., 1999, Quantum Chaos—An Introduction (Cambridge University Press, Cambridge, England), 1st ed.
  248. Szanyi, S., 2012, in Proceedings of the Conference of Junior Researchers in Civil Engineering, edited by J. Jozsa, T. Lovas, and R. Nemeth (Budapest University of Technology and Economics, Budapest), pp. 225–232.
  249. Taddese, B. T., T. M. Antonsen, E. Ott, and S. M. Anlage, 2010, J. Appl. Phys. 108, 114911.
  250. Takamura, S., Y. Shen, H. Yamada, M. Miyake, T. Tamakoshi, and T. Okuda, 1989, J. Nucl. Mater. 162–164, 643.
  251. Tanaka, H., and A. Shudo, 2006, Phys. Rev. E 74, 036211.
  252. Tanaka, T., M. Hentschel, T. Fukushima, and T. Harayama, 2007, Phys. Rev. Lett. 98, 033902.
  253. Tanner, G., 2009, J. Sound Vib. 320, 1023.
  254. Tél, T., 1987, Phys. Rev. A 36, 1502.
  255. Tél, T., A. P. S. de Moura, C. Grebogi, and G. Károlyi, 2005, Phys. Rep. 413, 91.
  256. Tél, T., and M. Gruiz, 2006, Chaotic Dynamics: An Introduction Based on Classical Mechanics (Cambridge University Press, Cambridge, England).
  257. Tuval, I., J. Schneider, O. Piro, and T. Tél, 2004, Europhys. Lett. 65, 633.
  258. Ullmann, K., and I. L. Caldas, 2000, Chaos, Solitons & Fractals 11, 2129.
  259. Vahala, K., 2003, Nature (London) 424, 839.
  260. Venegeroles, R., 2009, Phys. Rev. Lett. 102, 064101.
  261. Viana, R. L., E. C. D. Silva, T. Kroetz, I. L. Caldas, M. Roberto, and M. A. F. Sanjuán, 2011, Phil. Trans. R. Soc. A 369, 371.
  262. Vicentini, E., and V. B. Kokshenev, 2001, Physica (Amsterdam) 295A, 391.
  263. Vivaldi, F., G. Casati, and I. Guarneri, 1983, Phys. Rev. Lett. 51, 727.
  264. Waltner, D., M. Gutierrez, A. Goussev, and K. Richter, 2008, Phys. Rev. Lett. 101, 4.
  265. Weiss, M., L. Hufnagel, and R. Ketzmerick 2002, Phys. Rev. Lett. 89, 239401.
  266. Wesson, J., 1987, Tokamaks (Oxford University Press, Oxford, UK).
  267. Wiersig, J., and M. Hentschel, 2008, Phys. Rev. Lett. 100, 033901.
  268. Wiersig, J., and J. Main, 2008, Phys. Rev. E 77, 036205.
  269. Wimberger, S., A. Krug, and A. Buchleitner, 2002, Phys. Rev. Lett. 89, 263601.
  270. Wingen, A., M. Jakubowski, K. H. Spatschek, S. S. Abdullaev, K. H. Finken, M. Lehnen, and TEXTOR team, 2007, Phys. Plasmas 14, 042502.
  271. Wojtkowski, M., 1986, Commun. Math. Phys. 105, 391.
  272. Xeridat, O., C. Poli, O. Legrand, F. Mortessagne, and P. Sebbah, 2009, Phys. Rev. E 80, 035201.
  273. Yan, C., Q. J. Wang, L. Diehl, M. Hentschel, J. Wiersig, N. Yu, C. Pflügl, M. A. Belkin, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, 2009 Appl. Phys. Lett. 94, 251101.
  274. Zaslavsky, G. M., 2002, Phys. Rep. 371, 461.
  275. Zaslavsky, G. M., 2005, Hamiltonian Chaos and Fractional Dynamics (Oxford University Press, Oxford).
  276. Życzkowski, K., and E. M. Bollt, 1999, Physica (Amsterdam) 132D, 392.

Outline

Information

Rev. Mod. Phys. 85, 869– Published 29 May, 2013
  • Received 13 January 2012

© 2013 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation