Prevalence of marginally unstable periodic orbits in chaotic billiards

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 77, Issue 1, 2008, Pages

  Altmann, E G ^a   Friedrich, T ^b   Motter, A E ^c   Kantz, H ^a   Richter, A ^b  

^a MAX PLANCK INSTITUT FÜR PHYSIK KOMPLEXER SYSTEME   (Germany)

^b DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

^c NORTHWESTERN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

MICROWAVES; ORBITS; PERTURBATION TECHNIQUES; QUANTUM THEORY;

CHAOTIC BILLIARDS; PERIODIC ORBITS; QUANTUM REGIME;

HIGH ENERGY PHYSICS;

EID: 38749095501     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.77.016205     Document Type: Article

References (34)

1. L. A. Bunimovich and Y. Sinai, Commun. Math. Phys. CMPHAY 0010-3616 10.1007/BF02046760 78, 479 (1980).
2. L. A. Bunimovich, Commun. Math. Phys. CMPHAY 0010-3616 10.1007/BF01197884 65, 295 (1979).
3. M. Brack and R. K. Bhaduri, Semiclassical Physics (Addison-Wesley, Reading, Massachusetts, 1997).
4. W. Fang, A. Yamilov, and H. Cao, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.72.023815 72, 023815 (2005).
5. A. Richter, in Emerging Applications of Number Theory, edited by, D. A. Hejhal, J. Friedman, M. C. Gutzwiller, and, A. M. Odlyzko, The IMA Volumes in Mathematics and its Applications Vol. 109 (Springer, New York, 1999), p. 479.
6. H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, 1999).
7. F. Vivaldi, G. Casati, and I. Guarneri, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.51.727 51, 727 (1983);
8. P. Gaspard and J. R. Dorfman, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.52.3525 52, 3525 (1995).
9. A. Bäcker, R. Schubert, and P. Stifter, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/30/19/017 30, 6783 (1997).
10. H. Primack and U. Smilansky, J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/27/13/018 27, 4439 (1994);
11. E. E. Narimanov and A. D. Stone, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.80.49 80, 49 (1998).
12. N. Saitô, H. Hirooka, J. Ford, F. Vivaldi, and G. H. Walker, Physica D PDNPDT 0167-2789 10.1016/0167-2789(82)90022-7 5, 273 (1982).
13. L. A. Bunimovich, Chaos CHAOEH 1054-1500 10.1063/1.1418763 11, 802 (2001).
14. Our proof of the existence of infinitely many MUPOs holds when the scatterers (1) induce chaotic dynamics and (2) have a finite number of points at the critical caustic [circle of radius a in Fig. 1 in the case of circularlike billiards].
15. E. G. Altmann, A. E. Motter, and H. Kantz, Chaos CHAOEH 1054-1500 10.1063/1.1979211 15, 033105 (2005).
16. E. G. Altmann, A. E. Motter, and H. Kantz, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.73.026207 73, 026207 (2006).
17. E. G. Altmann, Ph.D. thesis, Wuppertal University, Germany (unpublished).
18. The width in where such star polygons may exist is fixed while the distance between two adjacent vertices is 2πq, i.e., goes to zero with increasing q [see Fig. 2].
19. A. Ya. Khinchin, Continued Fractions (Dover, New York, 1992).
20. C. Dembowski, H.-D. Gräf, A. Heine, R. Hofferbert, H. Rehfeld, and A. Richter, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.84.867 84, 867 (2000);
21. R. Hofferbert, H. Alt, C. Dembowski, H.-D. Gräf, H. L. Harney, A. Heine, H. Rehfeld, and A. Richter, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.71.046201 71, 046201 (2005).
22. B. Dietz, T. Friedrich, M. Miski-Oglu, A. Richter, T. H. Seligman, and K. Zapfe, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.74.056207 74, 056207 (2006);
23. B. Dietz, T. Friedrich, M. Miski-Oglu, A. Richter, T. H. Seligman, and K. Zapfe, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.75.035203 75, 035203 (R) (2007).
24. F. von Oppen and E. K. Riedel, Phys. Rev. B PRBMDO 0163-1829 10.1103/PhysRevB.48.9170 48, 9170 (1993);
25. D. S. Saraga and T. S. Monteiro, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.81.5796 81, 5796 (1998).
26. O. Bohigas, D. Bosé, R. Egydio de Carvalho, and V. Marvulle, Nucl. Phys. A NUPABL 0375-9474 10.1016/0375-9474(93)90092-C 560, 197 (1993);
27. S. D. Frischat and E. Doron, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.57.1421 57, 1421 (1998);
28. R. W. Robinett, Am. J. Phys. AJPIAS 0002-9505 10.1119/1.19193 67, 67 (1999).
29. M. Hentschel and K. Richter, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.66.056207 66, 056207 (2002).
30. G. Gouesbet, S. Meunier-Guttin-Cluzel, and G. Grehan, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.65.016212 65, 016212 (2001).
31. A. Mouchet, C. Eltschka, and P. Schlagheck, Phys. Rev. E PLEEE8 1063-651X 10.1103/PhysRevE.74.026211 74, 026211 (2006).
32. A. H. Barnett and T. Betcke, e-print arXiv:nlin/0611059.
33. G. Vidmar, H. -J. Stöckmann, M. Robnik, U. Kuhl, R. Höhmann, and S. Grossmann, J. Phys. A JPHAC5 0305-4470 40, 13883 (2007);
34. A. Bäcker, R. Ketzmerick, S. Löck, M. Robnik, G. Vidmar, R. Höhmann, U. Kuhl, and H. -J. Stöckmann (to be published).