16
Views
1
CrossRef citations to date
0
Altmetric
Original Articles

Off-Center Reflections: Caustics and Chaos

&
Pages 287-301 | Published online: 04 Apr 2012

REFERENCES

  • Arnold , V. I. 1961 . “Small denominators, I: Mapping the circle onto itself” . Izv. Akad. Nauk SSSR Ser. Mat. , 25 : 21 – 86 . [Arnold 1961], In Russian; translation in Amer. Math. Soc. Translations, Ser. 2, 46 (1965), 213–254
  • Arnold , V. I. 1983 . Geometrical methods in the theory of ordinary differential equations New York : Springer. . [Arnold 1983a], Grundlehren der Math. Wissenschaften 250, Translated from the Russian; 2nd edition, 1988; also published in German by Birkhäuser, 1987
  • Arnold , V. I. 1983 . “Remarks on the perturbation theory for problems of Mathieu type” . Uspekhi Mat. Nauk , 38 ( 4 ) : 189 – 203 . [Arnold 1983b], In Russian; translation in Russian Math. Surveys, 38:4 (1983), 215–233
  • Arnold , V. I. 1994 . Topological invariants of plane curves and caustics , Univ. Lecture Series 5 Providence , RI : Amer. Math. Soc. . [Arnold 1994]
  • Arnold , V. I. 1996 . “Topological problems of the theory of wave propagation” . Uspekhi Mat. Nauk , 51 ( 1 ) : 3 – 50 . [Arnold 1996], In Russian; translation in Russian Math. Surveys, 51:1 (1996), 215–233
  • Au , T. K. K. 1999 . “The dynamics of off-center reflection” Chinese University of HongKong. . [Au 1999], research report, See xxx.arxiv.org/math.DS/9908057
  • Bak , P. , Bohr , T. and Jensen , M. H. 1988 . “Circle maps, mode-locking and chaos”. ” . In Directions in chaos Edited by: Bai-lin , H. vol. 2 , 16 – 45 . River Edge , NJ : World Scientific. . [Bak et al. 1988]
  • Bruce , J. W. and Giblin , P. J. 1984 . “One-parameter families of caustics by reflexion in the plane” . Quart. J. Math. Oxford Ser. (2) , 35 : 243 – 251 . [Bruce and Giblin 1984]
  • Bruce , J. W. and Giblin , P. J. 1985 . “Two-parameter families of plane caustics by reflexion” . Proc. London Math. Soc. (3) , 50 ( 3 ) : 571 – 576 . [Bruce and Giblin 1985]
  • Bruce , J. W. , Giblin , P. J. and Gibson , C. G. 1981 . “On caustics of plane curves” . Amer. Math. Monthly , 88 ( 9 ) : 651 – 667 . [Bruce et al. 1981]
  • Bruce , J. W. , Giblin , P. J. and Gibson , C. G. 1982 . “Source genericity of caustics by reflexion in R3” . Proc. Roy. Soc. London Ser. A , 381 : 83 – 116 . [Bruce et al. 1982]
  • Devaney , R. L. 1989 . An introduction to chaotic dynamical systems, , 2nd ed. Redwood City , CA : Addison-Wesley. . [Devaney 1989]
  • Ding , E. J. and Hemmer , P. C. 1988 . “Winding numbers for the supercritical sine circle map” . Phys. D , 32 ( 1 ) : 153 – 160 . [Ding and Hemmer 1988]
  • Feudel , U. , Kurths , J. and Pikovsky , A. S. 1995 . “Strange non-chaotic attractor in a quasiperiodically forced circle map” . Phys. D , 88 ( 3–4 ) : 176 – 186 . [Feudel et al. 1995]
  • Giblin , P. J. and Kingston , J. G. 1986 . “Caustics by reflexion in the plane with stable triple crossings” . Quart. J. Math. Oxford Ser. (2) , 37 ( 145 ) : 17 – 25 . [Giblin and Kingston 1986]
  • Herman , M.-R. 1977 . “Mesure de Lebesgue et nombre de rotation”. ” . In Geometry and topology (Rio de Janeiro, 1976) 271 – 293 . Berlin : Springer. . [Herman 1977], Lecture Notes in Math. 597
  • Herman , M.-R. 1979 . “Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations” . Inst. Hautes Études Sci. Publ. Math. , 49 : 5 – 233 . [Herman 1979]
  • Jensen , M. H. , Bak , P. and Bohr , T. 1984 . “Transition to chaos by interaction of resonances in dissipative systems, I: Circle maps” . Phys. Rev. A (3) , 30 ( 4 ) : 1960 – 1969 . [Jensen et al. 1984]
  • Kaneka , K. 1984 . “Supercritical behavior of disordered orbits of a circle map” . Prog. Theoretical Phys. , 72 ( 6 ) : 1089 – 1103 . [Kaneka 1984]
  • Piña , E. 1986 . “Kneading theory of the circle map” . Phys. Rev. A (3) , 34 ( 1 ) : 574 – 581 . [Piña 1986]
  • Tabachnikov , S. L. 1990 . “Around four vertices” . Uspekhi Mat. Nauk , 45 ( 1 ) : 191 – 192 . [Tabachnikov 1990], In Russian; translation in Russian Math. Surveys, 45:1 (1990), 229–230
  • Tabachnikov , S. 1995 . “The four-vertex theorem revisited—two variations on the old theme” . Amer. Math. Monthly , 102 ( 10 ) : 912 – 916 . [Tabachnikov 1995]
  • Yau , S.-T. 1993 . “Open problems in geometry”. ” . In Partial differential equations on manifolds (Los Angeles, 1990) Edited by: Greene , R. and Yau , S. T. vol. 1 , 1 – 28 . Providence , RI : Amer. Math. Soc. . [Yau 1993], Proc. Symp. Pure Math. 54,'
  • Zheng , W. M. 1991 . “Symbolic dynamics for the circle map” . Internat. J. Modem Phys. B , 5 ( 3 ) : 481 – 495 . [Zheng 1991]

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.