Chaotic evolution of dynamical systems is caused by the divergence of nearby orbits, i. e., by the intrinsic instability of the dynamics. The best way to see how the divergence of orbits may occur is to consider the orbits as the rays of light, i. e., within the framework of the geometrical optics. We discuss the basic mechanisms of chaos and demonstrate how the discovery of these mechanisms allowed one to enrich the geometrical optics by some new fundamental ideas and notions.
Similar content being viewed by others
References
J. H. Poincaré, Les Methodes Nouvelles de la Mechnique Celeste, Gauthier–Villars, Paris (1892–1899).
J. Hadamard, J. Math. Pures Appl., 4, 27–74 (1898).
G. A. Hedlund, Ann. Math., 35, 787–808 (1934). https://doi.org/10.2307/1968495
E. Hopf, Ber. Verh. Sachs. Akad. Wiss. Leipzig, 91, 261–304 (1939).
E. Hopf, AMS Bull., 77, 863–877 (1971). https://doi.org/10.1090/S0002-9904-1971-12799-4
E. Hopf, Math. Ann., 117, 590–608 (1940). https://doi.org/10.1007/BF01450032
N. S. Krylov, Works on the Foundations of Statistical Physics, Princeton Univ. Press, Princeton, N. J. (1979).
Ya. G. Sinai, Russian Math. Surveys, 25, No. 2, 137–197 (1970). https://doi.org/10.1070/RM1970v025n02ABEH003794
Ya. G. Sinai, in: N. S. Krylov, Works on the Foundations of Statistical Physics, Princeton Univ. Press (2014), pp. 239–281. https://doi.org/10.1515/9781400854745.239
L. A. Bunimovich, Funct. Anal. Appl., 8, 254–255 (1974). https://doi.org/10.1007/BF01075700
L. A. Bunimovich, Sbornik Math., 94, 45–67 (1974). https://doi.org/10.1070/SM1974V023N01ABEH001713
L. A. Bunimovich, Commun. Math. Phys., 65, 295–312 (1979). https://doi.org/10.1007/BF01197884
K. Burns and M. Gerber, Ergod. Theory Dyn. Syst., 9, 27–45 (1989). https://doi.org/10.1017/S0143385700004806
V. Donnay, Ergod. Theory Dyn. Syst., 8, 531–553 (1988). https://doi.org/10.1017/S0143385700004685
V. Donnay, Lect. Notes Math., 1342, 112–153 (1988). https://doi.org/10.1007/BFb0082827
L. A. Bunimovich, Izv. Vyssh. Uchebn. Zaved., Radiofiz ., 28, No. 12, 1601–1602 (1985).
R. Markarian, Commun. Math. Phys., 118, 87–97 (1988). https://doi.org/10.1007/BF01218478
M. Wojtkowski, Commun. Math. Phys., 105, 391–414 (1986). https://doi.org/10.1007/BF01205934
L. A. Bunimovich, Physica D, 33, 58–64 (1988). https://doi.org/10.1016/S0167-2789(98)90009-4
L. A. Bunimovich, Lect. Notes Math., 1514, 62–82 (1992). https://doi.org/10.1007/BFb0097528
L. A. Bunimovich, Commun. Math. Phys., 130, 599–621 (1990). https://doi.org/10.1007/BF02096936
G. Del Magno and R. Markarian, Commun. Math. Phys., 350, 917–955 (2017). https://doi.org/10.1007/s00220-017-2828-7
V. Donnay, Commun. Math. Phys., 141, 225–257 (1991). https://doi.org/10.1007/BF02101504
L. A. Bunimovich and A. Grigo, Commun. Math. Phys., 293, 127–143 (2010). https://doi.org/10.1007/s00220-009-0927-9
L. A. Bunimovich, H.-K. Zhang, and P. Zhang, Commun. Math. Phys., 341, 781–803 (2016). https://doi.org/10.1007/s00220-015-2539-x
L. A. Bunimovich and J. Rehacek, Commun. Math. Phys., 197, 277–301 (1998). https://doi.org/10.1007/s002200050451
L. A. Bunimovich and J. Rehacek, Ann. Inst. H. Poincar’e, 68, 421–448 (1998).
L. A. Bunimovich, J. Stat. Phys., 101, 373–384 (2000). https://doi.org/10.1023/A:1026405920274
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 64, No. 10, pp. 769–776, October 2021. Russian DOI: https://doi.org/10.52452/00213462_2021_64_10_769
Rights and permissions
About this article
Cite this article
Bunimovich, L.A. Chaos and Geometrical Optics. Radiophys Quantum El 64, 693–699 (2022). https://doi.org/10.1007/s11141-022-10171-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11141-022-10171-6