ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Najdenov S. V., Yanovsky V. V. Geometrical nonlinear dynamics features of systems with elastic reflections. Izvestiya VUZ. Applied Nonlinear Dynamics, 2002, vol. 10, iss. 1, pp. 113-126. DOI: 10.18500/0869-6632-2002-10-1-113-126

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon and_2002-1-2_naydenov_etal.pdf
(downloads: 0)
Language: 
Russian
Heading: 
Deterministic Chaos
Article type: 
Article
UDC: 
514.8; 517.938; 530.182
DOI: 
10.18500/0869-6632-2002-10-1-113-126

Geometrical nonlinear dynamics features of systems with elastic reflections

Autors: 
Najdenov Sergej Vjacheslavovich, Institute of Monocrystals of NAS of Ukraine
Yanovsky Vladimir Vladimirovich, Institute of Monocrystals of NAS of Ukraine
Abstract: 

Within the framework оf the new geometrical-dynamic approach а special class оf dynamic systems - reversible mappings with projective involutions оп «symmetric» phase space - is linked with billiard systems. Basic geometrical features оf locally smooth billiard involutions - projectivity and piecewise discontinuity are explored and their role in making оf one оr another (regular and random) nonlinear billiard dynamics is indicated. Billiard involutions for simple algebraic curves are obtained and their common properties are set.

Key words: 
-
Reference: 
  1. Schuster HG. Deterministic Chaos. An Introduction. Weinheim: Physik-Verlag; 1984. 220 p.
  2. Lichtenberg AJ, Lieberman MA. Regular and Stochastic Motion. New York: Springer; 1983. 499 p. DOI: 10.1007/978-1-4757-4257-2.
  3. Sagdeev RZ, Zaslavsky GM. Nonlinear Physics: From the Pendulum to Turbulence and Chaos. Harwood Academic Publishers; 1988. 675 p.
  4. Sinai YG, editor. Modern Problems of Mathematics. Fundamental Directions. Dynamic Systems - 2. Vol. 2. Moscow: VINITI; 1985. 312 p. (in Russian).
  5. Birkhoff G. Dynamical Systems. American Mathematical Soc.; 1927. 305 p.
  6. Krylov NS. Works on the Substantiation of Statistical Physics. Moscow: Publishing House of the Academy of Sciences USSR; 1950. 208 p. (in Russian).
  7. Sinai YG. Towards the substantiation of the ergodic hypothesis for one dynamic system of statistical mechanics. Proc. Acad. Sci. USSR. 1963;153(6):1261-1264 (in Russian); Dynamical systems with elastic reflections. Russian Math. Surveys. 1970;25(2):137-189. DOI: 10.1070/RM1970v025n02ABEH003794.
  8. Bunimovich LA. On billiards close to dispersing. Math. USSR-Sb. 1974;23(1):45-67. DOI: 10.1070/SM1974v023n01ABEH001713.
  9. Lazutkin VF. Convex Billiards and Eigenfunctions of the Laplace Operator. Leningrad: Leningrad State University Publishing; 1981. 196 p. (in Russian)
  10. Arnold VI. Mathematical Methods of Classical Mechanics. New York: Springer; 1989. 520 p. DOI: 10.1007/978-1-4757-2063-1.
  11. Аlt Н, Gräf H-D, Hofferbert R, Rangacharyulu C, Rehfeld H, Richter A, Schardt P, Wirzba A. Chaotic dynamics in а three-dimensional superconductiving microwave billiard. Phys. Rev. E. 1996;54(3):2303-2312. DOI: 10.1103/PhysRevE.54.2303.
  12. Nocel JU, Stone AD, Chen G, Grossman HL, Chang RK. Directional emission from asymmetric resonant cavities. Optics Letters. 1996;21(19):1609-1611. DOI: 10.1364/ol.21.001609.
  13. Naydenov SV‚ Yanovsky VV. Stochastical theory оf light collection. I. Detectors and billiards. Functional Materials. 2000;7(4(2)):743.
  14. Guhr T, Muller-Groeling А, Weidenmuller HA. Random-matrix theories in quantum physics: common concepts. Phys. Rep. 1998;299(4-6):189-425. DOI: 10.1016/S0370-1573(97)00088-4.
  15. Bunimovich L, Casati G, Guarneri I. Chaotic Focusing Billiards in Higher Dimensions. Phys. Rev. Lett. 1996;77(14):2941-2944. DOI: 10.1103/PhysRevLett.77.2941.
  16. Tabachnikov SL. Dual billiards. Russian Math. Surveys. 1993;48(6):81-109. DOI: 10.1070/RM1993v048n06ABEH001092.
  17. Loskutov AY, Ryabov AB, Akinshin LG. Mechanism of Fermi acceleration in dispersing billiards with time-dependent. J. Exp. Theor. Phys. 1999;89(5):966-974. DOI: 10.1134/1.558939.
  18. Arnold VI, Sevryuk MB. Nonlinear Phenomena in Plasma Physics and Hydrodynamics. Moscow: Mir; 1986. P. 31-64.
  19. Roberts JAG,Qwispel GRW. Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems. Phys. Rep. 1992;216(2-3):63-77. DOI: 10.1016/0370-1573(92)90163-T.
  20. Naydenov SV, Yanovskii VV. Geometric-dynamic approach to billiard systems. I-II. Theoret. and Math. Phys. I. 2001;127(1):500-512. DOI: 10.1023/A:1010316025791. II. 2001;129(1):1408-1420. DOI: 10.1023/A:1012475713108.
  21. Pogorelov AV. Differential Geometry. Groningen: Noordhoff; 1959. 171 p.
  22. Dubrovin BA, Novikov SP, Fomenko AT. Modern Geometry. Methods and Applications. Moscow: Nauka; 1986. 760 p. (in Russian).
  23. Efimov HV. Higher Geometry. Moscow: FM; 1961. 580 p. (in Russian).
Received: 
04.09.2000
Accepted: 
13.03.2002
Available online: 
13.12.2023
Published: 
31.07.2002
  • 161 reads