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

For citation:

Abarbanel G. ., Rabinovich M. I., Cimring L. S. "Crystals" and "quasicrystals" in nonequilibrium media. Izvestiya VUZ. Applied Nonlinear Dynamics, 1994, vol. 2, iss. 1, pp. 5-23. DOI: 10.18500/0869-6632-1994-2-1-5-23

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Language: 
Russian
Heading: 
Review of Actual Problems of Nonlinear Dynamics
Article type: 
Article
UDC: 
530.182
DOI: 
10.18500/0869-6632-1994-2-1-5-23

"Crystals" and "quasicrystals" in nonequilibrium media

Autors: 
Abarbanel Genri , Lobachevsky State University of Nizhny Novgorod
Rabinovich Mihail Izrailevich, University of California, San Diego
Cimring Lev Shulimovich, Lobachevsky State University of Nizhny Novgorod
Abstract: 

The material offered to the reader represents two chapters of review character, revised specially for this issue, of the forthcoming book of the authors “Theory of Dynamical Disorder”. The formation of regular crystalline and quasicrystalline structures in various nonequilibrium media, in particular, quasi-dimensional hydrodynamic flows, is discussed from the standpoint of nonlinear dynamics.

Key words: 
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Reference: 
  1. Van Dyke М. An Album of Fluid Motion. Stanford: Parabolic Press; 1982. 176 p.
  2. Bodenschats E, de Bruyn JR, Ahlers G, Cannell D. Transitions between patterns in thermal convection. Phys. Rev. Lett. 1991; 67(22):3078-3081. DOI: 10.1103/PhysRevLett.67.3078.
  3. Blumel R, Davidson IH, Reinhardt WP, Lin H, Sharnoff M. Quasilinear ridge structures in water surface waves. Phys. Rev. A. 1992;45(4):2641-2644. DOI: 10.1103/physreva.45.2641.
  4. O’Connor P, Gehlen J, Heller EJ. Properties of random superpositions of plane waves. Phys. Rev. Lett. 1987;58(13):1296-1299. DOI: 10.1103/PhysRevLett.58.1296.
  5. Berry MV. In: Iooss G, Helleman R, Stora В. Chaotic Behaviour of Deterministic Systems. Amsterdam: Elsiever; 1983. P. 171.
  6. Christiansen B, Alstrom Preben, Levinsen MT. Ordered capillary-wave states: Quasicrystals, hexagons, and radial waves. Phys. Rev. Lett. 1992;68(14):2157-2160. DOI: 10.1103/PhysRevLett.68.2157.
  7. Edwards WS, Fauve S. Parametrically excited quasicrystalline surface waves. Phys. Rev. Е. 1993;47(2):R788-R791. DOI: 10.1103/PhysRevE.47.R788.
  8. Cross MC, Hohenberg РC. Pattern formation outside of equilibrium. Rev. Mod. Phys. 1991;65(3):851-1122. DOI: 10.1103/RevModPhys.65.851.
  9. Rabinovich MI, Fabrikant AL, Tsimring LSh. Finite-dimensional spatial disorder. Phys. Usp. 1992;35(8):629–649. DOI: 10.1070/PU1992v035n08ABEH002253.
  10. Gaponov-Grekhov AV, Lomov AS, Osipov GV, Rabinovich МI. In: Gaponov-Grekhov AV, Rabinovich MI, editors. Nonlinear Waves. М.: Nauka; 1987. P. 61.
  11. Ouyang Q, Swinney H. Transition to chemical turbulence. Chaos. 1991;1:411-420. DOI: 10.1063/1.165851.
  12. Bestehorn М. Pattern Formation in Fluids — Variational and Non-Variational Models. In: Friedrich R, Wunderlin A, editors. Evolution of Dynamical Structures in Complex Systems. Springer Proceedings in Physics. Vol. 69. Berlin: Springer; 1992. P. 165-196.
  13. Le Gal P, Pocheau A, Croquette VV. Square versus roll pattern at convective threshold. Phys. Rev. Lett. 1985;54(23):2501-2504. DOI: 10.1103/PhysRevLett.54.2501.
  14. Penrose R. The role of aesthetics in pure and applird mathematical research. Bull. Inst. Math. Appl. 1974;10:266-271. Penrose R. Pentaplexity A Class of Non-Periodic Tilings of the Plane. Math. Intelligencer. 1979;2:32-37. DOI: 10.1007/BF03024384.
  15. Shechtman D, Blech I, Gratias D, Cahn JW. Metallic Phase with Long-Range Orientational Order and No Translational Symmetry. Phys. Rev. Lett. 1984;53(20):1951-1953. DOI: 10.1103/PhysRevLett.53.1951.
  16. Zaslavskii GМ, Sagdeev RZ, Usikov DА, Chernikov АS. Weak Chaos and Quasi-Periodic Structures. М.: Nauka; 1991. 234 p.
  17. Manneville P. Dissipative Structures and Weak Turbulence. Boston: Academic Press; 1990. 448 p.
  18. Korepin VЕ. In: Notes of LOMI scientific seminars. Vol. 155. Leningrad: Nauka; 1986. P. 116.
  19. Malomed BА, Nepomyashchii АА, Tribelskii МI. Two-dimensional quasiperiodic structures in nonequilibrium systems. J. Exp. Theor. Phys. 1989;69(3):388-396.
  20. Landau LD. On the problem of turbulence. Soviet Phys. Doklady. 1944;44:311-314.
  21. Hopf Е. A mathematical example displaying features of turbulence. Comm. Pure Appl.Math. 1948;1(4):303-322.
  22. Lichtenberg АJ, Lieberman MA. Regular and Stochastic Motion. N.Y.: Springer; 1983. 499 p. DOI: 10.1007/978-1-4757-4257-2.
  23. Newell A, Pomo Y. Turbulent crystal. Preprint. Tucson; 1993.
  24. Kai S, Chizumi N, Kohno М. Spatial and temporal behavior of pattern formations and defect motions in the electrohydrodynamic instability of nematic liquid crystals. Phys. Rev. A. 1989;40(11):6554-6572. DOI: 10.1103/physreva.40.6554.
  25. Eckman JP, Procaccia I. The generation of spatio-temporal chaos in large aspect ratio hydrodynamics. Nonlinearity. 1991;4(2):567-581. DOI: 10.1088/0951-7715/4/2/016.
Received: 
17.12.1993
Accepted: 
29.04.1994
Published: 
27.06.1994
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