Minimal chaos, stochastic webs, and structures of quasicrystal symmetry

The relationship between the problem of the symmetry of a plane tiling and the properties of nonintegrable dynamic systems is reviewed. The formation of stochastic layers and a stochastic web in the motion of linear and nonlinear oscillators subjected to a perturbation is discussed in detail. Emphasis is placed on research on the symmetry properties of a stochastic web with a fractal structure of a quasicrystal type. Structures with a quasicrystal symmetry form as a result of an interaction of two types of symmetries: translational and rotational. Various characteristics of structures with a quasicrystal symmetry are discussed: the distributions of stable and unstable points, the state density, and the Fourier spectrum. Quasicrystal structures in solid state physics, hydrodynamics, botany, and ornamental art are discussed.

PACS: 61.44.Br, 05.45.Df (all)
DOI: 10.1070/PU1988v031n10ABEH005632
URL: https://ufn.ru/en/articles/1988/10/a/
Citation: Zaslavskii G M, Sagdeev R Z, Usikov D A, Chernikov A A "Minimal chaos, stochastic webs, and structures of quasicrystal symmetry" Sov. Phys. Usp. 31 887–915 (1988)

BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Заславский Г М, Сагдеев Р З, Усиков Д А, Черников А А «Минимальный хаос, стохастическая паутина и структуры с симметрией типа "квазикристалл"» УФН 156 193–251 (1988); DOI: 10.3367/UFNr.0156.198810a.0193

References (83) Cited by (60) Similar articles (20) ↓

M.V. Kalashnik, M.V. Kurgansky, O.G. Chkhetiani “Baroclinic instability in geophysical fluid dynamics” Phys. Usp. 65 1039–1070 (2022)
K.V. Koshel, S.V. Prants “Chaotic advection in the ocean” Phys. Usp. 49 1151–1178 (2006)
O.G. Onishchenko, O.A. Pokhotelov et al “Structure and dynamics of concentrated mesoscale vortices in planetary atmospheres” Phys. Usp. 63 683–697 (2020)
B.M. Smirnov “Electrical cycle in the Earth’s atmosphere” Phys. Usp. 57 1041–1062 (2014)
A.N. Vulfson, O.O. Borodin “The system of convective thermals as a generalized ensemble of Brownian particles” Phys. Usp. 59 109–120 (2016)
F.V. Dolzhanskii, V.A. Krymov, D.Yu. Manin “Stability and vortex structures of quasi-two-dimensional shear flows” Sov. Phys. Usp. 33 (7) 495–520 (1990)
O.G. Onishchenko, O.A. Pokhotelov, N.M. Astaf’eva “Generation of large-scale eddies and zonal winds in planetary atmospheres” Phys. Usp. 51 577–589 (2008)
Yu.A. Stepanyants, A.L. Fabrikant “Propagation of waves in hydrodynamic shear flows” Sov. Phys. Usp. 32 783–805 (1989)
G.M. Zaslavskii, B.V. Chirikov “Stochastic instability of non-linear oscillations” Sov. Phys. Usp. 14 549–568 (1972)
V.M. Fedorov “Problems of parameterization of the radiation block in physical and mathematical climate models and the possibility of their solution” Phys. Usp. 66 914–930 (2023)
Yu.Kh. Vekilov, M.A. Chernikov “Quasicrystals” Phys. Usp. 53 537–560 (2010)
L.Kh. Ingel, M.V. Kalashnik “Nontrivial features in the hydrodynamics of seawater and other stratified solutions” Phys. Usp. 55 356–381 (2012)
V.S. Anishchenko, T.E. Vadivasova et al “Statistical properties of dynamical chaos” Phys. Usp. 48 151–166 (2005)
S.S. Abdullaev, G.M. Zaslavskii “Classical nonlinear dynamics and chaos of rays in problems of wave propagation in inhomogeneous media” Sov. Phys. Usp. 34 (8) 645–664 (1991)
A. Loskutov “Fascination of chaos” Phys. Usp. 53 1257–1280 (2010)
P.V. Elyutin “The quantum chaos problem” Sov. Phys. Usp. 31 597–622 (1988)
V.I. Klyatskin “Integral characteristics: a key to understanding structure formation in stochastic dynamic systems” Phys. Usp. 54 441–464 (2011)
A.A. Chernyshov, K.V. Karelsky, A.S. Petrosyan “Subgrid-scale modeling for the study of compressible magnetohydrodynamic turbulence in space plasmas” Phys. Usp. 57 421–452 (2014)
Yu.L. Klimontovich “Nonlinear Brownian motion” Phys. Usp. 37 737–766 (1994)
A.L. Virovlyansky, D.V. Makarov, S.V. Prants “Ray and wave chaos in underwater acoustic waveguides” Phys. Usp. 55 18–46 (2012)

The list is formed automatically.