Geodesic
Dimensional Characteristics of Diffusion Chaos
Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 1, pp. 30-51.

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The phenomenon of multimode diffusion chaos is considered. For a number of examples it is shown that the Lyapunov dimension of the attractor of a distributed dynamical system increases as the diffusion coefficient tends to 0.
Mots-clés : diffusion chaos, bifurcation
Keywords: attractor, Lyapunov dimension, Ginzburg–Landau equation, Landau–Sell scenario. 
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S. D. Glyzin. Dimensional Characteristics of Diffusion Chaos. Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 1, pp. 30-51. http://geodesic.mathdoc.fr/item/MAIS_2013_20_1_a2/

[1] G. Nicolis, I. Prigogine, Self-Organization in Non-Equilibrium Systems, Wiley, 1977 | MR | Zbl

[2] T. S. Ahromeeva, S. P. Kurdyumov, G. G. Malinetskiy, A. A. Samarskiy, Struktury i khaos v nelineynykh sredakh, Fizmatlit, M., 2007 (in Russian) | Zbl

[3] E. F. Mishchenko, V. A. Sadovnichiy, A. Yu. Kolesov, N. Kh. Rozov, Avtovolnovye protsessy v nelineynykh sredakh s diffuziey, Fizmatlit, M., 2005 (in Russian)

[4] E. N. Lorenz, “Deterministic nonperiodic flow”, J. Atmos. Sci., 20 (1963), 130–141 | 2.0.CO;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[5] D. Ruelle, F. Takens, “On the nature of turbulence”, Comm. Math. Phys., 20 (1971), 167–192 | DOI | MR | Zbl

[6] Y. Kuramoto, “Diffusion-Induced Chaos in Reaction Systems”, Prog. Theor. Phys. Supplement, 1978, no. 64, 346–367 | DOI

[7] S. D. Glyzin, “Dynamic properties of the simplest finite-difference approximations of the “reaction-diffusion” boundary value problem”, Differential Equations, 33:6 (1997), 808–-814 | MR | Zbl

[8] A. Yu. Kolesov, “Description of Phase Instability of a System of Harmonic Oscillators Weakly Coupled by Diffusion”, Dokl. Akad. Nauk SSSR, 300 (1988), 831–835 | MR

[9] S. D. Glyzin, “Numerical Justification of the Landau-Kolesov Conjecture on the Nature of Turbulence”, Mathematical models in biology and medicine, 3 (1989), 31–36 (in Russian) | MR | Zbl

[10] S. D. Glyzin, “Difference approximations of “reaction – diffusion” equation on a segment”, Modeling and Analysis of Information Systems, 16:3 (2009), 96–116 (in Russian)

[11] A. V. Gaponov-Grekhov, M. I. Rabinovich, “Autostructures: chaotic dynamics of ensembles”, Nonlinear Waves. Structure and Bifurcations, Nauka, Moskva, 1987, 7–44 (in Russian) | MR

[12] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Finite-dimensional models of diffusion chaos”, Computational Mathematics and Mathematical Physics, 50:5 (2010), 816–830 | DOI | MR | Zbl

[13] Yu. S. Kolesov, Adequacy of Ecological Equations, Available from VINITI, No 1901-85, Yaroslavl, 1985 (in Russian)

[14] A. Yu. Kolesov; N. Kh. Rozov, “On the definition of chaos”, Russian Math. Surveys, 64:4 (2009), 701–744 | DOI | DOI | MR | Zbl

[15] P. Frederickson, J. Kaplan, J. Yorke, “The Lyapunov dimension of strange attractors”, J. Different. Equat., 49:2 (1983), 185–207 | DOI | MR | Zbl

[16] A. Yu. Kolesov, N. Kh. Rozov, V. A. Sadovnichii, “Mathematical aspects of the theory of development of turbulence in the sense of Landau”, Russian Math. Surveys, 63:2 (2008), 246–251 | DOI | DOI | MR | Zbl

[17] V. I. Arnold, B. A. Khesin, Topological Methods in Hydrodynamics, Springer, 1999, 391 pp. | MR

[18] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “On the Realizability of the Landau Scenario for the Development of Turbulence”, Theoretical and Mathematical Physics, 427 (2009), 292–311 | DOI | MR | Zbl

[19] J. R. Dormand, P. J. Prince, “A Family of Embedded Runge–Kutta Formulae”, J. Comp. Appl. Math., 6 (1980), 19–26 | DOI | MR | Zbl

[20] G. Benettin, L. Galgani, J. M. Strelcyn, “Kolmogorov entropy and numerical experiments”, Phys. Rev. A, 14 (1976), 2338–2345 | DOI

[21] A. Wolf, J. B. Swift, H. L. Swinney, J. A. Vastano, “Determining Lyapunov exponents from a time series”, Physica D, 16 (1985), 285–317 | DOI | MR | Zbl

[22] D. S. Glyzin, S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “The Dynamic Renormalization Method for Finding the Maximum Lyapunov Exponent of a Chaotic Attractor”, Differential Equations, 41:2 (2005), 284–289 | DOI | MR | Zbl

[23] D. S. Glyzin, S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Relaxation oscillations and diffusion chaos in the Belousov reaction”, Computational Mathematics and Mathematical Physics, 51:8 (2011), 1307–1324 | DOI | MR | Zbl

[24] A. Yu. Kolesov, Yu. S. Kolesov, V. V. Mayorov, “The Belousov Reaction: a Mathematical Model and Experimental Facts”, Dynamics of Biological Populations, GGU, Gorki, 1987, 43–51 (in Russian)

[25] J. Milnor, “On the concept of attractor”, Commun. Math. Phys., 99:2 (1985), 177–196 | DOI | MR