Geodesic
Analysis of bifurcations in double-mode approximation for Kuramoto — Tsuzuki system
Matematičeskoe modelirovanie i čislennye metody, no. 3 (2014), pp. 111-125 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The article discusses emergence of chaotic attractors in the system of three ordinary differential equations arising in the theory of reaction–diffusion models. We studied the dynamics of the corresponding one- and two-dimensional maps and Lyapunov exponents of such attractors. We have shown that chaos is emerging in an unconventional pattern with chaotic regimes emerging and disappearing repeatedly. We had already studied this unconventional pattern for one-dimensional maps with a sharp apex and a quadratic minimum. We applied numerical analysis to study characteristic properties of the system, such as bistability and hyperbolicity zones, crisis of chaotic attractors.
Keywords: Nonlinear dynamics, double-mode system, self-similarity, “cascade of cascades”, crisis of attractor, ergodicity, bistability.
Mots-clés : reaction–diffusion models, bifurcations 
@article{MMCM_2014_3_a6,
     author = {G. G. Malinetskii and D. S. Faller},
     title = {Analysis of bifurcations in double-mode approximation for {Kuramoto} {\textemdash} {Tsuzuki} system},
     journal = {Matemati\v{c}eskoe modelirovanie i \v{c}islennye metody},
     pages = {111--125},
     year = {2014},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MMCM_2014_3_a6/}
}
TY  - JOUR
AU  - G. G. Malinetskii
AU  - D. S. Faller
TI  - Analysis of bifurcations in double-mode approximation for Kuramoto — Tsuzuki system
JO  - Matematičeskoe modelirovanie i čislennye metody
PY  - 2014
SP  - 111
EP  - 125
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/MMCM_2014_3_a6/
LA  - ru
ID  - MMCM_2014_3_a6
ER  - 
%0 Journal Article
%A G. G. Malinetskii
%A D. S. Faller
%T Analysis of bifurcations in double-mode approximation for Kuramoto — Tsuzuki system
%J Matematičeskoe modelirovanie i čislennye metody
%D 2014
%P 111-125
%N 3
%U http://geodesic.mathdoc.fr/item/MMCM_2014_3_a6/
%G ru
%F MMCM_2014_3_a6
G. G. Malinetskii; D. S. Faller. Analysis of bifurcations in double-mode approximation for Kuramoto — Tsuzuki system. Matematičeskoe modelirovanie i čislennye metody, no. 3 (2014), pp. 111-125. http://geodesic.mathdoc.fr/item/MMCM_2014_3_a6/

[1] Nikolis G., Prigozhin I., Selforganization in nonequilibrium systems, Mir Publ., Moscow, 1979, 512 pp.

[2] Haken H., Synergetics, Springer-Velard, Berlin–Heidelberg, 1978, 406 pp.

[3] Kurdyumov S.P., Sharpening Regimes: Evolution of Ideas, Nauka Publ., Moscow, 1999, 302 pp.

[4] Turing A., “The chemical basis of morphogenesis”, Phil. Trans. Roy. Soc. London, 237 (1952), 37–72

[5] Kuramoto Y., Tsuzuki T., “On the formation of dissipative structures in reactondiffusion systems”, Prog. Theor. Phys., 54:3 (1975), 687–699

[6] Kashchenko S.A., “On the quasinormal forms for parabolic equations with small diffusion”, Reports of the USSR Academy of Sciences, 229:5 (1998), 1049–1052

[7] Akhromeeva T.S., Kurdyumov S.P., Malinetsky G.G., Samarskiy A.A., Structures and chaos in nonlinear media, Fizmatlit, Moscow, 2007, 488 pp.

[8] Benettin G., Galgani L., Giorgilli A., Stretcin J.M., “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them”, Mechanica, 15:1 (1980), 9–30

[9] Bokolishvili I.B., Malinetsky G.G., About scenarios of transition to chaos in one-dimensional maps with a sharp top, IPM, Moscow, 1987, 28 pp.

[10] Feigenbaum M.J., “Universal behavior in nonlinear systems”, Los Alamos Sci., 1:1 (1980), 4–27

[11] Haken H., Synergetics. The hierarchy of instabilities in selforganizing systems and devices., Mir Publ., Moscow, 1985, 419 pp.

[12] Malinetskii G.G., Faller D.S., “Stsenarii perekhoda k khaosu v dvukhmodovoi sisteme dlya sistem «reaktsiya–diffuziya»”, IPM im. M.V. Keldysha, Preprinty, Moskva, 2013, no. 67, 36 pp. library.keldysh.ru/preprint.asp?id=2013-67