517.9:519.6 Analysis of bifurcations in double-mode approximation for Kuramoto — Tsuzuki system

Malinetsky G. G. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Faller D. S. (Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes)

NONLINEAR DYNAMICS, DOUBLE-MODE SYSTEM, REACTION–DIFFUSION MODELS, BIFURCATIONS, SELF-SIMILARITY, “CASCADE OF CASCADES”, CRISIS OF ATTRACTOR, ERGODICITY, BISTABILITY.


doi: 10.18698/2309-3684-2014-3-111125


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.


[1] Nikolis G., Prigozhin I. Selforganization in nonequilibrium systems. Moscow, Mir Publ., 1979, 512 p.
[2] Haken H. Synergetics. Berlin, Heidelberg, N.Y., Springer-Velard, 1978, 406 p.
[3] Kurdyumov S.P. Sharpening Regimes: Evolution of Ideas. Malinetskiy G.G., ed. Moscow, Nauka Publ., 1999.
[4] Turing A. The chemical basis of morphogenesis. Phyl. Trans. Roy. Soc. London, 1952, vol. 237, рp. 37−72.
[5] Kuramoto Y., Tsuzuki T. On the formation of dissipative structures in reaction–diffusion systems. Prog. Theor. Phys., 1975, vol. 54, no. 3, рp. 687−699.
[6] Kashchenko S.A. On the quasinormal forms for parabolic equations with small diffusion. Doklady Akademii nauk SSSR — Reports of the USSR Academy of Sciences, 1998, no. 5, vol. 229, pp.1049–1052.
[7] Akhromeeva T.S., Kurdyumov S.P., Malinetsky G.G., Samarskiy A.A. Structures and chaos in nonlinear media. Moscow, Fizmatlit, 2007, 488 p.
[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. Part 1, 2. Mechanica, 1980, vol. 15, no. 1, pр. 9−30.
[9] Bokolishvili I.B., Malinetsky G.G. About scenarios of transition to chaos in one-dimensional maps with a sharp top. Moscow, IPM, 1987, .
[10] Feigenbaum M.J. Universal behavior in nonlinear systems. Los Alamos Sci., 1980, vol. 1, no. 1, рp. 4−27.
[11] Haken H. Synergetics. The hierarchy of instabilities in selforganizing
systems and devices. Moscow, Mir Publ., 1985, 419 p.


Malinetsky G., Faller D. Analysis of bifurcations in double-mode approximation for Kuramoto — Tsuzuki system. Маthematical Modeling and Coтputational Methods, 2014, №3 (3), pp. 111-125



Download article

Колличество скачиваний: 230