Abstract
We study infinite-dimensional systems of ordinary differential equations having applications in some popular and important physical problems. The appearance of infinite-dimensional space–time chaos is considered, namely, the bifurcations and critical phenomena that occur in the phase space of the systems and explain some physical problems are described.
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REFERENCES
Ya. G. Sinai, Theory of Phase Transitions: Rigorous Results [in Russian], Nauka, Moscow (1980); English transl., Pergamon, Oxford (1982).
P. Bak, C. Tang, and K. Wiesenfeld, Phys. R ev. Lett., 59, 381 (1987); Phys. R ev. A, 38, 364 (1988); P. Bak, How Nature Works, Springer, Berlin (1996); D. Dhar, Phys. Rev. Lett., 64, 1613 (1990); D. Dhar and S. N. Majumdar, J. Phys. A, 23, 4333 (1990); S. N. Majumdar and D. Dhar, Physica A, 185, 129 (1992); D. Dhar and R. Ramaswamy, Phys. R ev. Lett., 63, 1659 (1989); A. D´ýaz-Guilera, Europhys. Lett., 26, 177 (1994); Phys. Rev. A, 45, 8551 (1992); H. J. Jensen, Self-Organized Criticality: Emergent Complex Behavior in Physical and Biological Systems, Cambridge Univ. Press, Cambridge (1998); L. Pietronero, P. Tartaglia, and Y. C. Zhang, Physica A, 173, 22 (1991); R. Ca.ero, V. Loreto, L. Pietronero, A. Vespignani, and S. Zapperi, Europhys. Lett., 129, 111 (1995); L. Pietronero, A. Vespignani, and S. Zapperi, Phys. Rev. Lett., 72, 1690 (1994); A. Vespignani, S. Zapperi, and L. Pietronero, Phys. Rev. E, 51, 1711 (1995); H. Y. Zhang, Phys. R ev. Lett., 63, 470 (1988); S. N. Coppersmith and D. S. Fisher, “Threshold behavior of a driven incommensurate harmonic chain,” Preprint, Princeton Univ., Princeton, N. J. (1988); R. Burridge and L. Knopo., Bull. Seismol. Soc. Am., 57, 341 (1967).
V. I. Arnol'd (ed.), Dynamic Systems 5 [in Russian] (Advances in Science and Technology, Series Contemporary Problems in Mathematics: Fundamental Directions, Vol. 5, R. V. Gamkrelidze, ser. ed.), VINITI, Moscow (1986); English transl.: Dynamical Systems V: Bifurcation Theory and Catastrophe Theory (Ency. Math. Sci., Vol. 5), Springer, Berlin (1994).
L. D. Pustyl'nikov, Theor. Math. Phys., 92, 754 (1992).
L. D. Pustyl'nikov, “Critical phenomena and bifurcation of solutions of infinite-dimensional systems of ordinary diferential equations appearing in some physical problems [in Russian],” Preprint No. 43, Inst. Appl. Math., Russ. Acad. Sci., Moscow (2001); “Space-time chaos and the theory of infinite-dimensional systems of ordinary diferential equations appearing in some physical problems [in Russian],” Preprint No. 44, Inst. Appl. Math., Russ. Acad. Sci., Moscow (2001).
V. G. Gelfreich, V. F. Lazutkin, and N. V. Svanidze, “Refined formula to separatrix splitting for the standard map,” Warwick Preprints No. 59, Warwick Univ., Warwick, G. B. (1992).
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Pustyl'nikov, L.D. Space–Time Chaos, Critical Phenomena, and Bifurcations of Solutions of Infinite-Dimensional Systems of Ordinary Differential Equations. Theoretical and Mathematical Physics 133, 1348–1362 (2002). https://doi.org/10.1023/A:1020641913332
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DOI: https://doi.org/10.1023/A:1020641913332