Interactions between normal modes in nonlinear dynamical systems with discrete symmetry. Exact results

Physica D: Nonlinear Phenomena

Volumn 117, Issue 1-4, 1998, Pages 43-76

  Chechin, G M ^a   Sakhnenko, V P ^a  

^a SOUTHERN FEDERAL UNIVERSITY   (Russian Federation)

Author keywords

Discrete symmetry; Hamiltonian systems; Nonlinear dynamics; Normal mode interactions

Indexed keywords

EID: 0002831929     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(98)80012-2     Document Type: Article

References (41)

1. L.V. Ovsjannikov, Group Analysis in Differential Equations, in: A.F. Ames (Ed.), Academic Press, New York, 1982.
2. P.J. Olver, Applications of the Lie Groups to Differential Equations, Springer, New York, 1986.
3. E. Fermi, J.R. Pasta, S.M. Ulam, Collected Works of E. Fermi, vol. 2, University Chicago Press, Chicago, 1965, p. 978.
4. V.L. Gurevich, A.K. Tagantsev, Advances in Physics 40 (1991) 719.
5. Y. Yan, K.A. Nelson, J. Chem. Phys. 87 (1987) 6240.
6. A.D. Bruce, R.A. Cowley, Structural Phase Transitions, Taylor and Francis, London, 1981.
7. G.M. Chechin, T.I. Ivanova, V.P. Sakhnenko, Phys. Status Solidi (b) 152 (1989) 431.
8. D.A. Sadovskii, N.G. Fulton, J.R. Henderson, J. Tennyson, B.I. Zhilinskii, J. Chem. Phys. 99 (2) (1993) 906-918.
9. D.A. Sadovskii, B.I. Zhilinskii, Phys. Rev. A 47 (1993) 2653-2671.
10. R. Zurmuhlen, J. Petzelt, S. Kamba, V.V. Voitsekhovskii, E. Colla, N. Setter, Appl. Phys. 77 (1995) 5341;
11. Appl. Phys. 77 (1995) 5351.
12. H. Orihara, Y. Ishibashi, Journal of the Physical Society of Japan 64 (1995) 99.
13. Ta-Lun Yang, Int. J. Non-linear Mech. 3 (1968) 367.
14. L.I. Manevich, M.A. Pinski, Izvestiya Academii Nauk SSSR: MTT 2 (1972) 43.
15. A.K. Mishra, M.C. Singh, Int. J. Non-linear Mechanics 9 (1974) 463.
16. R.M. Rosenberg, J. Appl. Mech. 29 (1962) 7.
17. R.M. Rosenberg, Adv. Appl. Mech. 9, Academic Press, New York, 1966, p. 156.
18. J.A. Montaldi, R.M. Roberts, I.N. Stewart, Phil. Trans. Roy. Soc. London A 325 (1988) 237-293.
19. J.A. Montaldi, R.M. Roberts, I.N. Stewart, Pattern Formation in Physics, in: W. Guttinger, G. Dangelmayr (Eds.), Springer, Berlin, 1987, pp. 354-371.
20. J.A. Montaldi, R.M. Roberts, I.N. Stewart, Nonlinearity 3 (1990) 695-730.
21. J.A. Montaldi, R.M. Roberts, I.N. Stewart, Nonlinearity 3 (1990) 731-772.
22. V.P. Sakhnenko, G.M. Chechin, Dokl. Akad. Nauk 330 (1993) 308,
23. Phys. Dokl. 38 (1993) 219.
24. V.P. Sakhnenko, G.M. Chechin, Dokl. Akad. Nauk 338 (1994) 42,
25. Phys. Dokl. 39 (1994) 625.
26. A. Poincaré, Selected works, vol. 1, Nauka, Moscow, 1971.
27. J.P. Elliott, P.J. Dawber, Symmetry in Physics, vol. 1, Principles and Simple Applications, Macmillan, London, 1979.
28. L.D. Landau, E.M. Lifshitz, Statistical Physics, Part 1, Pergamon, Oxford, 1980.
29. O.V. Kovalev, Representations of the Crystallographic Space Groups: Irreducible Representations, Induced Representations and Corepresentations, in: H.T. Stokes, D.M. Hatch (Eds.), Gordon and Breach, Amsterdam, 1993.
30. V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1978.
31. V.F. Zhuravlev, D.M. Klimov, Applied Methods in Vibrational Theory, Nauka, Moscow, 1988.
32. A.P. Markeev, Libration Points in Celestial Mechanics, Nauka, Moscow, 1978.
33. M. Lakshmanan, R. Sahadevan, Physics Reports (Review Section of Physics Letters) 224 (1993) 1.
34. T.M. Belaya, S.A. Voloshanovskii, M.Yu. Zekhtser, A.V. Kukin, G.A. Malichev, G.M. Chechin, V.G. Yagubiants, Bushes of Interacting Modes in Systems with Crystallographic Point Symmetry. 1, Classes of Universality Manuscript deposited in VINITI, N 2708-B95, Moscow, 1995, 2. Normal Forms of Hamiltonians Manuscript deposited in VINITI, N 3467-B95, Moscow, 1995.
35. R. Blinc, B. Zeks, Soft Modes in Ferroelectrics and Antiferroelectrics, Oxford, New York, 1974.
36. J. Petzelt, V. Dvorak, J. Phys. C: Solid State Phys. 9 (1976) 1571.
37. Yu.A. Izyumov, V.N. Syromiatnikov, Phase Transitions and Crystal Symmetry, Nauka, Moscow, 1984.
38. G.M. Chechin, Comput. Math. Appl. 17 (1989) 255.
39. E. Wigner, Nachricht. Akad. Wiss. Göttingen, Math. Phys. Kl. 133 (1930).
40. A.J. Lichtenberg, M.A. Liebermann, Regular and Stochastic Motion, Springer, Berlin, 1983.
41. H.G. Schuster, Deterministic Chaos: An Introduction Physic-Verlag, Weinheim, 1984.