Using Lyapunov exponents to predict the onset of chaos in nonlinear oscillators

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 66, Issue 1, 2002, Pages

  Ryabov, Vladimir B ^a,b  

^a FUTURE UNIVERSITY HAKODATE   (Japan)

^b INSTITUTE OF RADIO ASTRONOMY   (Ukraine)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; COMPUTER SIMULATION; EIGENVALUES AND EIGENFUNCTIONS; ELEMENTARY PARTICLES; FUNCTIONS; MATHEMATICAL MODELS; PERTURBATION TECHNIQUES;

LATTICE MODELS; LINEAR GROWTH; LINEAR SPREADING SPEED; PARTICLE DENSITY;

QUANTUM THEORY;

EID: 37649031663     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.66.016214     Document Type: Article

References (56)

1. J.-P. Eckman and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
2. J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos (Wiley, New York, 1986).
3. G. Casati and B. Chirikov, Quantum Chaos (Cambridge University Press, Cambridge, England, 1995).
4. T. Bohr, M. H. Jensen, G. Paladin, and A. Vulpiani, Dynamical Systems Approach to Turbulence (Cambridge University Press, Cambridge, England, 1998).
5. J. W. Haefner, Modeling Biological Systems: Principles and Applications (Chapman & Hall, New York, 1996).
6. D. L. Turcotte, Fractals and Chaos in Geology and Geophysics, 2nd ed. (Cambridge University Press, Cambridge, England, 1997).
7. B. R. Hunt, E. Ott, and J. A. Yorke, Phys. Rev. E 55, 4029 (1997).
8. E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, England, 1993).
9. V. I. Oseledec, Trans. Mosc. Math. Soc. 19, 197 (1968).
10. H. Mori and Y. Kuramoto, Dissipative Structures and Chaos (Springer-Verlag, Berlin, 1998).
11. R. Badii and A. Politi, Complexity: Hierarchical Structures and Scaling in Physics (Cambridge University Press, Cambridge, England, 1997).
12. R. Zillmer, V. Ahlers, and A. Pikovsky, Phys. Rev. E 61, 332 (2000).
13. C. Ziehmann, L. A. Smith, and J. Kurths, Physica D 126, 49 (1999).
14. D. Beigie, A. Leonard, and S. Wiggins, Phys. Rev. Lett. 70, 275 (1993).
15. A. Prasad and R. Ramaswamy, Phys. Rev. E 60, 2761 (1999).
16. E. Barreto and P. So, Phys. Rev. Lett. 85, 2490 (2000).
17. P. So, E. Baretto, and B. R. Hunt, Phys. Rev. E 60, 378 (1999).
18. P. Grassberger and I. Procaccia, Physica D 13, 34 (1984).
19. H. Herzel and T. Schulmeister, in Dynamical Systems and Environmental Models (Akademie, Berlin, 1987).
20. D. M. Vavriv, G. A. Gromov, and V. B. Ryabov, Zh. Tekh. Fiz. 60, 1 (1990) [Sov. Phys. Tech. Phys. 35, 1231 (1990)].
21. J. M. Nese, Physica D 35, 237 (1989).
22. B. Eckhardt and D. Yao, Physica D 65, 100 (1993).
23. W. G. Hoover, C. G. Tull, and H. A. Posch, Phys. Lett. A 131, 211 (1988).
24. T. M. Janaki, G. Rangarajan, S. Habib, and R. D. Ryne, Phys. Rev. E 60, 6614 (1999).
25. G. Benettin, L. Galgani, A. Giorgilli, and J. M. Strelcyn, Meccanica 15, 9 (1980).
26. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, Physica D 16, 258 (1985).
27. I. Shimada and T. Nagashima, Prog. Theor. Phys. 61, 1605 (1979).
28. V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients and Their Applications (Nauka, Moscow, 1972).
29. D. M. Vavriv and V. B. Ryabov, USSR Comput. Math. Math. Phys. 32, 1259 (1992).
30. C. Hayashi, Nonlinear Oscillations in Physical Systems (McGraw-Hill, New York, 1964).
31. S. Wiggins, Global Bifurcations and Chaos: Analytical Methods (Springer-Verlag, Berlin, 1988).
32. D. M. Vavriv, V. B. Ryabov, S. A. Sharapov, and H. M. Ito, Phys. Rev. E 53, 103 (1996).
33. W. G. Hoover, C. G. Hoover, and H. A. Posch, Phys. Rev. A 41, 2999 (1990).
34. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, New York, 1990).
35. W. Walter, Ordinary Differential Equations (Springer-Verlag, New York, 1998).
36. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, England, 1986).
37. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).
38. Y. Ueda, The Road to Chaos (Aerial Press, Santa Cruz, 1994).
39. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, New York, 1979).
40. J. A. Murdock, Perturbations: Theory and Methods (Wiley, New York, 1991).
41. A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics. Analytical, Computational, and Experimental Methods (Wiley, New York, 1995).
42. J. G. Byatt-Smith, IMA J. Appl. Math. 37, 113 (1986).
43. Y. H. Kao, J. C. Huang, and Y. S. Gou, Phys. Lett. A 131, 91 (1988).
44. U. Parlitz and W. Lautemom, Phys. Lett. A 107, 351 (1985).
45. V. K. Melnikov, Trans. Mosc. Math. Soc. 12, 1 (1963);
46. P. Holmes and J. Marsden, Arch. Ration. Mech. Anal. 76, 135 (1981).
47. W. Szemplinska-Stupnicka, Ingenieur Archiv 58, 354 (1988).
48. C. S. Wang, Y. H. Kao, J. C. Huang, and Y. S. Gou, Phys. Rev. A 45, 3471 (1992).
49. T. Kapitaniak, Phys. Lett. A 144, 322 (1990).
50. J. Rudowsky and W. Szemplinska-Stupnicka, Schweiz. Ing. Archit. 57, 243 (1987).
51. B. A. Huberman, J. P. Crutchfield, and N. Packard, Appl. Phys. Lett. 37, 750 (1980).
52. D. D’Humieres, M. R. Beasley, B. A. Huberman, and A. Libchaber, Phys. Rev. A 26, 3483 (1988).
53. J. Miles, Physica D 31, 252 (1988).
54. M. Levi, Phys. Rev. A 37, 927 (1988).
55. A. D. Grishchenko and D. M. Vavriv, Tech. Phys. 42, 1115 (1997).
56. A. Litvak-Hinenzon and V. Rom-Kedar, Phys. Rev. E 55, 4964 (1997).