Universal and nonuniversal features in shadowing dynamics of nonhyperbolic chaotic systems with unstable-dimension variability

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 67, Issue 3 2, 2003, Pages

  Do, Younghae ^a   Lai, Ying Cheng ^a   Liu, Zonghua ^a   Kostelich, Eric J ^a  

^a ARIZONA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ASYMPTOTIC STABILITY; BOUNDARY CONDITIONS; COMPUTER SIMULATION; CONFORMAL MAPPING; LYAPUNOV METHODS; MATHEMATICAL MODELS; MATRIX ALGEBRA; ORDINARY DIFFERENTIAL EQUATIONS; PROBABILITY DISTRIBUTIONS; RANDOM PROCESSES;

COMPUTER GENERATED TRAJECTORIES; EXPONENTIAL DISTRIBUTION; NONHYPERBOLIC CHAOTIC SYSTEMS; NUMERICAL TRAJECTORIES;

CHAOS THEORY;

EID: 42749099162     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (25)

1. D.V. Anosov, Proc. Steklov Inst. Math. 90, 1 (1967)
2. R. Bowen, J. Diff. Eqns. 18, 333 (1975).
3. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical System (Cambridge University Press, Cambridge, 1995).
4. S.M. Hammel, J.A. Yorke, and C. Grebogi, J. Complexity 3, 136 (1987)
5. S.M. Hammel, J.A. Yorke, and C. Grebogi, Bull. Am. Math. Soc. 19, 465 (1988)
6. C. Grebogi, S.M. Hammel, J.A. Yorke, and T. Sauer, Phys. Rev. Lett. 65, 1527 (1990)
7. T. Sauer and J.A. Yorke, Nonlinearity 4, 961 (1991).
8. S.N. Chow and K.J. Palmer, Dyn. Diff. Eqn. 3, 361 (1991)
9. S.N. Chow and E.S. Van Vleck, SIAM J. Sci. Comput. (USA) 15, 959 (1994).
10. R. Abraham and S. Smale, Proc. Symp. Pure Math. 14, 5 (1970).
11. S.P. Dawson, C. Grebogi, T. Sauer, and J.A. Yorke, Phys. Rev. Lett. 73, 1927 (1994).
12. S.P. Dawson, Phys. Rev. Lett. 76, 4348 (1996)
13. E.J. Kostelich, I. Kan, C. Grebogi, E. Ott, and J.A. Yorke, Physica D 109, 81 (1997)
14. E. Barreto and P. So, Phys. Rev. Lett. 85, 2490 (2000).
15. T. Sauer, C. Grebogi, and J.A. Yorke, Phys. Rev. Lett. 79, 59 (1997).
16. Y.-C. Lai and C. Grebogi, Phys. Rev. Lett. 82, 4803 (1999)
17. Y.-C. Lai, D. Lerner, K. Williams, and C. Grebogi, Phys. Rev. E 60, 5445 (1999).
18. C. Grebogi, E.J. Kostelich, E. Ott, and J.A. Yorke, Physica D 25, 347 (1987).
19. Strictly, the random-walk model can be solved by the Fokker-Planck equation when Z is a zero mean, Gaussian random variable. For our shadowing problem, numerically, we find the distribution of Z is approximately Gaussian (by definition Z has a zero mean).
20. C.W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, New York, 1997)
21. H. Risken, The Fokker-Plank Equation (Springer-Verlag, Berlin, 1989).
22. J.F. Heagy, N. Platt, and S.M. Hammel, Phys. Rev. E 49, 1140 (1994)
23. D. Marthaler, D. Armbruster, Y.-C. Lai, and E.J. Kostelich, ibid. 64, 016220 (2001).
24. A. Čnys, A.N. Anagnostopoulos, and G.L. Bleris, Phys. Lett. A 224, 346 (1997).
25. The shadowing lemma of Anosov and Bowen [1], which holds for hyperbolic systems, has recently been extended to nonuniformly hyperbolic systems [2]. The nonhyperbolic systems studied here, i.e., dynamical systems with unstable-dimension variability, violate one of the essential conditions for hyperbolicity: the continuous splitting of the tangent space between the stable and unstable subspaces. Thus, the shadowing lemma in Ref. [2] does not hold for these severely nonhyperbolic systems. For them, shadowing of numerical trajectories, even of relatively short lengths, cannot be expected. Our method of the combination of algebraic (for short time) and exponential (for long time) behaviors in the statistical distribution of the shadowing time answers the question, "for how long a numerical trajectory can be expected to be valid?," in a quantitative way.