Shadowing orbits of ordinary differential equations on invariant submanifolds

Transactions of the American Mathematical Society

Volumn 349, Issue 1, 1997, Pages 203-216

  Coomes, Brian A ^a  

^a UNIVERSITY OF MIAMI   (United States)

Author keywords

First integrals; Hamiltonian systems; Invariant manifolds; Ordinary differential equations; Shadowing

Indexed keywords

EID: 21444459986     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-97-01783-2     Document Type: Article

References (19)

1. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Wiley, New York, 1975, Reprinted by Dover, New York, 1987.
2. D. V. Anosov, Geodesic flows and closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. Math. 90 (1967). MR 36:7157
3. R. Bowen, w-limit sets for Axiom A diffeomorphisms, J. Differential Equations 18 (1975), 333-339. MR 54:1300
4. S. N. Chow and K. J. Palmer, On the numerical computation of orbits of dynamical systems: the one-dimensional case, J. Dynamics and Differential Equations 3 (1991), 361-379. MR 92h:58057
5. _, On the numerical computation of orbits of dynamical systems: the higher dimensional case, J. Complexity 8 (1992), 398-423. MR 93k:65118
6. S. N. Chow and E. Van Vleck, A shadowing lemma approach to global error analysis for initial value ODE's, SIAM J. Sci. Comput. 15 (1994), 959-976. MR 95c:65096
7. B. A. Coomes, H. Koçak, and K. J. Palmer, Shadowing orbits of ordinary differential equations, J. Comp. Appl. Math. 53 (1994), 35-43. MR 95j:58145
8. _, Periodic shadowing, Chaotic Numerics, Proceedings of Chaotic Numerics: An International Workshop on the Approximation and Computation of Complicated Dynamical Behavior, Contemporary Mathematics series, vol. 172, American Mathematical Society, Providence, Rhode Island, 1994, pp. 115-130. MR 95i:58128
9. _, A shadowing theorem for ordinary differential equations, Z. Angew. Math. Phys. 46 (1995), 85-106. MR 96b:58085
10. _, Rigorous computational shadowing of orbits of ordinary differential equations, Numer. Math. 69 (1995), 401-421. MR 96g:34014
11. _, Shadowing in discrete dynamical systems, Six Lectures on Dynamical Systems, World Scientific, Singapore, 1996, pp. 163-212.
12. S. Dawson, C. Grebogi, T. Sauer, and J. A. Yorke, Obstructions to shadowing when a Lyapunov exponent fluctuates about zero, Phys. Rev. Lett. 73 (1994), 1927-1930.
13. J. E. Franke and J. F. Belgrade, Hyperbolicity and chain recurrence, J. Differential Equations 26 (1977), 27-36. MR 57:7685
14. E. A. Gonzalez Velasco, Generic properties of polynomial vector fields at infinity, Trans. Amer. Math. Soc. 143 (1969), 201-222. MR 40:6005
15. S. H. Hammel, J. A. Yorke, and C. Grebogi, Do numerical orbits of chaotic dynamical processes represent true orbits?, J. Complexity 3 (1987), 136-145. MR 88m:58115
16. _, Numerical orbits of chaotic dynamical processes represent true orbits, Bull. Amer. Math. Soc. 19 (1988), 465-470. MR 89m:58180
17. A. Nath and D. S. Ray, Horseshoe-shaped maps in chaotic dynamics of atom-field interaction, Phys. Rev. A 36 (1987), 431-434.
18. H. Poincaré, Mémoire sur les courbes définies par une équation différentielle, J. Mathématiques (3) 7 (1881), 375-422.
19. T. Sauer and J. A. Yorke, Rigorous verification of trajectories for computer simulations of dynamical systems, Nonlinearity'4 (1991), 961-979. MR 93a:58104