An open set of maps for which every point is absolutely nonshadowable

Yuan, Guo-Cheng; Yorke, James

doi:10.1090/S0002-9939-99-05038-8

An open set of maps for which every point is absolutely nonshadowable
HTML articles powered by AMS MathViewer

by Guo-Cheng Yuan and James A. Yorke PDF

Proc. Amer. Math. Soc. 128 (2000), 909-918 Request permission

Abstract:

We consider a class of nonhyperbolic systems, for which there are two fixed points in an attractor having a dense trajectory; the unstable manifold of one has dimension one and the other’s is two dimensional. Under the condition that there exists a direction which is more expanding than other directions, we show that such attractors are nonshadowable. Using this theorem, we prove that there is an open set of diffeomorphisms (in the $C^{r}$-topology, $r > 1$) for which every point is absolutely nonshadowable, i.e., there exists $\epsilon > 0$ such that, for every $\delta > 0$, almost every $\delta$-pseudo trajectory starting from this point is $\epsilon$-nonshadowable.

References

R. Abraham and S. Smale, Nongenericity of $\Omega$-stability, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 5–8. MR 0271986
D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature. , Proceedings of the Steklov Institute of Mathematics, No. 90 (1967), American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by S. Feder. MR 0242194
Michael Benedicks and Lennart Carleson, The dynamics of the Hénon map, Ann. of Math. (2) 133 (1991), no. 1, 73–169. MR 1087346, DOI 10.2307/2944326
Rufus Bowen, $\omega$-limit sets for axiom $\textrm {A}$ diffeomorphisms, J. Differential Equations 18 (1975), no. 2, 333–339. MR 413181, DOI 10.1016/0022-0396(75)90065-0
Shui-Nee Chow and Kenneth J. Palmer, On the numerical computation of orbits of dynamical systems: the one-dimensional case, J. Dynam. Differential Equations 3 (1991), no. 3, 361–379. MR 1118339, DOI 10.1007/BF01049737
Shui-Nee Chow and Kenneth J. Palmer, On the numerical computation of orbits of dynamical systems: the higher-dimensional case, J. Complexity 8 (1992), no. 4, 398–423. MR 1195260, DOI 10.1016/0885-064X(92)90004-U
Brian A. Coomes, Hüseyin Koçak, and Kenneth J. Palmer, Periodic shadowing, Chaotic numerics (Geelong, 1993) Contemp. Math., vol. 172, Amer. Math. Soc., Providence, RI, 1994, pp. 115–130. MR 1294411, DOI 10.1090/conm/172/01801
Brian A. Coomes, Hüseyin Koçak, and Kenneth J. Palmer, Shadowing in discrete dynamical systems, Six lectures on dynamical systems (Augsburg, 1994) World Sci. Publ., River Edge, NJ, 1996, pp. 163–211. MR 1441125
Ethan M. Coven, Ittai Kan, and James A. Yorke, Pseudo-orbit shadowing in the family of tent maps, Trans. Amer. Math. Soc. 308 (1988), no. 1, 227–241. MR 946440, DOI 10.1090/S0002-9947-1988-0946440-2
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
Jacques Hadamard, Sur l’itération et les solutions asymptotiques des équations différentielles, Bulletin de la Société Mathématique de France 29 (1901), 224–228.
Stephen M. Hammel, James A. Yorke, and Celso Grebogi, Do numerical orbits of chaotic dynamical processes represent true orbits?, J. Complexity 3 (1987), no. 2, 136–145. MR 907194, DOI 10.1016/0885-064X(87)90024-0
Stephen M. Hammel, James A. Yorke, and Celso Grebogi, Numerical orbits of chaotic processes represent true orbits, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 465–469. MR 938160, DOI 10.1090/S0273-0979-1988-15701-1
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
Yuri Kifer, Random perturbations of dynamical systems, Progress in Probability and Statistics, vol. 16, Birkhäuser Boston, Inc., Boston, MA, 1988. MR 1015933, DOI 10.1007/978-1-4615-8181-9
L. Gardini, J. C. Cathala, and C. Mira, Contact bifurcations of absorbing areas and chaotic areas in two-dimensional endomorphisms, Iteration theory (Batschuns, 1992) World Sci. Publ., River Edge, NJ, 1996, pp. 100–111. MR 1442274
P. J. Myrberg, Sur l’itération des polynomes réels quadratiques, J. Math. Pures Appl. (9) 41 (1962), 339–351 (French). MR 161968
Leon Poon, Celso Grebogi, Tim Sauer, and James A. Yorke, Limits to deterministic modeling, Preprint.
Clark Robinson, Dynamical systems, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. Stability, symbolic dynamics, and chaos. MR 1396532
Tim Sauer, Celso Grebogi, and James A. Yorke, How long do numerical chaotic solutions remain valid?, Physical Review Letters 79 (1997), 59.
Tim Sauer and James A. Yorke, Rigorous verification of trajectories for the computer simulation of dynamical systems, Nonlinearity 4 (1991), no. 3, 961–979. MR 1124343
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
S. Smale and R. F. Williams, The qualitative analysis of a difference equation of population growth, J. Math. Biol. 3 (1976), no. 1, 1–4. MR 414147, DOI 10.1007/BF00307853
Lai-Sang Young, Stochastic stability of hyperbolic attractors, Ergodic Theory Dynam. Systems 6 (1986), no. 2, 311–319. MR 857204, DOI 10.1017/S0143385700003473