Parameter-shifted shadowing property for geometric Lorenz attractors

Kiriki, Shin; Soma, Teruhiko

doi:10.1090/S0002-9947-04-03607-4

Parameter-shifted shadowing property for geometric Lorenz attractors

Authors: Shin Kiriki and Teruhiko Soma
Journal: Trans. Amer. Math. Soc. 357 (2005), 1325-1339
MSC (2000): Primary 37C50, 37D45, 37D50; Secondary 34C28
DOI: https://doi.org/10.1090/S0002-9947-04-03607-4
Published electronically: April 27, 2004
MathSciNet review: 2115368
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we will show that any geometric Lorenz flow in a definite class satisfies the parameter-shifted shadowing property.

1. V. S. Afraĭmovič, V. V. Bykov, and L. P. Sil′nikov, The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk SSSR 234 (1977), no. 2, 336–339 (Russian). MR 0462175
2. D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209 (Russian). MR 0224110
3. W. J. Colmenarez and C. A. Morales, Transverse surfaces and attractors for 3-flows, Trans. Amer. Math. Soc. 354 (2002), no. 2, 795–806. MR 1862568, https://doi.org/10.1090/S0002-9947-01-02822-7
4. C. M. Carballo, C. A. Morales, and M. J. Pacifico, Maximal transitive sets with singularities for generic 𝐶¹ vector fields, Bol. Soc. Brasil. Mat. (N.S.) 31 (2000), no. 3, 287–303. MR 1817090, https://doi.org/10.1007/BF01241631
5. 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, https://doi.org/10.1090/S0002-9947-1988-0946440-2
6. J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Springer-Verlag, New York, 1976. With contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S. Smale; Applied Mathematical Sciences, Vol. 19. MR 0494309
7. John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72. MR 556582
R. F. Williams, The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 73–99. MR 556583
8. S. Kiriki and T. Soma, Parameter-shifted shadowing property of Lozi maps, preprint.
9. Motomasa Komuro, Lorenz attractors do not have the pseudo-orbit tracing property, J. Math. Soc. Japan 37 (1985), no. 3, 489–514. MR 792989, https://doi.org/10.2969/jmsj/03730489
10. E.N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci., 20 (1963) 130-141.
11. C. A. Morales, Lorenz attractor through saddle-node bifurcations, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 5, 589–617 (English, with English and French summaries). MR 1409664
12. C. A. Morales and M. J. Pacifico, Mixing attractors for 3-flows, Nonlinearity 14 (2001), no. 2, 359–378. MR 1819802, https://doi.org/10.1088/0951-7715/14/2/310
13. C. A. Morales, M. J. Pacifico, and E. R. Pujals, Singular hyperbolic systems, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3393–3401. MR 1610761, https://doi.org/10.1090/S0002-9939-99-04936-9
14. C. A. Morales, M. J. Pacifico, and E. R. Pujals, Strange attractors across the boundary of hyperbolic systems, Comm. Math. Phys. 211 (2000), no. 3, 527–558. MR 1773807, https://doi.org/10.1007/s002200050825
15. Helena E. Nusse and James A. Yorke, Is every approximate trajectory of some process near an exact trajectory of a nearby process?, Comm. Math. Phys. 114 (1988), no. 3, 363–379. MR 929137
16. Sergei Yu. Pilyugin, Shadowing in dynamical systems, Lecture Notes in Mathematics, vol. 1706, Springer-Verlag, Berlin, 1999. MR 1727170
17. Warwick Tucker, A rigorous ODE solver and Smale’s 14th problem, Found. Comput. Math. 2 (2002), no. 1, 53–117. MR 1870856
18. Marcelo Viana, What’s new on Lorenz strange attractors?, Math. Intelligencer 22 (2000), no. 3, 6–19. MR 1773551, https://doi.org/10.1007/BF03025276
19. R. F. Williams, The structure of Lorenz attractors, Turbulence Seminar (Univ. Calif., Berkeley, Calif., 1976/1977) Springer, Berlin, 1977, pp. 94–112. Lecture Notes in Math., Vol. 615. MR 0461581
20. John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72. MR 556582
R. F. Williams, The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 73–99. MR 556583
21. James A. Yorke and Ellen D. Yorke, Metastable chaos: the transition to sustained chaotic behavior in the Lorenz model, J. Statist. Phys. 21 (1979), no. 3, 263–277. MR 542050, https://doi.org/10.1007/BF01011469