Parameter-shifted shadowing property for geometric Lorenz attractors
HTML articles powered by AMS MathViewer
- by Shin Kiriki and Teruhiko Soma
- Trans. Amer. Math. Soc. 357 (2005), 1325-1339
- DOI: https://doi.org/10.1090/S0002-9947-04-03607-4
- Published electronically: April 27, 2004
- PDF | Request permission
Abstract:
In this paper, we will show that any geometric Lorenz flow in a definite class satisfies the parameter-shifted shadowing property.References
- 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
- D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209 (Russian). MR 0224110
- 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, DOI 10.1090/S0002-9947-01-02822-7
- C. M. Carballo, C. A. Morales, and M. J. Pacifico, Maximal transitive sets with singularities for generic $C^1$ vector fields, Bol. Soc. Brasil. Mat. (N.S.) 31 (2000), no. 3, 287–303. MR 1817090, DOI 10.1007/BF01241631
- 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
- J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Applied Mathematical Sciences, Vol. 19, 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. MR 0494309
- John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72. MR 556582
- S. Kiriki and T. Soma, Parameter-shifted shadowing property of Lozi maps, preprint.
- Motomasa Komuro, Lorenz attractors do not have the pseudo-orbit tracing property, J. Math. Soc. Japan 37 (1985), no. 3, 489–514. MR 792989, DOI 10.2969/jmsj/03730489
- E. N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci., 20 (1963) 130–141.
- C. A. Morales, Lorenz attractor through saddle-node bifurcations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 13 (1996), no. 5, 589–617 (English, with English and French summaries). MR 1409664, DOI 10.1016/S0294-1449(16)30116-0
- C. A. Morales and M. J. Pacifico, Mixing attractors for 3-flows, Nonlinearity 14 (2001), no. 2, 359–378. MR 1819802, DOI 10.1088/0951-7715/14/2/310
- 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, DOI 10.1090/S0002-9939-99-04936-9
- 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, DOI 10.1007/s002200050825
- 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
- Sergei Yu. Pilyugin, Shadowing in dynamical systems, Lecture Notes in Mathematics, vol. 1706, Springer-Verlag, Berlin, 1999. MR 1727170
- Warwick Tucker, A rigorous ODE solver and Smale’s 14th problem, Found. Comput. Math. 2 (2002), no. 1, 53–117. MR 1870856, DOI 10.1007/s002080010018
- Marcelo Viana, What’s new on Lorenz strange attractors?, Math. Intelligencer 22 (2000), no. 3, 6–19. MR 1773551, DOI 10.1007/BF03025276
- R. F. Williams, The structure of Lorenz attractors, Turbulence Seminar (Univ. Calif., Berkeley, Calif., 1976/1977) Lecture Notes in Math., Vol. 615, Springer, Berlin, 1977, pp. 94–112. MR 0461581
- John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72. MR 556582
- 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, DOI 10.1007/BF01011469
Bibliographic Information
- Shin Kiriki
- Affiliation: Department of Mathematical Sciences, Tokyo Denki University, Hatoyama, Hiki, Saitama-ken, 350-0394, Japan
- Email: ged@r.dendai.ac.jp
- Teruhiko Soma
- Affiliation: Department of Mathematical Sciences, Tokyo Denki University, Hatoyama, Hiki, Saitama-ken, 350-0394, Japan
- MR Author ID: 192547
- Email: soma@r.dendai.ac.jp
- Received by editor(s): April 10, 2003
- Received by editor(s) in revised form: July 31, 2003
- Published electronically: April 27, 2004
- Additional Notes: The first author was supported in part by Research Institute for Science and Technology at TDU Grant Q02J-02, Q03J-08
- © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2115368