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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
Published electronically:
April 27, 2004
MathSciNet review:
2115368
Full-text PDF Free Access
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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.
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M. Komuro, Lorenz attractors do not have the pseudo-orbit tracing property, J. Math. Soc. Japan, 37 (1985) 489-514. MR 87a:58106
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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) 589-617. MR 97f:58084
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C.A. Morales and M.J. Pacifico, Mixing attractors for
 -flows, Nonlinearity, 14 (2001) 359-378. MR 2002a:37036
- 13.
C.A. Morales, M.J. Pacifico and E.R. Pujals, Singular hyperbolic systems, Proc. Amer. Math. Soc., 127 (1999) 3393-3401. MR 2000c:37034
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C.A. Morales, M.J. Pacifico and E.R. Pujals, Strange attractors across the boundary of hyperbolic systems, Comm. Math. Phys., 211 (2000) 527-558. MR 2001g:37036
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H.E. Nusse and J.A. Yorke, Is every approximate trajectory of some process near an exact trajectory of a near process? Comm. Math. Phys., 114 (1988) 363-379. MR 89g:58108
- 16.
S.Y. Pilyugin, Shadowing in dynamical systems, Lecture Notes in Math. 1706 Springer-Verlag (1999). MR 2001b:37030
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W. Tucker, A rigorous ODE solver and Smale's
 th problem, Found. Comput. Math., 2 (2002) 53-117. MR 2003b:37055
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M. Viana, What's new on Lorenz strange attractors? Math. Intelligencer, 22 (2000) 6-19. MR 2001h:37068
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R. Williams, The structure of Lorenz attractors, Turbulence Seminar (Univ. Calif., Berkeley, Calif., 1976/1977), Lecture Notes in Math. 615 (pp. 94-112), Springer (1977). MR 57:1566
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R. Williams, The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979) 73-99. MR 82b:58055b
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- 1.
- V.S. Afraimovich, V.V. Bykov, and L.P. Shil'nikov, On the appearance and structure of the Lorenz attractor, Dokl. Acad. Sci. USSR, 234 (1977) 336-339. MR 57:2150
- 2.
- D.V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature, Proc. Steklov Math. Inst., 90 (1967). MR 36:7157
- 3.
- W.J. Colmenarez and C.A. Morales, Transverse surfaces and attractors for -flows, Trans. Amer. Math. Soc., 354 (2002) 795-806. MR 2002h:37043
- 4.
- C.M. Carballo, C.A. Morales and M.J. Pacifico, Maximal transitive sets with singularities for generic vector fields, Bol. Soc. Brasil. Mat. (N.S.), 31 (2000) 287-303. MR 2002b:37075
- 5.
- E.M. Coven, I. Kan and J.A. Yorke, Pseudo-orbit shadowing in the family of tent maps, Trans. Amer. Math. Soc., 308 (1988) 227-241. MR 90b:58236
- 6.
- J. Guckenheimer, A strange, strange attractor, in The Hopf bifurcation and its applications, (J.E. Marsden and M. McCracke eds.), Springer-Verlag, New York (1976). MR 58:13209
- 7.
- J. Guckenheimer and R.F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979) 59-72. MR 82b:58055a
- 8.
- S. Kiriki and T. Soma, Parameter-shifted shadowing property of Lozi maps, preprint.
- 9.
- M. Komuro, Lorenz attractors do not have the pseudo-orbit tracing property, J. Math. Soc. Japan, 37 (1985) 489-514. MR 87a:58106
- 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) 589-617. MR 97f:58084
- 12.
- C.A. Morales and M.J. Pacifico, Mixing attractors for -flows, Nonlinearity, 14 (2001) 359-378. MR 2002a:37036
- 13.
- C.A. Morales, M.J. Pacifico and E.R. Pujals, Singular hyperbolic systems, Proc. Amer. Math. Soc., 127 (1999) 3393-3401. MR 2000c:37034
- 14.
- C.A. Morales, M.J. Pacifico and E.R. Pujals, Strange attractors across the boundary of hyperbolic systems, Comm. Math. Phys., 211 (2000) 527-558. MR 2001g:37036
- 15.
- H.E. Nusse and J.A. Yorke, Is every approximate trajectory of some process near an exact trajectory of a near process? Comm. Math. Phys., 114 (1988) 363-379. MR 89g:58108
- 16.
- S.Y. Pilyugin, Shadowing in dynamical systems, Lecture Notes in Math. 1706 Springer-Verlag (1999). MR 2001b:37030
- 17.
- W. Tucker, A rigorous ODE solver and Smale's th problem, Found. Comput. Math., 2 (2002) 53-117. MR 2003b:37055
- 18.
- M. Viana, What's new on Lorenz strange attractors? Math. Intelligencer, 22 (2000) 6-19. MR 2001h:37068
- 19.
- R. Williams, The structure of Lorenz attractors, Turbulence Seminar (Univ. Calif., Berkeley, Calif., 1976/1977), Lecture Notes in Math. 615 (pp. 94-112), Springer (1977). MR 57:1566
- 20.
- R. Williams, The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math., 50 (1979) 73-99. MR 82b:58055b
- 21.
- J.A. Yorke and E.D. Yorke, Metastable chaos: the transition to sustained chaotic behavior in the Lorenz model, J. Statist. Phys., 21 (1979) 263-277. MR 80f:76023
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Additional 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
Email:
soma@r.dendai.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002-9947-04-03607-4
PII:
S 0002-9947(04)03607-4
Keywords:
Geometric Lorenz model,
strange attractor,
shadowing property
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
Article copyright:
© Copyright 2004 American Mathematical Society
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