## On various types of shadowing for geometric Lorenz flows.(English)Zbl 1352.37067

Summary: We show that Lorenz flows have neither limit shadowing property nor average shadowing property nor the asymptotic average shadowing property where the reparametrizations related to these concepts relies on the set of increasing homeomorphisms with bounded variation.

### MSC:

 37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 37C10 Dynamics induced by flows and semiflows
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### References:

 [1] V.S. Afraimovich, V.V. Bykov and L.P. Shil’nikov, On attracting structurally unstable limit sets of Lorenz attractor type , Trudy Mosk. Mat. Obsh. 44 (1982), 150–212. · Zbl 0506.58023 [2] Vítor Araújo and Maria José Pacífico, Three-dimensional flows , Ergeb. Math. Grenz 3 , Springer, Heidelberg, 2010. · Zbl 1202.37002 [3] M.L. Blank, Metric properties of minimal solutions of discrete periodical variational problems , Nonlinearity 2 (1989), 1–22. · Zbl 0676.49001 [4] Timo Eirola, Olavi Nevanlinna and Sergei Yu Pilyugin, Limit shadowing property , Num. Funct. Anal. Optim. 18 (1997), 75–92. · Zbl 0881.58049 [5] Rongbao Gu, The asymptotic average-shadowing property and transitivity for flows , Chaos Solit. Fract. 41 (2009), 2234–2240. · Zbl 1113.15302 [6] John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields , revised and corrected reprint of the 1983 original, Appl. Math. Sci. 42 , Springer-Verlag, New York, 1990. · Zbl 0515.34001 [7] John Guckenheimer and R.F. Williams, Structural stability of Lorenz attractors , Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72. · Zbl 0436.58018 [8] Shin Kiriki and Teruhiko Soma, Parameter-shifted shadowing property for geometric Lorenz attractors , Trans. Amer. Math. Soc. 357 (2005), 1325–1339 (electronic). · Zbl 1073.37023 [9] Motomasa Komuro, Lorenz attractors do not have the pseudo-orbit tracing property , J. Math. Soc. Japan 37 (1985), 489–514. · Zbl 0552.58020 [10] E.N. Lorenz, Deterministic non-periodic flow , J. Atmos. Sci. 20 (1963), 130-141. · Zbl 1417.37129 [11] C. Robinson, Differentiability of the stable foliation of the model Lorenz equations , Lect. Notes Math. 898 , Springer, Berlin, 1981. · Zbl 0485.58012 [12] Michael Shub, Global stability of dynamical systems , Springer-Verlag, New York, 1987. · Zbl 0606.58003 [13] Shlomo Sternberg, On the structure of local homeomorphisms of Euclidean $$n$$-space , II, Amer. J. Math. 80 (1958), 623–631. · Zbl 0083.31406 [14] R.F. Williams, The structure of Lorenz attractors , Lect. Notes Math. 615 , Springer, Berlin, 1977. · Zbl 0363.58005 [15] —-, The structure of Lorenz attractors , Inst. Hautes Études Sci. Publ. Math. 50 (1979), 73–99. · Zbl 0484.58021 [16] James A. Yorke and Ellen D. Yorke, Metastable chaos : The transition to sustained chaotic behavior in the Lorenz model , J. Stat. Phys. 21 (1979), 263–277.
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