Parameter-shifted shadowing property for geometric Lorenz attractors

Kiriki, Shin; Soma, Teruhiko

doi:10.1090/s0002-9947-04-03607-4

Cited by 9 publications

(4 citation statements)

References 20 publications

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“…It is worth to mention studies of the shadowing property for the Lorenz attractor. Komuro [13] proved that (except very special case) geometric Lorenz flow do not have shadowing property (see also [2]), however geometric Lorenz flow has parameter-shifted shadowing property for a wide range of parameters [11].…”

Section: Introductionmentioning

confidence: 94%

Oriented Shadowing Property and $$\Omega $$ Ω -Stability for Vector Fields

Gan

Tikhomirov

2014

J Dyn Diff Equat

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We call that a vector field has the oriented shadowing property if for any ε > 0 there is d > 0 such that each d-pseudo orbit is ε-oriented shadowed by some real orbit. In this paper, we show that the C 1 -interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with the Ω-stability.2010 Mathematics Subject Classification. Primary 37C50.

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Section: Introductionmentioning

confidence: 94%

Oriented Shadowing Property and $$\Omega $$ Ω -Stability for Vector Fields

Gan

Tikhomirov

2014

J Dyn Diff Equat

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“…is the geometric Lorenz flow if it is generated by a geometric Lorenz vector field X (see e.g. [20,21,26] for more details). Let T X be the closure of the set…”

Section: Now We Briefly Describe the Geometric Lorenz Attractormentioning

confidence: 99%

Specification and partial hyperbolicity for flows

2015

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We prove that if a flow exhibits a partially hyperbolic attractor Λ with splitting T Λ M = E s ⊕ E c and two periodic saddles with different indices such that the stable index of one of them coincides with the dimension of E s then it does not satisfy the specification property. In particular, every singular-hyperbolic attractor with the specification property is hyperbolic. As an application, we prove that no Lorenz attractor satisfies the specification property.1 for t ≥ 0. Now, take a small t 2 > 0 such thatz)) ≤ 2L 0 ε} and t 0 = inf I. By (3.21), (3.25) and (3.26) we have T ∈ I. Assume that t 0 > 0. Sinceby (3.21) and (3.26) we have t 0 − t 2 ∈ I, which is a contradiction. Thus 0 = inf I. ThereforeThe proposition follows taking L = 3L 0 . Lemma 3.10. Let p ∈ Λ be a hyperbolic critical element with dim W ss (p) = dim E s p and U be a subdisk of W u (p) with U ⊂ W u (p). Then there exists µ > 0 such that A(U ) := z∈U F s µ (z) is homeomorphic to U × [−µ, µ] dim E s .

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“…Komuro [9] showed that geometric Lorenz flows do not satisfy the (parameter-fixed) shadowing property excepted in very restricted cases. Even so, in [8] it was shown that the geometric Lorenz attractors have the parameter-shifted shadowing property. However, this notion is very technical.…”

Section: Introductionmentioning

confidence: 99%