Hartman’s theorem for hyperbolic sets

Kurata, Masahiro

doi:10.1017/s0027763000022534

Cited by 4 publications

(4 citation statements)

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“…In particular, we have omitted a result concerning the existence and Hölder continuity of some conjugacies needed in section 4.3. The existence follows from the extension of the Hartman-Grobman Theorem mentioned above ( [16]), and it should be plausible at least that the conjugacies implied by this theorem are Hölder continuous (this is certainly true for the standard Hartman-Grobman Theorem). All the relevant details can be found in [21].…”

Section: Analytic Resultsmentioning

confidence: 98%

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Convergence properties of gradient descent noise reduction

Ridout

Judd

2002

Physica D: Nonlinear Phenomena

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Gradient descent noise reduction is a technique that attempts to recover the true signal, or trajectory, from noisy observations of a non-linear dynamical system for which the dynamics are known. This paper provides the first rigorous proof that the algorithm will recover the original trajectory for a broad class of dynamical systems under certain conditions. The proof is obtained using ideas from linearisation theory. Since the first introduction of the algorithm it has been recognised that the algorithm can fail to recover the true trajectory, and it has been suggested that this is a practical or numerical limitation that is a consequence of near tangencies between stable and unstable manifolds. This paper demonstrates through numerical experiments and details of the proof that the situation is worse than expected in that near tangencies impose essential limitations on noise reduction, not just practical or numerical limitations. That is, gradient descent noise reduction will sometimes fail to recover the true trajectory, even with unlimited, perfect computation. On the other hand, the numerical experiments suggest that the gradient descent noise reduction algorithm will always recover a trajectory that is entirely consistent with the evidence provided by the observations, that is, it attains the best that can be achieved given the observations. It is argued that near tangencies will therefore impose the same limitations on any noise reduction algorithm.

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Section: Analytic Resultsmentioning

confidence: 98%

“…Instead, we introduce a semi-conjugacy whose properties are more amenable to analysis. A generalisation of the Hartman-Grobman Theorem ( [20,16,21]) and Condition 7 below, then provide this information.…”

Section: Analytic Resultsmentioning

confidence: 99%

Convergence properties of gradient descent noise reduction

Ridout

Judd

2002

Physica D: Nonlinear Phenomena

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“…ficiently large L But this is impossible by (1) and (2). Thus S o does not contain any periodic point of h.…”

Section: Proof Of the Main Theoremmentioning

confidence: 93%

“…Also, by (4.2), S o = p{σ*(a w ); w e (0,1), k ^ 0} 3 pjα^; M; 6 (0,1)}. By (1), (2), and (4.1), it is clear that p{a w ; w e (0,1)} is uncountable. Hence S o is an uncountable subset of M.…”

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confidence: 95%

A generalization of a theorem of marotto

Shiraiwa¹,

Kurata²

1981

Nagoya Mathematical Journal

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In 1975, Li and Yorke [3] found the following fact. Let f: I→ I be a continuous map of the compact interval I of the real line R into itself. If f has a periodic point of minimal period three, then f exhibits chaotic behavior. The above result is generalized by F.R. Marotto [4] in 1978 for the multi-dimensional case as follows. Let f: Rn → Rn be a differentiate map of the n-dimensional Euclidean space Rn (n ≧ 1) into itself. If f has a snap-back repeller, then f exhibits chaotic behavior.In this paper, we give a generalization of the above theorem of Marotto. Our theorem can also be regarded as a generalization of the Smale’s results on the transversal homoclinic point of a diffeomorphism.

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Hyperbolic Sets

Anosov¹,

Solodov²

1995

Encyclopaedia of Mathematical Sciences

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Hartman’s theorem for hyperbolic sets

Cited by 4 publications

References 5 publications

Convergence properties of gradient descent noise reduction

Convergence properties of gradient descent noise reduction

A generalization of a theorem of marotto

Hyperbolic Sets

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