Covering relations and the existence of topologically
normally hyperbolic invariant sets

Capiński, Maciej J.

doi:10.3934/dcds.2009.23.705

Cited by 25 publications

(37 citation statements)

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“…We believe that this approach is very well suited for computer assisted (rigorous, interval arithmetic based) validation of the needed assumptions. Similar approach has already been successfully applied in [5,8] in the setting of the rotating Hénon map, in [6] to establish the center manifold in the restricted three body problem, in [7] in the setting of a driven logistic map or in [21] to establish a hyperbolic attractor in the Kuznetsov system. All these results follow from verification of cone conditions based on the estimates of the derivative.…”

Section: Introductionmentioning

confidence: 92%

Geometric proof for normally hyperbolic invariant manifolds

Capiński

Zgliczyński

2015

Journal of Differential Equations

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We present a new proof of the existence of normally hyperbolic manifolds and their whiskers for maps. Our result is not perturbative. Based on the bounds on the map and its derivative, we establish the existence of the manifold within a given neighbourhood. Our proof follows from a graph transform type method and is performed in the state space of the system. We do not require the map to be invertible. From our method follows also the smoothness of the established manifolds, which depends on the smoothness of the map, as well as rate conditions, which follow from bounds on the derivative of the map. Our method is tailor made for rigorous, interval arithmetic based, computer assisted validation of the needed assumptions.

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Section: Introductionmentioning

confidence: 92%

Geometric proof for normally hyperbolic invariant manifolds

Capiński

Zgliczyński

2015

Journal of Differential Equations

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“…We refer to the work of [18,20,21,19,36,69] for a general theory of validated computation for stable/unstable (and other types of normally hyperbolic) invariant manifolds based on the topological notion of covering relations and cone conditions. The computer assisted topological arguments are carried out in phase space using polygonal elements.…”

Section: Computing Invariant Manifolds: a Brief Overviewmentioning

confidence: 99%

Polynomial approximation of one parameter families of (un)stable manifolds with rigorous computer assisted error bounds

James

2015

Indagationes Mathematicae

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This work describes a method for approximating a branch of stable or unstable manifolds associated with a branch of hyperbolic fixed points or equilibria in a one parameter family of analytic dynamical systems. We approximate the branch of invariant manifolds by polynomials and develop a-posteriori theorems which provide mathematically rigorous bounds on the truncation error. The hypotheses of these theorems are formulated in terms of certain inequalities which are checked via a finite number of calculations on a digital computer. By exploiting the analytic category we are able to obtain mathematically rigorous bounds on the jets of the manifolds, as well as on the derivatives of the manifolds with respect to the parameter. A number of example computations are given.

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“…(See, e.g., [8,9] for general numerical methods to compute normally hyperbolic invariant manifolds.) An alternative topological approach for validating the existence of invariant sets of normally hyperbolic type has been considered in [11], which is based on the method of covering relations [65]. These methods work for more general dynamical systems but cannot be used to prove the (local) uniqueness of the invariant sets.…”

Section: The Methodologymentioning

confidence: 99%

Reliable Computation of Robust Response Tori on the Verge of Breakdown

Figueras¹,

Haro²

2012

SIAM J. Appl. Dyn. Syst.

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Abstract. We prove the existence and local uniqueness of invariant tori on the verge of breakdown for two systems: the quasi-periodically driven logistic map and the quasi-periodically forced standard map. These systems exemplify two scenarios: the Heagy-Hammel route for the creation of strange nonchaotic attractors and the nonsmooth bifurcation of saddle invariant tori. Our proofs are computerassisted and are based on a tailored version of the Newton-Kantorovich theorem. The proofs cannot be performed using classical perturbation theory because the two scenarios are very far from the perturbative regime, and fundamental hypotheses such as reducibility or hyperbolicity either do not hold or are very close to failing. Our proofs are based on a reliable computation of the invariant tori and a careful study of their dynamical properties, leading to the rigorous validation of the numerical results with our novel computational techniques. Key words. normally hyperbolic invariant manifolds, computer validations, invariant tori, strange nonchaotic attractors AMS subject classifications. 37C55, 37D10, 65P99DOI. 10.1137/1008092221. Introduction. The goal of this paper is to present a new methodology to provide rigorous proofs of the existence and local uniqueness of (fiberwise hyperbolic) invariant tori in quasi-periodic systems, even in cases in which the available perturbative theory does not apply. The methodology is suitable for computer-assisted proofs and consists in checking the hypotheses of a validation result based on the Newton-Kantorovich theorem [27]. As an application of the methodology, we prove the existence and local uniqueness of invariant tori on the verge of breakdown in two scenarios: the Heagy-Hammel route to strange nonchaotic attractors (SNA) [32] in a quasi-periodically driven logistic map and the breakdown of saddle tori [26] in a quasi-periodically forced standard map.

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Covering relations and the existence of topologically normally hyperbolic invariant sets

Cited by 25 publications

References 0 publications

Geometric proof for normally hyperbolic invariant manifolds

Geometric proof for normally hyperbolic invariant manifolds

Polynomial approximation of one parameter families of (un)stable manifolds with rigorous computer assisted error bounds

Reliable Computation of Robust Response Tori on the Verge of Breakdown

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