A computer-assisted proof of symbolic dynamics in Hyperion’s rotation

Gierzkiewicz, Anna; Zgliczyński, Piotr

doi:10.1007/s10569-019-9910-8

Cited by 12 publications

(30 citation statements)

References 13 publications

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“…For full description of h-sets and their covering relations see [17]. In [5] one may also find a 2D simplified version, which may be useful for understanding the general idea.…”

Section: Topological Covering and Periodic Orbitsmentioning

confidence: 99%

A computer-assisted proof of the existence of Smale horseshoe for the folded-towel map

Gierzkiewicz

2021

Preprint

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The paper contains a rigorous proof of existence of symbolic dynamics chaos in the generalized Hénon map's 4th iterate H 4 , which was conjectured in the paper A 3D Smale Horseshoe in a Hyperchaotic Discrete-Time System of Li and Yang, 2007. We prove also the uniform hyperbolicity of the invariant set with symbolic dynamics. The proofs are computer-assisted with the use of C++ library CAPD for interval arithmetic, differentiation and integration.

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“…For full description of h-sets and their covering relations see [17]. In [5] one may also find a 2D simplified version, which may be useful for understanding the general idea.…”

Section: Topological Covering and Periodic Orbitsmentioning

confidence: 99%

A computer-assisted proof of the existence of Smale horseshoe for the folded-towel map

Gierzkiewicz

2021

Preprint

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“…The purpose of this paper is to show that, if the parameters are chosen about (1.20) and the energy is fixed to the level of a suitable initial datum pG ‹ , R ‹ , g ‹ , r ‹ q satisfying (1.16) (see Appendix B.1 for the exact values), then, in the system (1.14) a topological horseshoe wakes up in the plane pg, G{Λq. The analysis will be purely numerical, based on techniques developed in Gierzkiewicz and Zgliczyński (2019), Zgliczynski and Gidea (2004). More details on the methodological strategy are given along the following sections.…”

Section: Purpose Of the Papermentioning

confidence: 99%

“…In this section we prove the existence of symbolic dynamics for the considered model. The tools rely on ad-hoc covering relations that we present briefly following Gierzkiewicz and Zgliczyński (2019), in particular for the case n " 2.…”

Section: Symbolic Dynamics Via Covering Relationsmentioning

confidence: 99%

“…Finally, let SpN q l c " p´8, ´1q ˆR, SpN q r c " p1, 8q ˆR, and SpN q l " c ´1 N pSpN q l c q, SpN q r " c ´1 N pSpN q r c q be, respectively, the left and the right side of N . The general definition of covering relation can be found in Gierzkiewicz and Zgliczyński (2019). Here we provide a simplified notion, suited to the case that N is two-dimensional, based on 2 (Zgliczynski and Gidea, 2004, Theorem 16).…”

Section: Symbolic Dynamics Via Covering Relationsmentioning

confidence: 99%

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Symbolic dynamics in a binary asteroid system

Di Ruzza,

Daquin,

Pinzari

2020

Preprint

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We highlight the existence of a topological horseshoe arising from a a-priori stable model of the binary asteroid dynamics. The inspection is numerical and uses correctly aligned windows, as described in a recent paper by A. Gierzkiewicz and P. Zgliczyński, combined with a recent analysis of an associated secular problem. Contents 1. Purpose of the paper 1 2. Poincaré mapping 6 2.1. Hyperbolic structures and heteroclinic intersections 9 3. Symbolic dynamics via covering relations 10 3.1. Covering relations and topological horseshoe 10 3.2. Existence of a topological horseshoe 13 4. Conclusions and open problems 13 Acknowledgments 14 A. The Hamiltonian 15 B. Numerical setups and results 15 B.1. Choice of the parameters 15 B.2. Flow 15 B.3. Poincaré mapping P 16 B.4. Coordinates of the fixed-points of P 16 C. The Fast Lyapunov Indicator & dynamical timescales 17 References 18 This research is funded by the ERC project 677793 StableChaoticPlanetM. 1 See, e.g., Paez and Locatelli (2015) for a study based on a restricted model.

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“…(2) We apply the function covers2D, which checks the conditions from Definition 3 for the estimated images of the h-sets and their vertical edges; (3) If the conditions are fulfilled, then the function returns true. See also a similar method for 2-dimensional covering described in [6].…”

mentioning

confidence: 99%

Periodic orbits in Rössler system

Gierzkiewicz,

Zgliczyński

2021

Preprint

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A computer-assisted proof of symbolic dynamics in Hyperion’s rotation

Cited by 12 publications

References 13 publications

A computer-assisted proof of the existence of Smale horseshoe for the folded-towel map

A computer-assisted proof of the existence of Smale horseshoe for the folded-towel map

Symbolic dynamics in a binary asteroid system

Periodic orbits in Rössler system

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