Outer Billiards with the Dynamics of a Standard Shift on a Finite Number of Invariant Curves

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Outer Billiards with the Dynamics of a Standard Shift on a Finite Number of Invariant Curves. / Bialy, Misha; Mironov, Andrey E.; Shalom, Lior.

In: Experimental Mathematics, Vol. 30, No. 4, 2021, p. 469-474.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{51acc732011144e5abbb04f1735b68e4,

title = "Outer Billiards with the Dynamics of a Standard Shift on a Finite Number of Invariant Curves",

abstract = "We give a beautiful explicit example of a convex plane curve such that the outer billiard has a given finite number of invariant curves. Moreover, the dynamics on these curves is a standard shift. This example can be considered as an outer analog of the so-called Gutkin billiard tables. We test total integrability of these billiards, in the region between the two invariant curves. Next, we provide computer simulations on the dynamics in this region. At first glance, the dynamics looks regular but by magnifying the picture we see components of chaotic behavior near the hyperbolic periodic orbits. We believe this is a useful geometric example for coexistence of regular and chaotic behavior of twist maps.",

keywords = "chaotic behavior, Gutkin billiards, Outer billiards, total integrability",

author = "Misha Bialy and Mironov, {Andrey E.} and Lior Shalom",

note = "Funding Information: M.B. and L.S were supported in part by ISF grant 162/15 and A.E.M. was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation). It is our pleasure to thank these funds for the support. This work was started during the XXXVII Workshop on Geometric Methods in Physics, BIA?OWIE?A, POLAND. We would like to thank the organizers for this opportunity. Publisher Copyright: {\textcopyright} 2019 Taylor & Francis Group, LLC.",

year = "2021",

doi = "10.1080/10586458.2018.1563514",

language = "English",

volume = "30",

pages = "469--474",

journal = "Experimental Mathematics",

issn = "1058-6458",

publisher = "Taylor and Francis Inc.",

number = "4",

}

RIS

TY - JOUR

T1 - Outer Billiards with the Dynamics of a Standard Shift on a Finite Number of Invariant Curves

AU - Bialy, Misha

AU - Mironov, Andrey E.

AU - Shalom, Lior

N1 - Funding Information: M.B. and L.S were supported in part by ISF grant 162/15 and A.E.M. was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation). It is our pleasure to thank these funds for the support. This work was started during the XXXVII Workshop on Geometric Methods in Physics, BIA?OWIE?A, POLAND. We would like to thank the organizers for this opportunity. Publisher Copyright: © 2019 Taylor & Francis Group, LLC.

PY - 2021

Y1 - 2021

N2 - We give a beautiful explicit example of a convex plane curve such that the outer billiard has a given finite number of invariant curves. Moreover, the dynamics on these curves is a standard shift. This example can be considered as an outer analog of the so-called Gutkin billiard tables. We test total integrability of these billiards, in the region between the two invariant curves. Next, we provide computer simulations on the dynamics in this region. At first glance, the dynamics looks regular but by magnifying the picture we see components of chaotic behavior near the hyperbolic periodic orbits. We believe this is a useful geometric example for coexistence of regular and chaotic behavior of twist maps.

AB - We give a beautiful explicit example of a convex plane curve such that the outer billiard has a given finite number of invariant curves. Moreover, the dynamics on these curves is a standard shift. This example can be considered as an outer analog of the so-called Gutkin billiard tables. We test total integrability of these billiards, in the region between the two invariant curves. Next, we provide computer simulations on the dynamics in this region. At first glance, the dynamics looks regular but by magnifying the picture we see components of chaotic behavior near the hyperbolic periodic orbits. We believe this is a useful geometric example for coexistence of regular and chaotic behavior of twist maps.

KW - chaotic behavior

KW - Gutkin billiards

KW - Outer billiards

KW - total integrability

UR - http://www.scopus.com/inward/record.url?scp=85098426805&partnerID=8YFLogxK

U2 - 10.1080/10586458.2018.1563514

DO - 10.1080/10586458.2018.1563514

M3 - Article

AN - SCOPUS:85098426805

VL - 30

SP - 469

EP - 474

JO - Experimental Mathematics

JF - Experimental Mathematics

SN - 1058-6458

IS - 4

ER -

ID: 34952154