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Dynamics of the boundary map of a system with spherical noise.

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Title: Dynamics of the boundary map of a system with spherical noise.
Authors: Pochinka, O. V.1 (AUTHOR) opochinka@hse.ru, Yagilev, A. A.1 (AUTHOR)
Source: Theoretical & Mathematical Physics. Jul2025, Vol. 224 Issue 1, p1271-1279. 9p.
Subjects: Random dynamical systems, Attractors (Mathematics), Set-valued maps, Diffeomorphisms, Dynamical systems, Invariant sets, Mathematical mappings
Abstract: Random dynamical systems with bounded noise are studied. In such systems, all trajectories are typically attracted to minimal sets, which are attractors. The problem of directly determining a minimal set is nontrivial, because one has to deal with a poorly investigated object, namely, with a set-valued map. However, there is an approach that allows reducing this problem to finding the invariant set of an ordinary discrete dynamical system, namely, of a boundary map. The minimal invariant sets are considered for the class of random dynamical systems consisting of invertible linear maps with bounded spherical noise. An exhaustive description of boundary maps is given in the case of typical linear contraction. It is found that the boundary map is then a Morse–Smale diffeomorphism whose global attractor uniquely determines the boundary of the minimal set of a random system. [ABSTRACT FROM AUTHOR]
Copyright of Theoretical & Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="JN" term="%22Theoretical+%26+Mathematical+Physics%22">Theoretical & Mathematical Physics</searchLink>. Jul2025, Vol. 224 Issue 1, p1271-1279. 9p.
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  Data: Random dynamical systems with bounded noise are studied. In such systems, all trajectories are typically attracted to minimal sets, which are attractors. The problem of directly determining a minimal set is nontrivial, because one has to deal with a poorly investigated object, namely, with a set-valued map. However, there is an approach that allows reducing this problem to finding the invariant set of an ordinary discrete dynamical system, namely, of a boundary map. The minimal invariant sets are considered for the class of random dynamical systems consisting of invertible linear maps with bounded spherical noise. An exhaustive description of boundary maps is given in the case of typical linear contraction. It is found that the boundary map is then a Morse–Smale diffeomorphism whose global attractor uniquely determines the boundary of the minimal set of a random system. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Theoretical & Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Text: English
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              Text: Jul2025
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