TY - JOUR

T1 - On the Liouville Phenomenon in Estimates of Fractal Dimensions of Forced Quasi-Periodic Oscillations

AU - Anikushin, M. M.

N1 - Funding Information: This work was supported by the German-Russian Interdisciplinary Science Center (G-RISC) funded by the German Federal Foreign Office via the German Academic Exchange Service (DAAD) (projects M-2017a-5, M-2017b-9) and with the support of the Leading Scientific Schools of Russia (project NSh-2858.2018.1). Publisher Copyright: © 2019, Pleiades Publishing, Ltd. Copyright: Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019/7/1

Y1 - 2019/7/1

N2 - Abstract: The method for studying the fractal dimensions of forced almost periodic oscillations with various differential equations is described in this paper. The method is based on the concept previously introduced of the Diophantine dimension of an almost periodic function, which is closely related to the Diophantine approximations of its frequencies. Diophantine dimensions for some classes of quasi-periodic functions are estimated. The application of this method is demonstrated by the example of a single class of control systems studied by V. A. Yakubovich. As a result, one can observe a number theoretic phenomenon (the Liouville phenomenon), which does not make it possible to control the fractal dimension of forced oscillations with well-approximated frequencies.

AB - Abstract: The method for studying the fractal dimensions of forced almost periodic oscillations with various differential equations is described in this paper. The method is based on the concept previously introduced of the Diophantine dimension of an almost periodic function, which is closely related to the Diophantine approximations of its frequencies. Diophantine dimensions for some classes of quasi-periodic functions are estimated. The application of this method is demonstrated by the example of a single class of control systems studied by V. A. Yakubovich. As a result, one can observe a number theoretic phenomenon (the Liouville phenomenon), which does not make it possible to control the fractal dimension of forced oscillations with well-approximated frequencies.

KW - and fractal dimension

KW - dimension theory

KW - Diophantine approximation

KW - quasi-periodic function

UR - http://www.scopus.com/inward/record.url?scp=85067961641&partnerID=8YFLogxK

U2 - 10.1134/S1063454119030038

DO - 10.1134/S1063454119030038

M3 - Article

AN - SCOPUS:85067961641

VL - 52

SP - 234

EP - 243

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 3

ER -