Analytical and numerical estimates of the fractal dimension of forced quasiperiodic oscillations in control systems

Standard

Analytical and numerical estimates of the fractal dimension of forced quasiperiodic oscillations in control systems. / Anikushin, Mikhail M.; Reitmann, Volker ; Romanov, Andrey O.

In: Differencialnie Uravnenia i Protsesy Upravlenia, Vol. 2019, No. 2, 2019, p. 162-183.

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

BibTeX

@article{e417eaf76b2f4b5cb0f3794d362abd0b,

title = "Analytical and numerical estimates of the fractal dimension of forced quasiperiodic oscillations in control systems",

abstract = "We consider a class of nonlinear feedback control systems with monotone nonlinearities and several stationary states. If the system is under an almost periodic perturbation, one can obtain conditions for existence of almost periodic oscillations. Our purpose is to estimate the fractal dimension of the trajectory closure of forced almost periodic oscillations obtained by the mentioned way. We show that within the result of I. M. Burkin and V. A. Yakubovich, which extends the result of M. A. Krasnoselskii et al. on the existence of exactly two almost periodic solutions (the stable one and the unstable one) in the case of two stationary states, it is possible to obtain some estimates of the fractal dimension. This estimate depends on some properties of Diophantine approximations for the frequencies of the almost periodic perturbation. We also apply a similar approach to study almost periodic oscillations in the perturbed Chua circuit, where the unperturbed system has three stationary states. We provide some analytical upper estimates of the fractal dimension and some numerical simulations conrming that upper estimates provided can be exact.",

keywords = "Control system, Dimension theory, Forced almost periodic oscillation, Fractal dimension",

author = "Anikushin, {Mikhail M.} and Volker Reitmann and Romanov, {Andrey O.}",

year = "2019",

language = "русский",

volume = "2019",

pages = "162--183",

journal = "ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ И ПРОЦЕССЫ УПРАВЛЕНИЯ",

issn = "1817-2172",

publisher = "Электронный журнал {"}Дифференциальные уравнения и процессы управления{"}",

number = "2",

}

RIS

TY - JOUR

T1 - Analytical and numerical estimates of the fractal dimension of forced quasiperiodic oscillations in control systems

AU - Anikushin, Mikhail M.

AU - Reitmann, Volker

AU - Romanov, Andrey O.

PY - 2019

Y1 - 2019

N2 - We consider a class of nonlinear feedback control systems with monotone nonlinearities and several stationary states. If the system is under an almost periodic perturbation, one can obtain conditions for existence of almost periodic oscillations. Our purpose is to estimate the fractal dimension of the trajectory closure of forced almost periodic oscillations obtained by the mentioned way. We show that within the result of I. M. Burkin and V. A. Yakubovich, which extends the result of M. A. Krasnoselskii et al. on the existence of exactly two almost periodic solutions (the stable one and the unstable one) in the case of two stationary states, it is possible to obtain some estimates of the fractal dimension. This estimate depends on some properties of Diophantine approximations for the frequencies of the almost periodic perturbation. We also apply a similar approach to study almost periodic oscillations in the perturbed Chua circuit, where the unperturbed system has three stationary states. We provide some analytical upper estimates of the fractal dimension and some numerical simulations conrming that upper estimates provided can be exact.

AB - We consider a class of nonlinear feedback control systems with monotone nonlinearities and several stationary states. If the system is under an almost periodic perturbation, one can obtain conditions for existence of almost periodic oscillations. Our purpose is to estimate the fractal dimension of the trajectory closure of forced almost periodic oscillations obtained by the mentioned way. We show that within the result of I. M. Burkin and V. A. Yakubovich, which extends the result of M. A. Krasnoselskii et al. on the existence of exactly two almost periodic solutions (the stable one and the unstable one) in the case of two stationary states, it is possible to obtain some estimates of the fractal dimension. This estimate depends on some properties of Diophantine approximations for the frequencies of the almost periodic perturbation. We also apply a similar approach to study almost periodic oscillations in the perturbed Chua circuit, where the unperturbed system has three stationary states. We provide some analytical upper estimates of the fractal dimension and some numerical simulations conrming that upper estimates provided can be exact.

KW - Control system

KW - Dimension theory

KW - Forced almost periodic oscillation

KW - Fractal dimension

UR - http://www.scopus.com/inward/record.url?scp=85069524422&partnerID=8YFLogxK

M3 - статья

AN - SCOPUS:85069524422

VL - 2019

SP - 162

EP - 183

JO - ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ И ПРОЦЕССЫ УПРАВЛЕНИЯ

JF - ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ И ПРОЦЕССЫ УПРАВЛЕНИЯ

SN - 1817-2172

IS - 2

ER -

ID: 73405726