Dynamics of a System of Two Simplest Oscillators with Finite Non-linear Feedbacks
In this paper, we consider a singularly perturbed system of two differential equations with delay which simulates two coupled oscillators with nonlinear feedback. Feedback function is assumed to be finite, piecewise continuous, and with a constant sign. In this paper, we prove the existence of relaxat...
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| Format: | Article |
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Yaroslavl State University
2016-12-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/418 |
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| author | A. A. Kashchenko |
| author_facet | A. A. Kashchenko |
| author_sort | A. A. Kashchenko |
| collection | DOAJ |
| description | In this paper, we consider a singularly perturbed system of two differential equations with delay which simulates two coupled oscillators with nonlinear feedback. Feedback function is assumed to be finite, piecewise continuous, and with a constant sign. In this paper, we prove the existence of relaxation periodic solutions and make conclusion about their stability. With the help of the special method of a large parameter we construct asymptotics of the solutions with the initial conditions of a certain class. On this asymptotics we build a special mapping, which in the main describes the dynamics of the original model. It is shown that the dynamics changes significantly with the decreasing of coupling coefficient: we have a stable homogeneous periodic solution if the coupling coefficient is of unity order, and with decreasing the coupling coefficient the dynamics become more complex, and it is described by a special mapping. It was shown that for small values of the coupling under certain values of the parameters several different stable relaxation periodic regimes coexist in the original problem. |
| format | Article |
| id | doaj-art-c4cdc432ffcc43a9bcff29fbb137ea13 |
| institution | DOAJ |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2016-12-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-c4cdc432ffcc43a9bcff29fbb137ea132025-08-20T03:01:14ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172016-12-0123684184910.18255/1818-1015-2016-6-841-849347Dynamics of a System of Two Simplest Oscillators with Finite Non-linear FeedbacksA. A. Kashchenko0P.G. Demidov Yaroslavl State UniversityIn this paper, we consider a singularly perturbed system of two differential equations with delay which simulates two coupled oscillators with nonlinear feedback. Feedback function is assumed to be finite, piecewise continuous, and with a constant sign. In this paper, we prove the existence of relaxation periodic solutions and make conclusion about their stability. With the help of the special method of a large parameter we construct asymptotics of the solutions with the initial conditions of a certain class. On this asymptotics we build a special mapping, which in the main describes the dynamics of the original model. It is shown that the dynamics changes significantly with the decreasing of coupling coefficient: we have a stable homogeneous periodic solution if the coupling coefficient is of unity order, and with decreasing the coupling coefficient the dynamics become more complex, and it is described by a special mapping. It was shown that for small values of the coupling under certain values of the parameters several different stable relaxation periodic regimes coexist in the original problem.https://www.mais-journal.ru/jour/article/view/418asymptoticsstabilitylarge parameterrelaxation oscillationperiodic solution |
| spellingShingle | A. A. Kashchenko Dynamics of a System of Two Simplest Oscillators with Finite Non-linear Feedbacks Моделирование и анализ информационных систем asymptotics stability large parameter relaxation oscillation periodic solution |
| title | Dynamics of a System of Two Simplest Oscillators with Finite Non-linear Feedbacks |
| title_full | Dynamics of a System of Two Simplest Oscillators with Finite Non-linear Feedbacks |
| title_fullStr | Dynamics of a System of Two Simplest Oscillators with Finite Non-linear Feedbacks |
| title_full_unstemmed | Dynamics of a System of Two Simplest Oscillators with Finite Non-linear Feedbacks |
| title_short | Dynamics of a System of Two Simplest Oscillators with Finite Non-linear Feedbacks |
| title_sort | dynamics of a system of two simplest oscillators with finite non linear feedbacks |
| topic | asymptotics stability large parameter relaxation oscillation periodic solution |
| url | https://www.mais-journal.ru/jour/article/view/418 |
| work_keys_str_mv | AT aakashchenko dynamicsofasystemoftwosimplestoscillatorswithfinitenonlinearfeedbacks |