A Family of Non-rough Cycles in a System of Two Coupled Delayed Generators
In this paper, we consider the nonlocal dynamics of the model of two coupled oscillators with delayed feedback. This model has the form of a system of two differential equations with delay. The feedback function is non-linear, finite and smooth. The main assumption in the problem is that the couplin...
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| Format: | Article |
| Language: | English |
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Yaroslavl State University
2017-10-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/586 |
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| author | Aleksandra A. Kashchenko |
| author_facet | Aleksandra A. Kashchenko |
| author_sort | Aleksandra A. Kashchenko |
| collection | DOAJ |
| description | In this paper, we consider the nonlocal dynamics of the model of two coupled oscillators with delayed feedback. This model has the form of a system of two differential equations with delay. The feedback function is non-linear, finite and smooth. The main assumption in the problem is that the coupling between the generators is sufficiently small. With the help of asymptotic methods we investigate the existence of relaxation periodic solutions of a given system. For this purpose, a special set is constructed in the phase space of the original system. Then we build an asymptotics of the solutions of the given system with initial conditions from this set. Using this asymptotics, a special mapping is constructed. Dynamics of this map describes the dynamics of the original problem in general. It is proved that all solutions of this mapping are non-rough cycles of period two. As a result, we formulate conditions for the coupling parameter such that the initial system has a two-parameter family of nonrough inhomogeneous relaxation periodic asymptotic (with respect to the residual) solutions. |
| format | Article |
| id | doaj-art-7df1460c470f4d1ebde3907ea7948d79 |
| institution | DOAJ |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2017-10-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-7df1460c470f4d1ebde3907ea7948d792025-08-20T03:01:14ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172017-10-0124564965410.18255/1818-1015-2017-5-649-654428A Family of Non-rough Cycles in a System of Two Coupled Delayed GeneratorsAleksandra A. Kashchenko0P.G. Demidov Yaroslavl State UniversityIn this paper, we consider the nonlocal dynamics of the model of two coupled oscillators with delayed feedback. This model has the form of a system of two differential equations with delay. The feedback function is non-linear, finite and smooth. The main assumption in the problem is that the coupling between the generators is sufficiently small. With the help of asymptotic methods we investigate the existence of relaxation periodic solutions of a given system. For this purpose, a special set is constructed in the phase space of the original system. Then we build an asymptotics of the solutions of the given system with initial conditions from this set. Using this asymptotics, a special mapping is constructed. Dynamics of this map describes the dynamics of the original problem in general. It is proved that all solutions of this mapping are non-rough cycles of period two. As a result, we formulate conditions for the coupling parameter such that the initial system has a two-parameter family of nonrough inhomogeneous relaxation periodic asymptotic (with respect to the residual) solutions.https://www.mais-journal.ru/jour/article/view/586large parameterrelaxation oscillationperiodic solutionasymptoticsdelay |
| spellingShingle | Aleksandra A. Kashchenko A Family of Non-rough Cycles in a System of Two Coupled Delayed Generators Моделирование и анализ информационных систем large parameter relaxation oscillation periodic solution asymptotics delay |
| title | A Family of Non-rough Cycles in a System of Two Coupled Delayed Generators |
| title_full | A Family of Non-rough Cycles in a System of Two Coupled Delayed Generators |
| title_fullStr | A Family of Non-rough Cycles in a System of Two Coupled Delayed Generators |
| title_full_unstemmed | A Family of Non-rough Cycles in a System of Two Coupled Delayed Generators |
| title_short | A Family of Non-rough Cycles in a System of Two Coupled Delayed Generators |
| title_sort | family of non rough cycles in a system of two coupled delayed generators |
| topic | large parameter relaxation oscillation periodic solution asymptotics delay |
| url | https://www.mais-journal.ru/jour/article/view/586 |
| work_keys_str_mv | AT aleksandraakashchenko afamilyofnonroughcyclesinasystemoftwocoupleddelayedgenerators AT aleksandraakashchenko familyofnonroughcyclesinasystemoftwocoupleddelayedgenerators |