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О журнале История Журнала Издатель Адреса Тематика Редакторский состав Рецензирование Для авторов Издательская этика Коллекция номеров English version

Метод "перехода в пространство производных". 40 лет эволюции

Автор(ы):

Игорь Михайлович Буркин

профессор, заведующий кафедрой математического анализа
механико-математического факультета Тульского государственного университета,
доктор физ.-мат. наук.

i-burkin@yandex.ru

Аннотация:

В 1975 году был предложен "метод перехода в пространство производных" – эффективно проверяемый частотный критерий существования нетривиального периодического решения у многомерных моделей систем автоматического регулирования с одной дифференцируемой нелинейностью. Предложенный метод, использующий классический принцип тора, позволил, с одной стороны, вообще избежать каких-либо построений в фазовом пространстве исследуемой системы, а с другой – расширить класс систем, для исследования которых он может быть применен. В работе дан обзор результатов, представляющих обобщение и развитие этого метода. Продемонстрирована связь метода перехода в пространство производных с широко известным в настоящее время обобщенным принципом Пуанкаре-Бендиксона, предложенным R. A. Smith, а также результатами современных авторов, работающих в области теории колебаний в многомерных системах. Изложены полученные в последние годы автором статьи частотные критерии существования орбитально устойчивых циклов в многосвязных системах автоматического регулирования (MIMO системы), а также методы синтезирования многомерных систем, имеющих единственное состояние равновесия и обладающих любым наперед заданным числом орбитально устойчивых циклов. Приведено распространение обобщенного принципа Пуанкаре-Бендиксона на многомерные системы с угловой координатой. Показано применение излагаемых методов исследования колебательных процессов в многомерных динамических системах к решению известной задачи С.Смейла из теории химической кинетики биологических клеток, а также к поиску скрытых аттракторов обобщенной системы Чуа и минимального глобального аттрактора системы с полиномиальной нелинейностью. Изложение иллюстрируется многочисленными примерами.

Ключевые слова

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