Quasi-stability Method in Study of Asymptotic Behavior of Dynamical Systems
DOI:
https://doi.org/10.15407/mag15.04.448Ключові слова:
нескінченновимірні динамічні системи, асимптотичне поводження, глобальні атрактори, фрактальні експоненціальні атрактори, детермінуючи функціонали, фінітний фрактальний вимір, квазістійкість, стійкість, диференціальні рівняння з частинними похіднимиАнотація
В огляді здійснено спробу представити сучасні ідеї та методи дослідження якісної динаміки нескінченновимірних дисипативних систем. Представлено такі основні поняття, як дисипативність та асимптотична гладкість динамічних систем, глобальний та фрактальний атрактори, визначальні функціонали, регулярність асимптотичної динаміки. Акцент зроблено на методі квазістійкості, розробленому І. Чуєшовим та І. Лашецькою. Цей метод базується на відповідному розкладі різниці траєкторій на стійку та компактну частини. Існування такого розкладу має багато важливих наслідків: асимптотичну гладкість, існування та скінченновимірність атракторів, існування скінченної множини визначальних функціоналів та існування (за деяких додаткових умов) фрактального експоненціального атрактора. Решта статті ілюструє застосування абстрактної теорії до конкретних проблем. Основну увагу приділено демонстрації області застосування методу квазістійкості.Mathematics Subject Classification: 35-02, 35B40, 35B41, 37-02, 37L05, 37L30.
Посилання
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