Нелинейная динамика и гиперхаос в двух биофизических моделях связанных нелинейных осцилляторов тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Гаращук Иван Русланович

• Гаращук Иван Русланович
• кандидат наук
• 2023

Оглавление диссертации кандидат наук Гаращук Иван Русланович

1.2.1 Система уравнений

1.2.2 Разнообразие динамических режимов

1.2.3 Сценарий возникновения гинерхаотического аттрактора Шилышкова

1.2.4 Сценарий появления гинерхаотического аттрактора на основе разрушения синхронизации

1.2.5 Устойчивость к разрушению симметрии

1.3 Модель взаимодействующих нейронов

1.3.1 Модель двух взаимодействующих нейронов

1.3.2 Возбуждение системы из двух нейронов сигналом от внешнего нейрона

2 Заключение 31 Список литературы

3 Приложения

3.1 Приложение 1: Статья "Hyperehaos and multistabilitv in the model of two interacting mierobubble contrast agents"

3.2 Приложение 2: Статья "Synchronous oscillations and symmetry breaking in a model of two interacting ultrasound contrast agents"

3.3 Приложение 3: Статья "Bubbling transition as a mechanism of destruction of synchronous oscillations of identical mierobubble contrast agents"

3.4 Приложение 4: Статья "Asynchronous Chaos and Bifurcations in a Model of Two Coupled Identical Hindmarsh-Rose Neurons, Russian Journal of Xonlinear Dynamics"

3.5 Приложение 5: Статья "Excitation of a Group of Two Hindmarsh-Rose Neurons with a Xeuron-Cenerated Signal, Russian Journal of Xonlinear Dynamics"

Введение

Введение диссертации (часть автореферата) на тему «Нелинейная динамика и гиперхаос в двух биофизических моделях связанных нелинейных осцилляторов»

Актуальность

Динамические системы описывают процессы с детермепироваппым законом эволюции и возникают при моделировании разнообразных явлений во многих областях науки, таких как физика, химия и биология |1|,

В зависимости от требований конкретных приложений и сложности модели, её исследование может проводиться как с использованием аналитических, так и численных методов. Независимо от подхода, интерес представляет проведение бифуркационного анализа, т.о. изучения изменений качественных свойств поведения системы с изменениями управляющих параметров. В связи с этим при численном моделировании, как правило, проводится не только расчёт эволюции фазовых переменных, но и вычисление спектра показателей Ляпунова, который позволяет определить тин динамики на аттракторе |2; 3|. Например, присутствие в спектре положительного показателя Ляпунова можно использовать в качестве критерия хаотической динамики |4|, а построение карт ляпуиовских показателей но управляющим параметрам можно использовать в качестве инструмента при проведении бифуркационного анализа,

В данной работе изучаются две модели связанных нелинейных осцилляторов, возникающие в биофизических приложениях. Первой является модель двух взаимодействующих инкапсулированных газовых пузырьков в жидкости. Её исследование представляет интерес, т.к. такие пузырьки применяют в современной медицине в качестве контрастных агентов при проведении ультразвуковых исследований с цолыо визуализации некоторых органов или областей кровеносной системы, а также в задачах, связанных с направленной доставкой лекарственных препаратов |5—8|,

Отклик контрастных агентов формируется засчёт радиальных осцилляций пузырьков иод воздействием внешнего ультразвукового ноля. Нелинейность их колебаний приводит к тому, что возможны разнообразные режимы динамики, например субгармонические (но отношению к периоду ультразвукового ноля), квазинериодические или хаотические. В свою очередь, от режима колебаний пузырьков зависят свойства отклика, а также долгосрочная стабильность самих контрастных агентов. В зависимости от приложений, могут требоваться различные акустические свойства контрастных агентов. Например, считается, что дня задач, связанных с ультразвуковой визуализацией хорошо подходят субгармонические и хаотические режимы колебаний контрастных агентов, потому что спектр излучаемого ноля обратного рассеяния, хорошо подходит дня его эффективного распознавания, т.к. он резко отличается от спектра акустических волн, создаваемых окружающими тканями, колебания которых иод воздействием ультразвукового ноля близки к линейным |5; 9; 10|. Гинерхаотические режимы колебаний пузырьков также хорошо подходят дня эффективного распознавания контрастных агентов на фоне окружающих тканей, однако могут приводить к более быстрому разрыву оболочки и растворению пузырьков. С другой стороны, это свойство может быть полезным в некоторых задачах направленной доставки препаратов, в которых требуется быстрый контролируемый разрыв оболочки.

Таким образом, интерес представляет исследование режимов динамики колебаний контрастных агентов, т.к. они тесно связаны с их акустическими свойствами. В основе

моделей микропузырьковых контрастных агентов .нежит уравнение Рэлея-Плеееета, полученное дня описания сферических пузырьков газа в несжимаемой жидкости |11|, Дня формулировки корректной модели контрастных агентов, в уравнении необходимо учесть вязко-эластичную оболочку пузырьков. Существует ряд различных подходов к описанию оболочки, применимость которых к конкретным тинам контрастных агентов зависит от материала и толщины оболочки, необходимости учёта её разрыва при колебаниях большой амплитуды и т.д. (см. обсуждение в |9|, а также |5; 10; 12; 13|), В данной работе дня описания контрастных агентов Sono Vue с топкой .нинидпой оболочкой используется модель де-Йопга 112: 14|. Кроме того, в модели динамики пузырьков принимаются во внимание поправки па сжимаемость и вязкость жидкости |9|, Дня учёта сжимаемости использован подход Келлер-Микеиса, демонстрирующий наибольшую точность |15|, Поскольку в кровяной ноток вводится ансамбль контрастных агентов, акустические волны, излучаемые каждым из них, воздействуют па другие пузырьки. Влияние такого взаимодействия па колебания пузырьков описывается силой Бьёркпеса |16|, и может значительно изменять динамику колебаний пузырьков в кнастере |16—20|.

Вторым видом моделей, исследуемых в работе, является модель взаимодействующих нейронов. Возбудимость нейронов играет ключевую роль в передаче сигналов в нервной системе. Электрические механизмы возбуждения нервной клетки и генерации потенциала действия хорошо описываются моделью Ходжкииа-Хакели |21|, Модель Хиндмарш-Роуза, выводимая в рамках формализма Ходжкииа-Хакели, была предложена в работе |22| дня воспроизведения бёрстинговых паттернов активности потенциала нейронной мембраны, наблюдаемой в экспериментах. Она применяется дня описания возбудимых клеток, в которых такие режимы активности играют важную биологическую роль |23|, В организме человека такие клетки формируют сети взаимодействующих элементов, и прикладной интерес представляет моделирование их поведения. Более того, но современным представлениям некоторые специфические режимы динамики, такие как синхронная периодическая активность, в некоторых группах нейронов, ассоциируются с патологическим поведением |24|, и отыскание областей устойчивости таких режимов в пространстве параметров может играть важную роль.

Кроме того, изучение таких динамических явлений, как гинерхаотическая динамика и синхронизация, возникающих в данных моделях, интересно с токи зрения динамических систем. Динамика на аттракторе называется гииерхаотической, если присутствуют гиперболические неустойчивости размерности два или более. При численном исследовании это выражается в том, что два или более показателя Ляпунова являются положительными. Поскольку дивергенция фазового потока на аттракторе гладкой диссииативной динамической системы, соответствующая сумме показателей Ляпунова, должна быть отрицательной, гииерхаотические аттракторы могут наблюдаться только в динамических системах размерности четыре или выше.

Первый пример гииерхаотического аттрактора был найден в четырёхмерной системе Рёеелера |25|, В дальнейшем, присутствие гииерхаотических режимов наблюдалось в разнообразных динамических системах (см., например, |26—28|). Экспериментальное подтверждение гинерхаотического поведения в физических системах было получено в работах но исследованию динамики в электрических цепях |29|, лазерных системах с ядерным магнитным резонансом 1301 и германиевых полупроводниках р-тина |31|, Однако, бифур-

кациоппые сценарии появления гинерхаотических странных аттракторов в многомерных системах недостаточно исследованы |32|,

Другим интересным эффектом, возникающим в изучаемых моделях, является синхронизация колебаний в системах связанных нелинейных осцилляторов. Синхронные режимы имеют качественные отличия от асинхронных как с точки зрения динамики |33|, так и с точки зрения приложений. С точки зрения динамических систем представляет интерес изучение явления разрушения синхронизации, наблюдаемого в рассматриваемых моделях. Более того, в некоторых случаях, этот феномен связан с возникновением гинер-хаотических режимов динамики.

Степень разработанности проблемы

В области моделирования колебаний микронузырьковых контрастных агентов существует ряд работ но исследованию динамики отдельных пузырьков без учёта взаимодействия. Так, в работе |34| проведено детальное исследование бифуркационной структуры в модели пузырьков без оболочки. Более сложные модели динамики отдельных контрастных агентов с оболочкой были рассмотрены в работах |10; 35; 36|, Однако, в работе |36| при анализе бифуркаций, не была корректно учтена мультистабилыюсть в системе. Существует несколько работ, посвященных динамике двух или нескольких взаимодействующих пузырьков |17; 20; 37|, Однако, в работе |17| рассматривались пузырьки без оболочки. В то время как, в исследованиях 120; 37| использовалась некорректная модель оболочки (см. обсуждение в |9|), В работе |38| рассматривается большой кластер контрастных агентов, и дня понижения размерности применяется редукция, позволяющая эффективно искать синхронные режимы осцилляций пузырьков. Тем не менее, как показано в данной работе, синхронные режимы динамики могут быть трансвереалыю неустойчивы, или неустойчивы к возмущениям, разрушающим симметрию, а мультистабилыюсть является очень существенным фактором в динамике нескольких взаимодействующих контрастных агентов. С другой стороны, важный случай двух пузырьков с корректным учётом как оболочки, так и взаимодействия, не был исследован ранее. Рассмотрение такой системы является первым шагом к детальному моделированию поведения контрастных агентов в кластерах. Её изучение также необходимо дня исследования влияния силы взаимодействия на синхронизацию колебаний и возникновение мультистабильных состояний. В связи с чем, в диссертационной работе предлагается модель двух взаимодействующих контрастных агентов инканус.нированных в оболочки и проводится исследование динамики их колебаний.

Теперь остановимся на моделировании поведения отдельных и взаимодействующих нейронов. Динамика одного нейрона, описываемого системой Хиндмарш-Роуза, изучено достаточно подробно (см. работы 139—421), Ряд работ посвящен исследованию активности в ансамблях из двух и более элементов |43—■451, а также эффектам синхронизации в больших сетях |46; 47|, Однако, в моделях сетей взаимодействующих нейронов недостаточно подробно исследована динамика в малых ансамблях, в частности недостаточно изучен вопрос об устойчивости синхронных режимов, соответствующим режимам динамики отдельных нейронов. Помимо этого, интерес представляет исследование новых асинхронные режимов, возникающих в малых нейронных ансамблях. В связи с этим, в данной работе проводится исследование динамики и синхронизации в минимальном ансамбле из двух

взаимодействующих нейронов, а также рассматриваются механизмы возбудимости такой системы и проводится моделирование её возбуждения сигналом, поступающим от внешнего нейрона,

В области динамических систем давно известны примеры гинерхаотических аттракторов |25; 48|, Не смотря на это, до недавнего времени механизмы их возникновения подробно не изучались. Исключение составляют системы слабо связанных осцилляторов, в которых появление гинерхаотической динамики объяснялось возникновением хаотических аттракторов в каждой из подсистем |49; 50|, В работах |51; 52| авторы отмечали, что дня перехода от хаотической динамики к гинерхаотической необходимо включение множества еедловых орбит с двумерными неустойчивыми многообразиями в аттрактор. Однако, бифуркационные механизмы появления таких траекторий объяснены не были. Помимо этого, изучались сценарии возникновение гинерхаотических аттракторов в отображениях типа Энб |53| и отмечалась гинерхаотическая природа дискретных аттракторов Шилышкова |54; 55|, В связи с этим, важную роль играет изучение бифуркационных механизмов, .нежащих в основе возникновения гинерхаотической динамики в физических моделях связанных нелинейных осцилляторов, рассматриваемых в диссертационной работе.

Цели и задачи исследования

Цолыо работы является изучение динамики в двух прикладных системах связанных нелинейных осцилляторов и выявление новых бифуркационных сценариев возникновения гинерхаотических колебаний. Для достижения этой цели решены следующие задачи:

1. Формулировка математической модели двух газовых пузырьков при учёте их оболочек, вязкости и сжимаемости жидкости, взаимодействия посредством сипы Бьёрк-неса и внешнего ультразвукового ноля.

2. Разработка модификации алгоритма вычисления спектра показателей Ляпунова дня детального изучения изменений динамики в условиях разрушения синхронизации и реализация программного комплекса для численного исследования изучаемых моделей.

3. Исследование динамики в модели двух взаимодействующих контрастных агентов с учётом мультиетабилыюети и выявление новых сценариев возникновения гинерхао-тической динамики.

4. Исследование синхронизации в модели двух взаимодействующих контрастных агентов, устойчивости синхронных режимов и сценариев разрушения синхронизации.

5. Исследование динамики в модели двух взаимодействующих нейронов и проведение анализа устойчивости синхронных режимов колебаний.

6. Изучение механизмов возбудимости группы из двух связанных нейронов и моделирование возбуждения внешними сигналами, поступающими от отдельного нейрона.

Методы исследования

Для решения поставленных задач применялось сочетание численных и аналитических методов. Для поиска стационарных точек и проведения анализа устойчивости в линейном приближении использовались аналитические методы.

Для вычисления траекторий динамических систем применяются адаптивные методы с переменным шагом, подходящие для интегрирования жёстких систем. Для нахождения начальных условий при изменении управляющих параметров, используется метод наследования начальных данных, основанный на идее продолжимости по параметру. Использование данного подхода необходимо из-за мультиетабильноети, характерной для исследуемых систем.

Для установления типов динамики используется комбинация методов построения отображений Пуанкаре и вычисления спектра показателй Ляпунова, Для анализа бифуркаций строятся карты спектра показателей Ляпунова по управляющим параметрам. Для исследования трансверсальной устойчивости синхронных режимов используется метод вычисления старшего траневереального показателя Ляпунова, Также, для детального изучения динамики в процессе разрушения синхронизации и потери трансверсальной устойчивости в работе реализован метод раздельного вычисления спектра ляпуновеких показателей для синхронной и асинхронной компонент траектории. Для ускорения вычислений при построении карт показателей Ляпунова использовались методы параллельных вычислений. Для спектрального анализа численных решений применялось быстрое преобразование Фурье,

Научная новизна

• Предложена математическая модель, описывающая динамику двух газовых пузырьков в оболочке, взаимодействующих посредством силы Бьёркнеса, Построены двумерные и одномерные карты режимов динамики, исследованы бифуркационные сценарии переходов между различными типами динамических режимов,

• Выявлен новый сценарий возникновения гиперхаотических аттракторов в системах связанных осцилляторов с внешним воздействием, сопровождающийся вторичной бифуркацией Неймарка-Сакера и возникновением гомоклинического аттрактора Шильникова,

• Выявлен новый сценарий появления гиперхаотических аттракторов в системах связанных осцилляторов с внешним воздействием, связанный с разрушением синхронизации через «пузырьковый переход»,

• Найдены новые асинхронные режимы активности в модели, описывающей динамику двух взаимодействующих нейронов, и указаны области устойчивости синхронных режимов. Описаны механизмы возбудимости такой системы, и представлены результаты моделирования возбуждения этой системы нейронным сигналом.

Теоретическая значимость

Не смотря на то, что известны многочисленные примеры гиперхаотических аттракторов, в том числе в прикладных моделях, бифуркационные механизмы появления двумерных неуетойчивоетей были объяснены лишь в специфических случаях, В диссертационной работе выявлено два новых сценария возникновения гиперхаотической динамики в широком классе систем связанных осцилляторов с внешним воздействием. Предложенное объяснение бифуркационных механизмов является достаточно универсальным, и реализации данных сценариев были обнаружены в других системах. Также, была показана связь некоторых явлений, известных в теории синхронизации, с возникновением гиперхаотической динамики.

Практическая значимость

Не смотря на достаточно подробные исследования колебаний отдельных газовых пузырьков как без оболочки, так и с различными моделями оболочек, динамика в кластерах с учётом взаимодействия рассматривалась лишь в специфических случаях, а подробные исследования динамики в малых ансамблях с корректным учётом оболочки практически отсутствуют, В диссертационной работе предложена модель не изученного ранее случая двух взаимодействующих контрастных агентов в оболочках, и проведено подробное исследование динамики с учётом мультиетабильноети. Были обнаружены новые режимы, не наблюдаемые ранее в моделях контрастных агентов, и построены карты режимов динамики, показывающие связь амплитуды внешнего поля давлений и расстояния между пузырьками с режимами колебаний,

В рамках моделей связанных нейронов было проведено исследование динамики в полносвязной группе из двух элементов. Была рассмотрена устойчивость синхронных режимов и возникновение асинхронной динамики. Изучены механизмы возбудимости такой системы и проведено моделирование её возбуждения внешними сигналами, В качестве генератора естественных для нейронных систем возбуждающих сигналов использовался отдельный нейрон.

Результаты, выносимые на защиту

1, Предложена математическая модель двух взаимодействующих газовых пузырьков с учётом их оболочек, В рамках данной модели построены одномерные и двумерные карты режимов динамики в физически релевантном диапазоне управляющих параметров с учётом мультиетабильноети, В пространстве параметров найдены области существования синхронных колебаний. Описаны бифуркационные сценарии возникновения регулярных, хаотических и гиперхаотических режимов динамики,

2, Выявлен сценарий возникновения гиперхаотической динамики, ключевыми этапами которого являются вторичная бифуркация Неймарка-Сакера и появление гомокли-нического аттрактора Шильникова при включении седло-фокусных орбит с двумерными неустойчивыми многообразиями в хаотический аттрактор. Численно подтвер-

ждена реализация предложенного сценария в модели взаимодействующих микропузырьковых контрастных агентов,

3, Выявлен сценарий возникновения гиперхаотического аттрактора, в основе которого лежит разрушение синхронизации. Установлено, что механизмом появления тране-вереапьно неустойчивых областей в синхронном хаотическом аттракторе является каскад бифуркаций, известный, как «пузырьковый переход». Реализация данного сценария в модели взаимодействующих микропузырьковых контрастных агентов подтверждена численно с помощью разработанной модификации алгоритма вычисления спектра показателей Ляпунова,

4, В модели двух взаимодействующих нейронов, описываемых системой Хиндмарш-Роуза, построены одномерные карты режимов динамики и описаны паттерны нейронной активности. Установлены области траневереальной устойчивости синхронных режимов и сценарий возникновения асинхронного хаотического аттрактора. Найдены области бистабильности. Изучены бифуркационные механизмы возбудимости такой системы и предложена модель её возбуждения сигналом, создаваемый поступающим от внешнего нейрона, В области устойчивости синхронного положения равновесия построены карты возбуждения различными сигналами.

Достоверность результатов

Достоверность результатов математического моделирования подтверждается тщательным тестированием численных алгоритмов, реализованных в рамках комплекса программ, на известных данных, результатах, полученных на других программных комплексах, и аналитических решениях. Основные результаты диссертационного исследования были апробированы на международных научных конференциях и опубликованы в рецензируемых научных журналах. Все результаты, представленные к защите, опубликованы в научных журналах, индексируемых в научных базах Web of Science или Scopus, Три статьи опубликованы в журналах квартиля Q1, и две работы в журналах квартиля Q3,

Апробация результатов исследования

Основные результаты диссертации докладывались на следующих международных конференциях:

1, Topological methods in dynamics and related topics, «Nonlinear dynamics and typical bifurcations in the model of three coupled ultrasound contrast agents», май 2022,

2, Shilnikov Workshop, «Asynchronous chaos and bifurcations in a model of two coupled Hindmarsh-Eose neurons», декабрь 2021,

3, SIAM Conference on Applications of Dynamical Systems, «Hvperehaos and Synchronization in a Model of Two Interacting Encapsulated Mierobubbles», май 2021,

4, International Conference 'Topological Methods in Dynamics and Related Topics, Shilnikov Workshop,', «Synchronization and symmetry breaking in a model of two interacting ultrasound contrast agents», декабрь 2020,

5, International Conference 'Topological Methods in Dynamics and Related Topics, Shilnikov Workshop,', «Symmetry breaking in a system of two coupled microbubble contrast agents», декабрь 2019,

6, International Conference 'Shilnikov Workshop', «Multistabilitv and Hvperehaos in the Dynamics of Two Coupled Contrast Agents», декабрь 2018,

Результаты, полученные в ходе работы над диссертацией использовались в следующих научных проектах:

• 2020-2022, Грант РФФИ 20-31-90122 для аспирантов «Гиперхаос и механизмы его возникновения, мультиетабильноеть и синхронизация в моделях взаимодействующих нейронов»,

• 2019-2022, Министерства науки и высшего образования РФ соглашение JV2 075-152019-1931.

• 2019-2022, Грант РИФ 19-71-10048 «Теория гиперхаоса и ее приложение к задачам биомедицины».

Список статей, представленных к защите по теме диссертации

Основные положения по теме диссертации изложены в 5 публикациях, проиндексированных в международной системе Scopus:

[1*] Garashehuk I.E., Sinelshchikov D.I, Kazakov А,О,, Kudrvashov N.A., Hyperchaos and multistability in the model of two interacting 'microbubble contrast agents, Chaos (2019), 29,1199-1213.

[2*] Garashehuk I.E., Sinelshchikov D.I, Kazakov А.О., Synchronous oscillations and symmetry breaking in a model of two interacting ultrasound contrast agents, Nonlinear Dynamics (2020), 101, 1199-1213.

[3*] Garashehuk I.E., Sinelshchikov D.I, Bubbling transition as a mechanism of destruction of synchronous oscillations of identical 'microbubble contrast agents, Chaos (2021), 31, 023130.

[4*] Garashehuk I.E., Asynchronous Chaos and Bifurcations in a Model of Two Coupled Identical Hindmarsh-Rose Neurons, Eussian Journal of Nonlinear Dynamics (2021), 17(3), 307-320.

[5*] Garashehuk I.E., . Sinelshchikov D.I, Excitation of a Group of Two Hindmarsh-Rose Neurons with a Neuron-Generated Signal, Eussian Journal of Nonlinear Dynamics (2023), 19(1), 19-34.

1 Краткое содержание работы: основные результаты

1.1 Основные понятия и методы исследования

Рассмотрим гладкую динамическую систему, задаваемую системой дифференциальных уравнений, правая часть которой зависит от управляющих параметров

x = f (x, а), (1)

где x = (xi,..., xn) - фазовые переменные, определённые в некоторой области D Ç Mn, f (x) = (f1,..., fn), f1,..., fn G Cr (D) r > 1 а G Mm - m управляющих параметров, от которых зависит правая часть. Напомним, что аттрактором такой динамической системы называют замкнутое, ограниченное иоложителыю-ишзариаитиое локально притягивающее множество K С D Ç Rn [ ],

Поскольку мультиетабилыюеть является неотъемлемым свойством рассматриваемых систем в физически релевантных областях значений управляющих параметров, в работе используется процедура получения начальных условий при изменении управляющих параметров. Предположим, что при некотором значении параметров а = а(0) существует два (или более) аттрактора K1,K2,..., Для исследования изменений интересующего аттрактора при переходе к значению параметра а = а(1), рассмотрим гомотопию : а(0) ^ G [0,1] [ ], В численной реализации, этот одно-параметрический марш-

рут в пространство параметров представляется дискретной последовательностью значений а(0), а(1),..., а(1), Для анализа изменений аттрактора K1 вдоль маршрута недостаточно использовать фиксированные начальные условия x0° и интегрировать систему до окончания переходного процесса па каждом шаге по параметру, т.к. при некотором значении параметра а на этом маршруте, x0° может оказаться в бассейне притяжения аттрактора K2, Если в точке а = а(:?-1), x0° лежит в бассейне притяжения аттрактора K1, а в точке а = а^ - в бассейне аттрактора K2, то при переходе от а(:?'-1) к а = а^, будет наблюдаться скачок в спектре показателей Ляпунова, из-за резкого изменения динамики системы при переходе на другой аттрактор, хотя в этой точке не происходит бифуркаций, связанных

K1

наследования начальных условий, основанный на идее продолжимости но параметру. На каждом шаге по параметру, последнее значение x(t) на аттрактope K1 выбирается в качестве новых начальных условий x0+1, которые будут использованы при следующем значении параметра а(г+1). Если шаг по параметру а достаточно мал, это позволяет избегать перехода начальной точки в фазовом пространстве через сепаратрису.

При численном исследовании, основным способом определения типа динамики на аттракторе в данной работе является вычисление спектра показателей Ляпунова, Они характеризуют рост малых возмущений в линейном приближении |3|, Пусть гладкая траектория x(t) С Rn динамической системы (1) начинается в точке x(0) = x0 G Rn, Тогда возмущённую траекторию можно представить в виде x(t) = x(t)+5x(t), оде ¿x(t) : Vt > 0, ||5x(t)|| ^ 1 описывает эволюцию малого возмущения, и возмущённой траектории соответствуют начальные условия x(0) = x0 + ¿x0,5x0 G Rn : ||5x01| ^ 1, Уравнение (1) для возмущённой траектории принимает вид x(t) = f (x + ¿x), раскладывая правую часть которого в ряд Тейлора, получаем, что в линейном приближение эволюция возмущения описывается ли-

пейпой системой уравнений

Sx (t) = JSx, (2)

где J = ,i = 1... n, j = 1... n - матрица Якоби на невозмущённой траектории x(t). Показатели Ляпунова определяются, как верхний продол

Список литературы диссертационного исследования кандидат наук Гаращук Иван Русланович, 2023 год

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3 Приложения

3.1 Приложение 1: Статья "Hyperchaos and multistability in the model of two interacting microbubble contrast agents"

Статья 1.

Garashehuk I.R., Sinelshehikov D.I, Kazakov A.O., Kudryashov X.A., Hyperchaos and multistability in the model of two interacting microbubble contrast agents, Chaos (2019), 29, 1199-1213.

https://aip.scitation.org/doi/10.1063/1.5098329

Hyperchaos and multistability in the model of two interacting microbubble contrast

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Cite as: Chaos 29, 063131 (2019); https://doi.org/10.1063A5098329 Submitted: 01 April 2019 . Accepted: 11 June 2019 . Published Online: 28 June 2019

Ivan R. Garashchuk Dmitry I. Sinelshchikov Alexey O. Kazakov and Nikolay A. Kudryashov o COLLECTIONS

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Chaos 29, 063131 (2019); https://doi.org/10.1063A5098329 © 2019 Author(s).

Hyperchaos and multistability in the model of two interacting microbubble contrast agents

Cite as: Chaos 29,063131 (2019); doi: 10.1063/1.5098329 Submitted: 1 April 2019 • Accepted: 11 June 2019 • Published Online: 28 June 2019

Ivan R. Garashchuk,1a) Dmitry I. Sinelshchikov, ■ © Alexey O. Kazakov, ■ © and NikolayA. Kudryashov ■

AFFILIATIONS

1 National Research Nuclear University MEPhI, 31 Kashirskoe sh., 115409 Moscow, Russia

2National Research University Higher School of Economics, 34 Tallinskaya Ulitsa, 123592 Moscow, Russia

3National Research University Higher School of Economics, 25/12 Bolshaya Pecherskaya Ulitsa, 603155 Nizhny Novgorod, Russia

a)Electronic mail: [e-mail]

b)Electronic mail: [e-mail]

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d)Electronic mail: [e-mail]

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ABSTRACT

We study nonlinear dynamics of two coupled contrast agents that are micrometer size gas bubbles encapsulated into a viscoelastic shell. Such bubbles are used for enhancing ultrasound visualization of blood flow and have other promising applications like targeted drug delivery and noninvasive therapy. Here, we consider a model of two such bubbles interacting via the Bjerknes force and exposed to an external ultrasound field. We demonstrate that in this five-dimensional nonlinear dynamical system, various types of complex dynamics can occur, namely, we observe periodic, quasiperiodic, chaotic, and hypechaotic oscillations of bubbles. We study the bifurcation scenarios leading to the onset of both chaotic and hyperchaotic oscillations. We show that chaotic attractors in the considered system can appear via either the Feigenbaum cascade of period-doubling bifurcations or the Afraimovich-Shilnikov scenario of torus destruction. For the onset of hyperchaotic dynamics, we propose a new bifurcation scenario, which is based on the appearance of a homoclinic chaotic attractor containing a saddle-focus periodic orbit with its two-dimensional unstable manifold. Finally, we demonstrate that the dynamics of two bubbles can be essentially multistable, i.e., various combinations of the coexistence of the above mentioned attractors are possible in this model. These cases include the coexistence of a hyperchaotic regime with an attractor of any other remaining type. Thus, the model of two coupled gas bubbles provides a new example of physically relevant system with multistable hyperchaos.

Published under license by AIP Publishing. https://doi.org/10.1063/L5098329

We study nonlinear dynamics of two identical micrometer size gas bubbles interacting via the Bjerknes force and exposed to an external ultrasound periodic pressure field. Such bubbles are currently used as contrast agents for ultrasound visualization and there are also some other promising applications of the bubbles such as targeted drug delivery and noninvasive therapy. Various complex regimes can occur in the system describing dynamics of gas bubbles. First, we show that oscillations of bubbles can be periodic, quasiperiodic, chaotic, and even hyperchaotic. We study routes to the onset of both chaotic and hyperchaotic attrac-tors. We demonstrate that chaotic attractors appear via either the Feigenbaum cascade of period-doubling bifurcations or the Afraimovich-Shilnikov scenario of the destruction of invariant tori. For the onset of hyperchaotic regimes, we propose a new phenomenological scenario which is based on the appearance

of a discrete Shilnikov attractor containing a saddle-focus periodic orbit with its two-dimensional unstable manifold. Orbits in such attractors can pass arbitrarily close to this saddle-focus orbit, where two-dimensional areas are expanded. As a result, two Lyapunov exponents become positive, i.e., a hyperchaotic attrac-tor is born. We suppose that this scenario may be typical also for other multidimensional systems demonstrating transition to hyperchaos via the destruction of an invariant torus. We also show that the dynamics in the system under consideration can be essentially multistable. Regular, chaotic, and hyperchaotic attrac-tors corresponding to asynchronous oscillations of bubbles can coexist with regular and chaotic synchronous regimes. We believe that quasiperiodic and hyperchaotic oscillations and also mul-tistability phenomenon in the dynamics of coupled bubbles are studied for the first time in this work. Finally, we discuss possible

connections of our results with biomedical applications of microbubbles.

I. INTRODUCTION

In this work, we study nonlinear dynamics of two coupled identical encapsulated gas bubbles in a liquid, which are driven by an external periodic pressure field. Investigation of oscillations of such bubbles is of interest due to their applications as contrast agents for ultrasound visualization - and future possible applications for noninvasive therapy and targeted drug delivery.4,5 Depending on applications, different types of bubbles' dynamics can be either beneficial or undesirable (see, e.g., Refs. 3 and 6). Therefore, it is important to study the variety of possible dynamical regimes and how the dynamics of the bubbles depend on both control parameters and initial conditions.

Typically, mathematical models of a single microbubble contrast agent are one-dimensional nonautonomous oscillators based on the Rayleigh-Plesset equation and its generalizations (see, e.g., Refs. 7 and 8 and references therein). Later, these models were extended to the coupled Rayleigh-Plesset equations which take into account bubble-bubble interactions via the Bjerknes force. - From a mathematical point of view, these models are described by systems of coupled nonlinear oscillators, with external periodic force, and, thus, various types of dynamics can be observed in them. Despite the great interest, there are only a few works devoted to studying nonlinear dynamics of gas bubbles. For example, nonlinear dynamics of a single bubble described by one of the Rayleigh-Plesset-like models was studied in Refs. 6 and 14-18, where it was shown that oscillations of a single bubble can be either regular or chaotic and routes to the corresponding attractors were studied. Some bifurcations of two and N coupled bubbles were studied in Refs. 9, 13, and 16. However, in Ref. 9, unencapsulated bubbles were considered, while in Refs. 13 and 6, an inappropriate model of the bubbles' shell was investigated (see discussion in Ref. 8).

After some simple transformations, the dynamics of two coupled bubbles is described by a system of five ordinary differential equations. In this work, we show that, in addition to regular and chaotic regimes which are typical for models of one bubble, ■ - the system exhibits quasiperiodic and, what is more interesting, hyper-chaotic types of motion, which, to the best of out knowledge, have not been previously observed in the models of gas bubbles.

Hyperchaotic behavior of trajectories is characterized by the presence of at least two directions of hyperbolic instability in an attractor. Such a phenomenon was observed for the first time by Rossler in Ref. 19 in a four-dimensional dynamical system. Later, hyperchaotic attractors were numerically found in dynamical systems from various applications (see, e.g., 20-26). Despite the fact that the hyperchaos phenomenon has been known for quite a long time, the mechanisms of emergence of hyperchaotic attractors as well as their properties have not been studied except for some special cases. For instance, in Ref. 27, the emergency of hyperchaotic attrac-tors in the system of two weakly coupled oscillators was explained by the transition to chaos via the cascade of period-doubling bifurcations in each oscillator, while in Ref. 28 for a specific family of two-dimensional endomorphisms, the authors proposed a scenario

for the onset of hyperchaos via a cascade of saddle-node and period-doubling bifurcations, which lead to the emergence of completely unstable periodic orbits inside a strange attractor.

In this paper, we propose a universal bifurcation scenario for the onset of hyperchaotic behavior with two positive Lyapunov exponents. The key part of this scenario is the appearance of a "homoclinic chaotic attractor" containing a saddle-focus periodic orbit with a two-dimensional unstable manifold, i.e., such an orbit which has a pair of complex conjugated multipliers with positive real parts while all the other multipliers have negative real part. Recall that the chaotic attractor is called homoclinic, if it contains a "selected" saddle fixed point29 together with its unstable invariant manifold. This notion can be naturally extended to homoclinic attractors with a selected periodic saddle orbit.

Trajectories on a homoclinic attractor can pass arbitrarily close to the selected saddle orbit belonging to it. The dynamics near this saddle orbit and, as a result, on the entire homoclinic attrac-tor significantly depends on the multipliers of the corresponding saddle orbit.30 In particular, in a small neighborhood of a homo-clinic attractor containing a saddle-focus periodic orbit with a two-dimensional unstable manifold, two-dimensional areas are expanded and Lyapunov exponents on the entire attractor "can feel" this expansion. As a result, two Lypaunov exponents become positive. Note that homoclinic attractors of this type were called "discrete Shilnikov attractors" in Refs. 29-31. This terminology comes from Ref. 32 where Shilnikov proposed a universal bifurcation scenario leading to the birth of a spiral attractor containing a saddle-focus equilibrium together with its two-dimensional unstable manifold for one-parametric families of three-dimensional flow systems. In Refs. 30 and 31, this scenario was transferred to the case of one-parametric families of three-dimensional diffeomorphisms (or four-dimensional flows, if the corresponding Poincare cross section is considered).

Another interesting property of the considered system is its multistability. It has recently been shown17,18 that the dynamics of a single encapsulated bubble can be multistable,33 i.e., several attractors can coexist at the same values of parameters. Thus, it is interesting to understand whether multistability persists in the dynamics of coupled bubbles or even more new multistable regimes can occur. We show that the dynamics of coupled bubbles is essentially multi-stable and various attractors can coexist. In particular, a hyperchaotic attractor can coexist with a synchronous chaotic one. Thus, we demonstrate the possibility of the existence of multistable hyperchaos in the dynamics of two interacting gas bubbles, i.e., the considered model is a new example of a physically relevant dynamical system with multistable hyperchaos.

Lastly, we would like to note that although in reality there are numerous interbubble interactions within an ensemble of microbub-bles, we believe that the consideration of two interacting microbub-bles sheds light on the dynamics of the whole ensemble. First, the model of two interacting bubbles can be considered as an approximation of the behavior of an average pair of the closest to each other bubbles within the ensemble. Since the influence of the further located bubbles quickly decreases with the distance between bubbles, it can be neglected in a first approximation. Therefore, we can consider oscillations of a bubbles pair as average behavior in the ensemble, and the overall acoustic response of the ensemble is

formed as an average response of a set of such pairs. Second, the consideration of a pair of interacting gas bubbles is the first step toward the understanding of the impact of the interbubble interactions within an ensemble on its dynamics. In fact, this approximation shows that this interaction has a considerable influence on bubbles' dynamics. Third, most of the existing studies deal with an isolated bubble neglecting the bubble-bubble interactions in the ensemble (see, e.g.,Refs. 6,14,15, and 17 and references therein) and in current work, we attempt to carry out a detailed study of the influence of the interbubble interactions on their dynamics. Finally, the investigation of the dynamics of larger sets of interacting bubbles is not the aim of this work and should be considered elsewhere.

The rest of this work is organized as follows. In Sec. II, we present the governing system of equations for the dynamics of two coupled bubbles and discuss some of its properties. In Sec. III, we present a two-parametric chart of Lyapunov exponents for the considered system and discuss possible types of dynamics. In Sec. IV, we focus on scenarios leading to the onset of both chaotic and hyper-chaotic oscillations of coupled bubbles. We demonstrate that chaotic attractors can appear via either the Feigenbaum cascade of period-doubling bifurcations or the Afraimovich-Shilnikov scenario of the destruction of invariant tori. We propose a new phenomenological scenario for the onset of hyperchaotic oscillations as well. In Sec. V, we discuss a possibility of the coexistence of several attractors in the systems under consideration and point out that chaotic attractor can coexist with hyperchaotic one. In Sec. VI, we briefly discuss our results emphasizing that the proposed scenario of the onset of hyper-chaotic attractors should also be typical for other multidimensional systems demonstrating hyperchaotic behavior.

II. MAIN SYSTEM OF EQUATIONS

In this section, we consider a model for the description of oscillations of two interacting gas bubbles in a liquid. Essentially, this model is formed by two generalized Raleigh-Plesset equations that are coupled via the Bjerknes forces (see, e.g., Refs. 9-13 and 16). In this work, we take into account liquid's compressibility in accordance with the Keller-Miksis model, liquid's viscosity on the gas-liquid interface, surface tension, and the impact of bubbles' shells, which is described by the de-Jong model.35,36 We also suppose that bubbles are exposed to the external periodic pressure field. Under these assumptions, the governing system of equations for oscillations of two coupled bubbles takes the following form (see also Fig. 1):

1 - 7 R'Rl + 31 - 7 R2

R1 R1 d

1 + + Tt e e dt

P1

d /R2R2

dt d

(1)

1 - 7) R2R2 + K1 - I' R 2

R2 R2 d 1 + + ~Tt e e dt

d

P2

r2R 1

dt d

FIG. 1. Schematic picture of two interacting bubbles oscillating in a liquid under the influence of an external pressure field.

where

2a

P> = Po + ^ ^ -

Rh

R¡

3y

4vlR¡ 2a

R¡

- ^T - Po - 4X ñ--—

R¡

Ri0

R ¡

- 4ks-2 - Pae Sin(«t), i = 1, 2. R;

Here, t is the time, dots denote derivatives with respect to t, R1(t) and R2(t) are radii of bubbles, d is the distance between the centers of bubbles, Pstat is the static pressure, Pv is the vapor pressure, P0 = Pstat — Pv, Pac is the magnitude of the external pressure field, and w is its cyclic frequency, a is the surface tension, p is the density of the liquid, nL is the viscosity of the liquid, c is the speed of sound in the liquid, y is the polytropic exponent, and x and ks denote the shell elasticity and shell surface viscosity, respectively.

It can be easily seen that by means of simple transformations, Eq. (1) can be rewritten in the form of a five-dimensional system of ordinary differential equations in terms of the following dependent variables R1, R2, R1, R2, and 9 e [0,2n]. We perform all our numerical experiments exactly with this five-dimensional system. However, due to its cumbersome form, we do not present it here.

In what follows, we assume that Pac, w, and d are the control parameters and the remaining parameters are fixed as follows: Pv = 2.33 kPa, a = 0.0725 N/m, p = 1000 kg/m3, nL = 0.001 Ns/m3, c = 1500 m/s, y = 4/3, x = 0.22 N/m, and ks = 2.5 • 10—9 kg/s. These values of the parameters correspond to the adiabatic oscillations of two interacting SonoVue contrast agents with equilibrium radii Ri0 = 1.72 ^m, i = 1,2.37

We suppose that the equilibrium radii of bubbles are the same (i.e., R10 = R20 = R0), because the injected ensemble of contrast agents is assumed to consist of bubbles of the same characteristics. Thus, system (1) is symmetric with respect to the following change of variables:

R1 ^ R2, R1 ^ R2.

(2)

This symmetry leads to several conclusions. First, there always exists a family of symmetrical solutions, for which Vt > 0, R1 (t) = R2(t), if R1(t = 0) = R2(t = 0) and R 1(t = 0) = R2(t = 0), that corresponds to fully synchronous in-phase oscillations of bubbles. Some of these solutions can be asymptotically stable (attractive). Since all symmetrical solutions lie in the invariant manifold R1 = R2, R1 = R2 and system (1) restricted to this manifold is three-dimensional and volume-contracting, such regimes can be of two possible types: periodic and simply chaotic (with only one positive

Lyapunov exponent). In other words, synchronous oscillations of two bubbles can be either periodic or chaotic (for more details, see Sec. III).

Second, asymptotically stable regimes (attractors) can exist outside the invariant manifold R1 = R2, R1 = R2. These regimes correspond to asynchronous oscillations of bubbles and, in contrast to synchronous ones, they can be of four possible types. In addition to periodic and simply chaotic regimes, asynchronous oscillations can also be quasiperiodic and even hyperchaotic. Moreover, the existence of such asymmetrical regimes leads to the trivial multistability in the system. Indeed, for each non-self-symmetrical asynchronous regime passing through point (R1, R2, R1, R2), there exists the symmetrical one passing through (R2, R1, R2, R1). In Sec. V, we show that in addition to this simple multistability, system (1) exhibits more complex types of this phenomenon. In this work, we discuss the coexistence of different attractors at the same values of the control parameters, which cannot be obtained via transformation (2). Below, we will refer to the term "multistability" describing a situation of the coexistence of several substantially different attractors that are not simply symmetrical with respect to the manifold R1 = R2, R1 = R2. Notice also that the coexistence of two synchronous periodic attractors both lying in the manifold R1 = R2, R1 = R2 is possible.

We perform all calculations in the following nondimen-sional variables: R; = R0r;, t = , where = 3kP0/(pR0) + 2(3k — 1)ct/R0 + 4x/R0 is the natural frequency of bubbles' oscillations. The nondimensional bubbles' speeds are given by U = dri/dr = Rj/(R0<i>0). We use the fourth-fifth order Runge-Kutta method38 for finding numerical solutions of the Cauchy problem for the considered system. For calculations of the Lyapunov spectra, we use the standard algorithm by Bennetin et a/.39 Notice that throughout this work, the Lyapunov exponents are computed for the autonomous five-dimensional system on a five-dimensional cylinder. On all the graphs of Lyapunov exponents presented below, we provide three exponents: X1,k2,kf, where X1 and k2 are the two largest exponents and kf = 0 is the referent exponent which is always zero and corresponds to the translations along an orbite. The Poincare map is constructed by taking values of phase variables at t = kT, T = 2n/«, k e N, where T is the period ofthe external pressure field.

III. VARIETY OF DYNAMICAL REGIMES IN THE MODEL

In this section, we demonstrate the diversity ofpossible dynamical regimes in system (1) and show that depending on the values of the control parameters d/R0 and Pac, two bubbles can exhibit periodic, quasiperiodic, chaotic, or hyperchaotic oscillations. Moreover, due to symmetry (2) of the system, periodic and chaotic regimes can be either synchronous or asynchronous. Hyperchaotic and quasiperiodic oscillations cannot be synchronous.

Figure 2(a) shows the chart of two maximal Lyapunov exponents k1 > k2 on the (d/R0, Pac) plane at « = 2.87 x 107 s—1. We choose this value of « due to the following reasons. First, it belongs to the range of frequencies which is relevant for biomedical applications. Second, system (1) also demonstrates quite rich dynamics at this frequency.

Depending on the values of k1 and k2, the corresponding pixel on the chart is painted with a certain color using the following scheme:

• k1 < 0, k2 < 0—periodic regime—blue color;

• k1 = 0, k2 < 0—quasiperiodic regime—green color;

• k1 > 0, k2 < 0—simple chaotic regime (strange attractor with one positive Lyapunov exponent)—yellow color; and

• k1 > 0, k2 > 0—hyperchaotic regime (strange attractor with two positive Lyapunov exponents)—red color.

The big blue region on the left in the chart of Lyapunov exponents corresponds to the stable periodic regimes [see Fig. 2(i)], when two bubbles perform in-phase synchronous oscillations. At the top part of the chart, this blue region adjoins the yellow domain corresponding to a simple strange attractor with one positive Lyapunov exponent [see Fig. 2(h)], which matches to synchronous chaotic oscillations of two bubbles. In Sec. IV A 1, we show that the transition to chaos in this part of the chart occurs via the Feigenbaum cascade of period-doubling bifurcations.

At the middle and bottom parts of the diagram from Fig. 2(a), the big blue region adjoins the green-colored region corresponding to a quasiperiodic regime, when a torus (invariant curve on the Poincare map) is an attractor in system (1) [see Fig. 2(f)]. The invariant torus in Fig. 2(f) is born from the asynchronous periodic regime [see Fig. 2(g)] via the Neimark-Sacker bifurcation, which in its turn appears via the saddle-node bifurcation. Notice also that in the bottom-left part of the chart, the synchronous periodic regime coexists with the asynchronous one.

Moving through the region corresponding to the quasiperi-odic regime from left to right, one can observe transition to chaotic (in the middle part of the chart) and even hyperchaotic (in the bottom part of the chart) attractors. Simple chaotic attractors, occurring after the destruction of an invariant torus [see Fig. 6(c)] as well as hyperchaotic attractors [see Fig. 9(e)], correspond to asynchronous oscillations of the bubbles. In Sec. IV A 2, we show that in the middle part of the chart, chaotic attractors appear in accordance with the Afraimovich-Shilnikov scenario ofthe destruction of an invariant torus. In Sec. IV B, we study the scenario for the onset of hyperchaotic oscillations in the bottom part of the chart.

From Fig. 2(a), one can see that yellow- and red-colored regions corresponding to chaotic [see, e.g., Figs. 2(c) and 2(h)] and hyperchaotic [see, e.g., Figs. 2(b) and 2(e)] attractors alternate with the so-called stability windows, inside which stable periodic orbits are observed (see blue-colored regions inside yellow- and red-colored ones). Some of these stability windows have shrimplike form.40 Such stability windows indicate the existence of the specific homo-clinic bifurcations (cubic homoclinic tangencies or symmetrical pairs of homoclinic tangencies) in the system.41 This implies that chaotic dynamics in system (1) are not hyperbolic and even not pseudohyperbolic. - In other words, in accordance with the PQ (pseudohyperbolic attractor or quasiattractor)-hypothesis,44 strange attractors in the system under consideration belong to the class of quasiattractors introduced by Afraimovich and Shilnikov.45 Stable periodic orbits of high periods and with narrow absorbing domains exist inside such attractors or appear under arbitrarily small perturbations. However, from a physical point of view, in most cases, quasiattractors do not differ from "genuine" (pseudohyperbolic) attractors since absorbing domains of periodic orbits belonging to them are extremely narrow. As far as we know, currently there are no known

FIG. 2. (a) Chart of Lyapunov exponents for m = 2.87 x 107 s-1, Pac e [1.142,1.89] MPa, d/R0 e [6,35]; (b)-(i) projections of phase portraits of steady state regimes (attractors) for some representative points of this chart. Here and below, we draw the projections of attractors on the Poincare map are as black dots and projections of the phase trajectories for periodic attractors as blue lines. The following attractors are shown here: (b) hyperchaotic attractor at d/R0 = 32, Pac = 1.68 MPa, with largest Lyapunov exponents of X1 = 0.0803, X2 = 0.0357; (c) synchronous chaotic attractor at d/R0 = 30, Pac = 1.4 MPa with X1 = 0.0684, X2 = -0.0268; (d) syncronous 12-periodic limit cycle at d/R0 = 28, Pac = 1.3 MPa with X1 = -0.0616, X2 = -0.0733; (e) hyperchaotic attractor at d/R0 = 22, Pac = 1.2 MPa with X1 = 0.0241, X2 = 0.0034; (f) quasiperiodic attractor at d/R0 = 14.5, Px = 1.2 MPa with X1 = 0, X2 = -0.0149; (g) asynchronous 4-periodic limit cycle at d/R0 = 10, Pac = 1.2 MPa with X1 = -0.2331, X2 = -0.2343; (h) synchronous chaotic attractor at d/R0 = 10, Pac = 1.6 MPa with X1 = 0.0802, X2 = -0.0826; and (i) synchronous 2-periodic limit cycle at d/R0 = 6.75, Pac = 1.7 MPa with X1 = -0.1437, X2 = -0.2057.

systems arising in applications and which demonstrate hyperbolic or even pseudohyperbolic hyperchaotic behavior.

IV. TRANSITION TO CHAOS AND HYPERCHAOS

As it can be clearly seen from the charts of Lyapunov exponents [Fig. 2(a)], the dynamics in system (1) can be either regular (periodic or quasiperiodic) or chaotic and even hyperchaotic.

In this section, we study the routes to chaos and hyperchaos in the system along few paths from the bottom part of this chart shown in Fig. 3(a). As a rule, chaotic attractors appear from simple (regular) attractors as a result of the implementation of some bifurcation scenario. The most known examples of such scenarios are (1) the Feigenbaum cascade of period-doubling bifurcations46 according to which a chaotic attractor appears from a stable periodic orbit via the infinite sequence of period-doubling bifurcations; (2) destruction of an invariant torus by the Afraimovich-Shilnikov scenario;47 and (3) the Shilnikov scenario32 due to which a spiral attractor containing a saddle-focus equilibrium with a two-dimensional unstable invariant manifold appears from a stable equilibrium as a result of certain local and global bifurcations. It is worth noting that all the above

mentioned scenarios can be observed in a flow system with dimension N > 3 or in diffeomorphisms or discrete systems (except for the Shilnikov scenario) with dimension N > 2 and generally lead to the appearance of chaotic attractors with only one positive Lyapunov exponent.

An important class of chaotic attractors of multidimensional flows (N > 4) and maps (N > 3), namely, the so-called "homoclinic attractors" containing a selected saddle fixed point (periodic orbit) with its homoclinic structure, was introduced in Refs. 30 and 31, where the classification of such attractors and also phenomenolog-ical scenarios of their appearance were proposed (see also Ref. 29 for more examples of such attractors in three-dimensional Henon maps). The classification of homoclinic attractors is based on the type of the selected saddle orbit belonging to an attractor. Two main attractors of this type are the "discrete Lorenz" and "figure-eight" attractors. They contain a saddle fixed point with a one-dimensional unstable manifold forming a homoclinic structure resembling a butterfly and figure-eight, respectively As it has been recently shown,43 these two attractors belong to a class of "pseudohyperbolic" ■ ("genuinely" chaotic) attractors. Shortly speaking, each orbit on a pseudohyperbolic attractor has a positive Lyapunov exponent and,

6 35

6 35

FIG. 3. Two sheets of the charts of Lyapunov exponents at m = 2.87 x 107 s—1 and for Px e [1.142,1.579] MPa and d/R0 e [6,35].

what is important from a physical point of view, this property persists after small perturbations (changing in parameters). However, both the discrete Lorenz and figure-eight attractors cannot be hyper-chaotic.

Another important example of a homoclinic strange attractor proposed in Refs. 3( and is a "discrete Shilnikov attractor." In contrast to all the above mentioned examples of strange attractors, the discrete Shilnikov attractor contains a saddle-focus fixed point with a two-dimensional unstable invariant manifold. In any small neighborhood of such a fixed point, two-dimensional areas are expanded. Lyapunov exponents on the entire attractor "can feel" this expansion, and thus two Lyapunov exponents can be positive. Unlike the discrete Lorenz and figure-eight attractors, the discrete Shilnikov attractor typically is a quasiattractor.45 Recall that bifurcations of homoclinic tangencies inevitably lead to the appearance of stable periodic orbits inside quasiattractors and, thus such attractors either contain stable periodic orbits with large periods and narrow absorbing domains or such orbits appear after arbitrary small perturbations (parameter changing).

Discrete Shilinikov attractors were found in different dynamical systems such as the generalized three-dimensional Henon maps,29,31 nonholonomic models of Chaplygin top,49 Celtic stone50, and the model of coupled identical oscillators.51 However, not in all cases they were identified as hyperchaotic attractors. Apparently, in some cases, the expansion of two-dimensional areas near a saddle-focus orbit with a two-dimensional unstable manifold is compensated by the stronger contraction of areas near other saddle periodic orbits that also belong to the attractor but have only a one-dimensional unstable manifold. Since Lyapunov exponents are average characteristic of an attractor, only one Lyapunov exponent can become positive in this case.

In Sec. IV B, we show that for system (1), discrete Shilnikov attractors containing a saddle-focus periodic orbit are hyperchaotic. We also propose a new phenomenological scenario which leads to the appearance of hyperchaotic attractor and demonstrate that exactly due to this scenario, hyperchaos appears in the system under consideration. However, first of all, we describe scenarios of transition to simple chaotic (with only one positive Lyapunov exponent) attractors in the model.

A. Transition to chaos in the model

1. The Feigenbaum cascade of period-doubling bifurcations

The Feigenbaum infinite sequence of period-doubling bifurcations46 is one of the typical scenarios of the chaos onset in two-dimensional diffeomorphisms and three-dimensional flows. However, such scenarios can also be observed in multidimensional maps (N > 3) and flows (N > 4) (see e.g., Ref. 52). On the other hand, since in multidimensional systems, period-doubling bifurcations compete with the Neimark-Sacker bifurcations and the transition to chaos via the Feigenbaum cascade for multidimensional systems is more rare than in the case of two-dimensional maps and three-dimensional flows.

Here, we show that strange attractors in system (1) can appear via the Feigenbaum cascade of period-doubling bifurcations. Let us fix Pac = 1.55 and move along the path AB: d/R0 e [7.2,10.0] [see Fig. 3(a)]. Figure 4(a) shows the corresponding bifurcation tree. Phase portraits of attractors at certain values of the parameter d/R0 are presented in Figs. 4(b)-4(e), where in blue color we show the projection of the phase curves onto the (r1, u1) plane and in black color we show projections of the corresponding Poincare map on the same plane. At the starting point of the path, a stable periodic orbit (point of period 2 on the Poincare map) is the attractor of system (1), see Fig. 4(b). While parameter d/R0 increases this periodic orbit undergoes the cascade of period-doubling bifurcations, see Figs. 4(c) and 4(d), and finally, at d/R0 ^ 9.05, Feigenbaum-like strange attractor emerges. The projection of the Poincare map of the synchronous chaotic attractor at d/R0 = 9.85 on (r1, u1) plane is presented in Fig. 4(e). The set of Lyapunov exponents for this attractor is k1 = 0.0560,k2 = —0.1257,k3 = —0.3716,andk4 = —0.4808.

In the general case, the Feigenbaum cascade gives rise to the onset of a strange attractor with only one positive Lyapunov exponent. However, it is important to note that at some specific cases, such scenarios can lead to the onset of hyperchaotic behavior. For example, if we take two identical oscillators demonstrating transition to chaos via cascade of period-doubling bifurcations and make them interact through a weak coupling, both elements will demonstrate chaotic behavior, which will lead to two positive Lyapunov exponents

I r, 2 i r¡ 2

FIG. 4. Transition from synchronous periodic to synchronous chaotic oscillations on path AB: Pac = 1.55 MPa, d/R0 e [7.2,10.0] via the Fegeinbaum cascade. (a) Bifurcation tree; (b) projection of the phase portrait of the 2-periodic limit cycle at d/R0 = 7.3; (c) 4-periodic limit cycle at d/R0 = 8.0; (d) 16-periodic limit cycle at d/R0 = 9.0; and (e) projection of the Poincare section of the chaotic attractor at d = 9.85R0.

in the coupled system. Such a transition to hyperchaos was observed in Refs. 27 and 53. Since we take two identical elements, the transition to hyperchaos in accordance with this scenario is also possible in our system if we suppose that the bubbles are quite distant (i.e., R0/d ^ 1). However, this case is less interesting from a physical point of view and, therefore, we do not consider it here.

2. The Afraimovich-Shilnikov scenario of torus destruction

As one can see from the chart of Lyapunov exponents [Figs. 2(a), 3(a), and 3(b)], some regions of parameters corresponding to stable periodic regimes adjoint the region with stable qusiperiodic regimes—invariant tori. The boundary between these regions is formed by the curve of the supercritical Neimark-Sacker bifurcation. Passing through this curve, a stable limit cycle loses its stability and becomes a saddle-focus with a two-dimensional unstable invariant manifold and a stable invariant torus appears.

From another side of the regions of the existence of quasiperi-odic regimes, the dynamics of system (1) can be chaotic. This means that in system (1) chaotic attractors can appear after destruction of a torus. There are several typical scenarios of transition to chaos through the destruction of an invariant torus. One of such scenarios was proposed by Afraimovich and Shilnikov.47

Here, we show that the Afraimovich-Shilnikov scenario is the second typical scenario (the first one is the Feigenabum cascade of period-doubling bifurcations) for the onset of chaos in system (1). But first of all, let us recall some important details concerning typical organization of a bifurcation diagram inside region Q of an invariant torus existence for two-dimensional maps [see Fig. 5(a)]. Resonance regions Pj, the so-called tongues of synchronization, alternate with quasiperiodic regions above the curve of Neimark-Sacker bifurcation NS and with chaotic regions in the upper part of the diagram. Note that in the tongues of synchronization resonant stable and saddle periodic orbits appear on torus (through the saddle-node bifurcations SN), while this torus still exists, but now it is formed by the closure of the unstable invariant manifold of the resonant saddle orbit,47 see Fig. 5(d).

Moving along the arbitrary path in the parameter plane, one can observe sequences of alternated regular, quasiperiodic, and chaotic regimes. Thus, it is important to note that bifurcations of an invariant torus and, in particular, transition to chaos depend on the path in the bifurcation diagram. Moreover, the parts of this path from resonance regions to chaotic ones are the most important. For example, moving along path MN, one can observe the following sequence of regimes: stable periodic orbit [Fig. 5(b)], stable torus [Fig. 5(c)], resonance torus existing in resonance region Pj [Fig. 5(d)], and, finally, strange attractor [Fig. 5(e)].

According to Afraimovich and Shilnikov,47 in two-parametric families of two-dimensional maps, destruction of an invariant torus inside resonance regions can happen due to the following scenarios: (1) period-doubling bifurcation with a stable resonant orbit (e.g., if we move upwards inside a resonance region); (2) homoclinic bifurcation, when an unstable invariant manifold of the resonant saddle periodic orbit touches (and than intersects) its stable manifold [e.g., if we move inside a resonance region through the curve SN to the chaotic region, see path MN in Fig. 5(a)]; and (3) a more complex and difficult to observe scenario associated with the increasing of oscillations of the unstable manifold of a resonant saddle orbit, see details in Ref. 47.

What is important, in all these cases, when leaving a resonance region, one can observe the chaotic regime associated with the previously existed invariant torus. Such chaotic attractors were called "torus-chaos attractors."

Figure 6 gives an illustrative example of the onset of a torus-chaos attractor in system (1) in accordance with the Afraimovich-Shilnikov scenario. Here, we consider the path CD from the chart of Lyapunov exponents in Fig. 3(a): Pac = 1.35 MPa, 13 < d/R0 < 18. Figures 6(a) and 6(b) show the corresponding bifurcation tree and the graph of the two largest Lyapunov exponents, respectively. Portraits of some attractors along this path are presented in Figs. 6(c)-6(h). The stable periodic orbit (stable fixed point of period four on the Poincare map) existing at d/R < 13.28 [see Fig. 6(c)] undergoes the Neimark-Sacker bifurcation at d/R ^ 13.28 after which a stable invariant torus (four-component invariant curve in

FIG. 5. (a) Sketch of the bifurcation diagram illustrating bifurcations of an invariant torus. P, Q, Pi, and C—regions of the existence of (b) stable periodic orbit, (c) stable invariant torus, (d) resonant periodic orbits, and (e) torus-chaos attrac-tors, respectively. MN—some path along which torus-chaos attractor appears in accordance with the Afraimovich-Shilnikov scenario.

the Poincare map) appears, see Fig. 6(d). Then, at d/R ^ 16.71, we get into a resonance region where stable periodic orbit appears, see Fig. 6(e). Moving inside this resonance region, a stable periodic orbit undergoes the Feigebaum cascade of period-doubling bifurcations [see Figs. 6(f) and 6(g) for details]. Finally, at d/R ^ 16.84, we get out of the resonance region and a torus-chaos attractor appears [see Fig. 6(h)].

In Sec. IV B, we will demonstrate that some paths out of resonance regions lead to the onset of hyperchaotic attractors. We will also give a bifurcation scenario of such a transition.

B. Transition to hyperchaos in system (1)

Starting from three-dimensional diffeomorphisms

(four-

dimensional flows), in addition to the Afraimovich-Shilnikov scenario, some other ways of the destruction of an invariant torus become possible (see e.g., Refs. 55-57). Here, we would like to mention the scenario of torus destruction via the secondary Neimark-Sacker bifurcation with a stable resonant periodic orbit inside a tongue of synchronization.

The typical bifurcation diagram near the Neimark-Sacker bifurcation for three-dimensional diffeomorphisms differs from the

FIG. 6. Afraimovich-Shilnikov scenario for the onset of the torus-chaos attrac-tor along path CD: Px = 1.35 MPa, d/R0 e [13,18]. (a) Bifurcation tree and (b) graph of the two largest Lya-punov exponents associated with this path; projections of attractors onto the (r1, r2)-plane: (c) stable periodic orbit (point of period 4 on the Poincare map) at d/R0 = 13; (d) four-component invariant curve after the Neimark-Sacker bifurcation at d/R0 = 14.5; (e) resonance on the torus, d/R0 = 16.825; (f) resonance after the first period-doubling bifurcation, d/R0 = 16.831; (g) d/R0 = 16.84, Feigenbaum-like strange attractor; and (h) torus-chaos attractor at d/R0 = 17.02 with the following Lyapunov exponents: X1 = 0.0139, X2 = —0.0538, A3 = —0.5571,and ).4 = —0.6131.

FIG. 7. Sketch of the bifurcation diagram illustrating the scenario of the onset of a hyperchaotic attractor in multidimensional maps (N > 3). P, Q, P,, C, and H—regions of the existence of (b) stable periodic orbit, (c) stable invariant torus, (d) resonant periodic orbits, (e) stable torus after the secondary Neimark-Sacker bifurcation NS2, (f) torus-chaos attractors, and (g) the hyperchaotic Shilnikov attractor, respectively. MN—some path along which the hyperchaotic attractor appears.

corresponding diagram in the two-dimensional case, see Fig. 7(a). One of the differences is that a resonant periodic orbit in the three-dimensional case can undergo the secondary Neimark-Sacker bifurcation NS2 instead of a typical for two-dimensional diffeomorphisms period-doubling bifurcation, see the right-top part in Fig. 7(a).

1. The scenario of the onset of hyperchaotic attractors

The secondary Neimark-Sacker bifurcation is the first (but not main) step in our scenario for the onset of hyperchaotic attractors. After this bifurcation, multiround stable invariant torus (multicom-ponent invariant curve in the Poincare map) is born in the system, while, what is very important, resonant periodic orbit becomes a saddle-focal with a two-dimensional unstable manifold, see Fig. 7(e), where SF denotes the saddle-focus periodic orbit.

The next step in the framework of this scenario is associated with the destruction of a multiround stable invariant torus. It does not matter how it happens: through the Afraimovich-Shilnikov scenario, cascade of torus period-doubling bifurcations or even via the tertiary Neimark-Sacker bifurcation. However, we suppose, and it is quite natural, that after this the corresponding bifurcations torus-chaos attractor with one positive Lyapunov exponent appears, see Fig. 7(f). It is worth noting that in this case, immediately after transition to chaos, saddle-focus SF does not belong to the torus-chaos attractor. It means that trajectories belonging to this chaotic attractor do pass near a small neighborhood of SFj, see Fig. 7(f).

The final and the key step in the scenario is the inclusion of the saddle-focus periodic orbit SFj, which appeared after the secondary Neimark-Sacker bifurcation, into the chaotic attractor. After this inclusion, the saddle-focus orbit SF together with its two-dimensional unstable manifold and its homoclinic structure starts to belong to the attractor, i.e., the discrete homoclinic Shilnikov attractor based on this saddle-focus orbit emerges, see Fig. 7(g). Orbits on this attractor can pass arbitrary close to SF, where two-dimensional areas are expanded. As a result, two Lyapunov exponents become positive, i.e., a hyperchaotic attractor is born.

In order to clarify how saddle-focus orbit SF is included into the chaotic attractor, let us recall some important steps in the framework of the Shilnikov scenario.31,32 Immediately after the Neimark-Sacker bifurcation, stable invariant curve has a nodal type, see Fig. 8(a). Then, this curve becomes of focal type, the so-called Shilnikov funnel

appears. After this, almost all orbits in the neighborhood of the saddle-focus are wound on the stable invariant curve, see Fig. 8(b). Furthermore, the size of the funnel is increased, it approaches stable invariant manifold WS1 (SF). During this transition, orbits of the attractor approach closer and closer to the saddle-focus. Finally, the homoclinic intersection between stable and unstable manifolds of SF occurs, discrete Shilnikov attractor appears, see Fig. 8(c), after which orbits of this attractor start pass arbitrarily close to the saddle-focus.

The inclusion of a saddle-focus periodic orbit to the chaotic attractor can occur in different ways. It depends on the transition from the stable multiround torus to the chaotic regime. In all known models demonstrating the onset of the discrete Shilnikov attrac-tor (in three-dimensional Henon maps,29 nonholonomic models of Chaplygin top,49 and Celtic stone50), this inclusion happens in a soft manner by a smooth transformation of a torus-chaos attrac-tors accompanied by a smooth increasing of the Shilnikov funnel. However, we also suppose that a saddle-focus orbit can be included into the chaotic attractor sharply due to the crisis of a multiround torus-chaos attractor.

2. Implementation of the proposed scenario

In order to support the proposed scenario, we show that the transition to hyperchaos along the paths EF, GH, and IJ [see Fig. 3(a)] happens in full compliance with this scenario. First, let us consider the path EF, corresponding to the following parameters interval:

,W U(SF|) \\, (b) ^ (C)

FIG. 8. Possible scenario of the inclusion of saddle-focus periodic orbit SF, to the chaotic attractor. (a) After the Neimark-Sacker bifurcation invariant curve LM has a nodal type. (b) Invariant curve LM becomes of a focal type, Shilnikov funnel is formed. (c) The size of Shilnikov funnel is increased, homoclinic intersection between stable WS1 (SF,) and unstable W"(SFi) invariant manifolds occurs.

FIG. 9. The implementation of the proposed scenario of the onset of a hyperchaotic attractor along the path EF: Pac = 1.2,13 < d/R0 < 25. (a) and (b) Bifurcation tree and the graph of two largest Lyapunov exponents associated with this path with the enlarged area for 17.5 < d/R0 < 19; projections of the Poincare maps for different attractors on the (r1, u1) plane: (c) four-component invariant curve at d/R0 = 16; (d) resonance orbit at d/R0 = 17.92; (e) multicomponent invariant curve after the secondary Neimark-Sacker bifurcation, d/R0 = 17.98; (f) high-period resonance orbit emerges on the multicomponent invariant curve, d/R0 = 18.01; (g) chaotic attractor after the period-doubling cascade at d/R0 = 18.35 with two largest Lyapunov exponents X1 = 0.0087, k2 = —0.0040; and (h) hyperchaotic Shilnikov attractor containing saddle-focus periodic orbit with a two-dimensional unstable manifold at d/R0 = 18.71.

FIG. 10. The implementation of the same scenario for the onset of the hyperchaotic attractor along path GH: Pac = 1.45 MPa, 9.85 < d/R0 < 11. (a) and (b) bifurcation tree and graph of the two largest Lyapunov exponents associated with this path; projections of the Poincare map of several attractors on the (r1, r2) plane: (c) four-component invariant curve at d/R0 = 10.15; (d) resonance orbit at d/R0 = 10.3961; (e) torus starts losing its smoothness, d/R0 = 10.4111; (f) torus-chaos attractor at d/R0 = 10.4687; (g) multicomponent invariant curve after the secondary Neimark-Sacker bifurcation, d/R0 = 10.59; and (h) hyperchaotic attractor containing saddle-focus periodic orbit with a two-dimensional unstable manifold at d/R0 = 10.6115.

Pac = 1.2 MPa and 13 < d/R0 < 25. The bifurcation tree, corresponding to this route, is shown in Fig. 9(a), and the graph of two largest Lyapunov exponents is presented in Fig. 9(b). We also show the enlarged part for both these graphs at the right panels

in Figs. 9(a) and 9(b) in order to explore some important details concerning the secondary Neimark-Sacker bifurcation and the transition to hyperchaos. Projections of the Poincarè sections for some representative attractors are shown in Figs. 9(c)-9(h). From Fig. 9,

one can observe the sequence of bifurcations starting from asynchronous orbit of period 4 at d/R < 13.89 and leading to the onset of hyperchaotic attractor. First, this periodic orbit undergoes the Neimark-Sacker bifurcation at d/R0 & 13.89 and the stable torus (4-component invariant curve in the Poincare map) appears, see Fig. 9(c). Then, a high-period resonance occurs on the torus, see Fig. 9(d). With a further increase in d/R0, the multiround torus emerges after the secondary Neimark-Sacker bifurcation while the stable resonance orbit becomes of the saddle-focal type with a two-dimensional unstable manifold [see Fig. 9(e)]. Soon this multiround torus gives rise to the torus-chaos attractor [see Fig. 9(g)], which appears via the cascade of period-doubling bifurcations happening with some stable resonant orbit emerging on this torus [see the long-periodic orbit emerging after few period-doubling bifurcations in Fig. 9(f)]. It is important to note that the saddle-focus orbit occurring after the secondary Neimark-Sacker bifurcation does not belong to this torus-chaos attractor. Finally, at d/R0 & 18.66, the saddle-focus orbit starts to belong to the chaotic attractor. As a result, the hyperchaotic attractor, containing this saddle-focus orbit appears, see Fig. 9(h). Lyapunov exponents at d = 18.71 are X1 = 0.0135, X2 = 0.0019, X3 = -0.5560, and X4 = -0.5607.

Furthermore, let us fix Pac = 1.45 and move along the path GH from the chart of Lyapunov exponents in Fig. 3(a). Figures 10(a) and 10(b) show the corresponding bifurcation tree and the graph of the two largest Lyapunov exponents along this path. From these figures, one can see that at d/R0 & 10.59, two Lyapunov exponents become positive, i.e., hyperchaotic attractor appears. Portraits of some representative attractors for several values of parameter d/R0 are presented in Figs. 10(c)-10(h), where projections of the Poincare map onto (r1, r2) plane are shown. The beginning of the route GH corresponds to an asynchronous 4-periodic limit cycle. At d/R0 & 9.98, it undergoes the Neimark-Sacker bifurcation, after which a stable invariant torus (4-component invariant curve on the Poincare map) appears, see Fig. 10(c). Increasing d/R0 we pass through few resonance regions [see Fig. 10(d)]. Soon after this, the invariant torus starts to lose its smoothness, see Fig. 10(e). Then, leaving one of the resonance regions, the torus-chaos attractor appears. This attractor has the following spectrum of Lyapunov exponents: X1 = 0.0090, X2 = -0.0404,

X3 = -0.5302, and X4 = -0.5507, see Fig. 10(f). With a further increase in d/R0, we again pass through a resonance region, but now the resonant orbit undergoes the secondary Neimark-Sacker bifurcation at d/R0 & 10.58 after which a multiround invariant torus (multicomponent invariant curve on the Poincare map) appears, while the stable resonance orbit becomes of the saddle-focal type with a two-dimensional unstable manifold, see Fig. 10(g). Then, the multiround torus gives rise to a chaotic attractor and, finally, the discrete hyperchaotic Shilnikov attractor containing the saddle-focus orbit appears [see Fig. 10(h)]. The set of Lyapunov exponents for this hyperchaotic attractor at (d/R, Pac) = (10.6115,1.45) is X1 = 0.012, k2 = 0.0022, X3 = -0.559, and X4 = -0.569.

Furthermore, let us briefly discuss bifurcations leading to the onset of hyperchaotic attractors along the path IJ: d/R0 = 27.9, Pac e [1.468,1.524] MPa. Unlike the previous routes, for this one we fix d/R0 and vary Pac. In Figs. 11(a)-11(c), we present the bifurcation diagram and the graph of two largest Lyapunov exponents associated with the route. The transition to hyperchaotic attractors along this route occurs due to the following sequence of bifurcations. At the beginning of IJ, as in the cases considered previously, a stable periodic orbit is an attractor of the system. Then, at Pac & 1.477, it undergoes the Neimark-Sacker bifurcation due to which a stable torus appears. Then, high-period resonance appears on this torus. At Pac & 1.49914, a stable resonance orbit undergoes the secondary Neimark-Sacker bifurcations due to which a multiround torus is born while the resonance periodic orbit becomes of the saddle-focal type with a two-dimensional unstable manifold, see Fig. 9(d). With a further increase in Pac, the multiround torus gives rise to torus-chaos attractor with only one positive Lyapunov exponent. Finally, at Pac & 1.4994, the saddle-focus periodic orbit emerging after the secondary Neimark-Sacker bifurcation is included into the attractor and, as a result, a hyperchaotic attractor appears.

Finally, we would like to raise an open question. In which cases a discrete Shilnikov attractor is hyperchaotic, in which cases it has only one positive Lyapunov exponent, and what exactly are the responses for it. For example, the discrete Shilnikov attractors from the non-holonomic model of Chaplygin top49 and Celtic stone50 have only one positive Lyapunov exponent. On the other hand, such attractors are hyperchaotic in three-dimensional Henon maps,29'58 the modified

FIG. 11. Transition to hyperchaos along route IJ: d = 27.9 • R0, Pac e [1.468, 1.524] MPa. (a) and (b) Bifurcation tree and graph of two largest Lyapunov exponents associated with this path;

(c) enlarged part of the graph of the Lyapunov exponents, corresponding to Px e [1.4966,1.5002] MPa; and

(d) projection of the invariant curve in the Poincare map corresponding to the multiround torus after the secondary Neimark-Sacker bifurcation on the (r1, r2) plane.

oscillator of Anishchenko-Astakhov,53 and the model under consideration. We suppose that in some cases two-dimensional expansion of areas near the saddle-focus orbit with two-dimensional unstable manifold is compensated by the stronger contraction (occurring with higher rate) near some other saddle periodic (quasiperiodic) orbits that also belongs to the attractor but which has only one-dimensional unstable manifold, and thus two-dimensional areas are contracted in their neighborhood. Since Lyapunov exponents are average characteristic of the attractor, only one Lyapunov exponent can become positive in this case.

V. MULTISTABILITY IN THE DYNAMICS OF TWO BUBBLES

In this section, we study the multistability phenomenon in the model under consideration. Recall that under multistability, here we mean the coexistence of two or more stable regimes which are nonsymmetric with respect to (2).

Figure 3 shows two charts of Lyapunov exponents obtained by a continuation of two different periodic regimes (one is synchronous while another is asynchronous) by parameters. One can see a lot of differences between Figs. 3(a) and 3(b) which confirms that dynamics in system (1) is essentially multistable. Moreover, not only different periodic orbits can coexist here. The coexistence of such types of attractors as periodic with quasiperiodic or chaotic, quasiperiodic with chaotic, and, even, chaotic with hyperchaotic is possible for the system. Further, we consider several routes for the detailed study of the multistability.

Let us start with the path EF from Fig. 3(b): Pac = 1.2 MPa, d/R0 e [13,25]. As it is shown in Sec. IV B, the hyperchaotic attractor of the Shilnikov type [see Fig. 7(h)] is born at the end of the path EF from the asynchronous periodic regime existing at point E in the chart of Lyapunov exponents from Fig. 3(a). However, at this point, there also exists a synchronous periodic regime which, on the same path [see EF in Fig. 3(b)], gives rise to a synchronous chaotic attractor. As Figs. 12(a) and 12(b) show, this strange attractor appears via the Feigenbaum cascade of period-doubling bifurcations and have only one positive Lyapunov exponent. Thus, by the end of path EF, the synchronous chaotic attractor coexists with an asynchronous chaotic one.

Route KL corresponds to Pac = 1.4 MPa and d/R0 e [8,13.5] (see Fig. 13). Note that in the left part of Figs. 13(c) and 13(d), there is a jump from the stable 4-periodic cycle to a synchronous chaotic attractor, which coincides with the chaotic presented in Figs. 13(a) and 13(b) at d/R0 & 9.1. Thus, we do not plot the graphs in Figs. 13(c) and 13(d) further left of this point. No hyperchaotic attractors are born on this route. However, we can observe here several examples of multistability. Namely, one can see the following pairs of coexisting attractors: quasiperiodic with periodic, quasiperiodic with chaotic, asynchronous periodic with synchronous chaotic, and quasiperiodic with synchronous periodic.

A lot of complicated behavior can be observed along the path MN: Pac = 1.68 MPa, d/R0 e [18.5,28] [see Figs. 14(a) and 14(b), where two fragments of the chart of Lyapunov exponents obtained by continuation of two different regimes are presented]. For this path, we present two bifurcation trees which correspond to different attractors [see Figs. 14(c) and 14(d)], and the merged graph of Lyapunov exponents for both attractors [see Fig. 14(e)]. Figure 14(e) explicitly shows the coexistence of attractors of different types. Here, X and X are the two largest Lyapunov exponents of the attractor from Fig. 14(c), and Xf, are the largest exponents of the attractor from Fig. 14(d). In the right part of the diagram in Fig. 14(c), one can observe an abrupt shift from hyperchaos to chaos occurring at d/R0 & 25.94 [see also the right part of the graph of Lyapunov exponents in Fig. 14(d)]. This represents the moment when the chaotic attractor undergoes crisis and the attractor from Fig. 14(d) remains the only one attractor, which we found in the corresponding region of parameter. Thus, in the remaining interval 25.94 < d/R0 < 28, the graphs of Lyapunov exponents overlap and the bifurcation trees in Figs. 14(c) and 14(d) look identical, because they correspond to the same hyperchaotic attractor. Similar situation can be observed at d/R0 & 19.42. Lyapunov exponents X2 and X2 shift at this point and begin to overlap with AJ, X to the left of this point [see the left side of Fig. 14(e)]. It represents the jump from the stable limit cycle existing in the left part of the bifurcation tree [Fig. 14(c)] to the chaotic attractor corresponding to Fig. 14(d). For all the lower values of d/R0, these attractors coincide and we do not draw all the Lyapunov exponents further left. Thus, two attractors discussed here coexist in the interval 19.42 < d/R0 < 25.94. A lot of different types of multistability can

FIG. 12. (a) and (b) Bifurcation tree and graph of two largest Lyapunov exponents for the bifurcation sequence of the synchronous attractor on the path EF: 13 • R0 < d < 25 • R0, Px = 1.2 MPa (cf. Fig. 9).

FIG. 13. Route KL: 8 • R0 < d < 13.5 • R0, Pac = 1.4 MPa. (a) and (b) Bifurcation tree and two largest Lyapunov exponents, corresponding to the synchronous attractor. (c) and (d) Bifurcation tree and two largest Lyapunov exponents, corresponding to the asynchronous attractor.

be observed in this interval. Here, one can see the following types of coexisting attractors: chaotic with periodic, chaotic with quasiperi-odic, hyperchaotic with periodic, hyperchaotic with quasiperiodic, and hyperchaotic with chaotic.

Note that the structure of chart of Lyapunov exponents near the routes MN in Fig. 14(b) looks very similar to the same structure around routes EF, GH, and IJ in Fig. 3(a). Thus, one can conclude that it is highly likely that the onset of the hyperchaotic attractor

FIG. 14. Multistability on the route MN and around it. (a) and (b) Fragments of chart of Lyapunov exponents obtained by a continuation of two different regimes. (c) and (d) Bifuraction trees corresponding to different attractors. (e) Graph of two largest Lyapunov exponents for both the attractors on the route MN: 18.5 • R0 < d < 28 • R0, Pac = 1.68 MPa. Lyapunov exponents ).] and ^2 correspond to the attractor associated with plot (a) and k\ and à2 correspond to the attractor associated with plot (b). A variety of types of multistability can be observed on a single path.

corresponding to the part MN from Fig. 14(b) happens according to the same scenario that was described in Sec. IV B.

VI. CONCLUSION

In this work, we have studied the nonlinear dynamics of two encapsulated interacting gas bubbles in a liquid. We have showed that the oscillations of bubbles can be periodic, quasiperiodic, chaotic, and hyperchaotic. Moreover, we have observed the multistability phenomenon in a wide region of the control parameters, which makes bubbles' dynamics even more complicated. We believe that both quasiperiodic and hyperchaotic oscillations along with multi-stability phenomenon in the dynamics of two coupled bubbles are reported for the first time.

Concerning the onset of chaotic dynamics, we have studied typical roots to chaos and hyperchaos in system (1). We have demonstrated that simple chaotic attractors (with only one positive Lya-punov exponent) occur either via the Feigenbaum cascade of period-doubling bifurcations or by the Afraimovich-Shilnikov scenario of the destruction of invariant tori. On the other hand, for the onset of hyperchaotic oscillations, we propose a new scenario which is based on the appearance of the discrete Shilnikov attractor containing a saddle-focus periodic orbit with its two-dimensional unstable manifold. Orbits on this attractor can pass arbitrary close to this saddle-focus orbit, where two-dimensional areas are expanded. As a result, two Lyapunov exponents become positive, i.e., a hyperchaotic attractor is born. To the best of our knowledge, the proposed scenario gives one of a few known explanations of the emergency of hyperchaotic behavior. Moreover, we believe that this scenario may be typical for other multidimensional systems demonstrating transition to hyperchaos via the destruction of a torus (see, e.g., Refs. 53 and 54).

We have also studied the multistability phenomenon in system (1). We have showed that various types of attractors, both synchronous and asynchronous, regular (periodic or quasiperiodic) and chaotic, and even hyperchaotic, can coexist at the same values of the control parameters. In particular, synchronous periodic attrac-tors can coexist with asynchronous periodic, quasiperiodic, asynchronous chaotic, or hyperchaotic states, and even coexistence of several synchronous regimes is possible (for example, two different synchronous periodic limit cycles). We have also demonstrated that hyperchaotic oscillations can coexist with regular and chaotic (both synchronous and asynchronous) ones, as well as with asynchronous quasiperiodic oscillations.

As far as applications are concerned, it is known3,6 that chaotic oscillations of bubbles can be beneficial for blood flow visualization. Thus, we believe that the regions of the control parameters where either one chaotic or hyperchaotic attractor exists or both of these attractors coexist may be recommended for this type of applications. On the other hand, the regions of the control parameters where different types of attractors coexist (e.g., periodic and quasipariodic) should be avoided in applications, since in this case the dynamics of bubbles becomes virtually unpredictable due to the fact that small perturbations in the initial conditions or control parameters may lead to a substantial change in bubbles' acoustic response.

ACKNOWLEDGMENTS

This paper (except Sec. III) was supported by the RSF (Grant No. 17-71-10241). The results in Sec. III were supported by the RFBR (Grant Nos. 18-31-20052 and 19-01-00607). A.K. also acknowledges Basic Research Program at NRU HSE in 2019 for the support of scientific researches. We thank an anonymous referee for valuable comments and suggestions that helped us to improve the manuscript.

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3.2 Приложение 2: Статья "Synchronous oscillations and symmetry breaking in a model of two interacting ultrasound contrast agents"

Статья 2,

Garashehuk I.R., Sinelshehikov D.I, Kazakov А.О., Synchronous oscillations and symmetry breaking in a model of two interacting ultrasound contrast agents, Nonlinear Dynamics (2020), 101, 1199-1213.

https://link.springer.com/article/10.1007/sl1071-020-05864-4

Nonlinear Dyn

https://doi.org/10.1007/s11071-020-05864-4 ORIGINAL PAPER