maisМоделирование и анализ информационных системModeling and Analysis of Information Systems1818-10152313-5417Yaroslavl State University10.18255/1818-1015-2015-5-595-608mais-280Research ArticleОригинальные статьиArticlesФазовая модель Курамото с инерцией: бифуркации потери синхронности и перехода к хаосуKuramoto Phase Model with Inertia: Bifurcations Leading to the Loss of Synchrony and to the Emergence of ChaosБелыхВ. Н.BelykhV. N.

д-р физ.-мат. наук., профессор, кафедра теории управления и динамики систем

Department of Control Theory

belykh@vgavt-nn.ruБолотовМ. И.BolotovM. I.

лаборант, кафедра теории управления и динамики систем

Department of Control Theory

bolotovmaximilich@mail.ruОсиповГ. В.OsipovG. V.

д-p физ.-мат. наук, кафедра теории управления и динамики систем

Department of Control Theory

grosipov@gmail.comНижегородский государственный университет им. Н.И. Лобачевского, пр. Гагарина, 23, Нижний Новгород, 603950, Россия Волжский государственный университет водного транспорта, ул. Нестерова, 5, Нижний Новгород, 603950, РоссияРоссияNizhny Novgorod University, Gagarin Ave., 23, Nizhny Novgorod, 603950, Russia, Volga State University of Water Transport, Nesterova str., 5, Nizhny Novgorod, 603950, RussiaRussian FederationНижегородский государственный университет им. Н.И. Лобачевского, пр. Гагарина, 23, Нижний Новгород, 603950, Россия,РоссияNizhny Novgorod University, Gagarin Ave., 23, Nizhny Novgorod, 603950, RussiaRussian FederationНижегородский государственный университет им. Н.И. Лобачевского, пр. Гагарина, 23, Нижний Новгород, 603950, РоссиРоссияNizhny Novgorod University, Gagarin Ave., 23, Nizhny Novgorod, 603950, RussiaRussian Federation201520102015225595608Copyright © Белых В.Н., Болотов М.И., Осипов Г.В., 20152015Белых В.Н., Болотов М.И., Осипов Г.В.Belykh V.N., Bolotov M.I., Osipov G.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.mais-journal.ru/jour/article/view/280

В данной работе рассматривается конечномерная модель Курамото с инерцией в случае топологии типа "звезда". Система уравнений сводится к нелинейно связанной системе маятниковых уравнений. Мы докажем, что переход от синхронных к асинхронным колебаниям происходит через седлоузловую бифуркацию состояния равновесия. Таким образом, асинхронный режим может представлять собой частично синхронные вращения. Обратный переход от асинхронного режима к синхронному происходит через бифуркацию гомоклинической орбиты как седлового состояния равновесия, так и седловой периодической орбиты. В случае гомоклинической петли седла синхронность возникает только из асинхронного режима без частично синхронных вращений. В случае гомоклинической кривой седловой периодической орбиты в системе имеет место хаотический режим вращения, который приводит к случайному возврату синхронности. Установлено, что переходы туда и обратно происходят с гистерезисом при большой инерции.

We consider a finite-dimensional model of phase oscillators with inertia in the case of star configuration of coupling. The system of equations is reduced to a nonlinearly coupled system of pendulum equations. We prove that the transition from synchronous to asynchronous oscillations occurs via bifurcation of saddle-node equilibrium. In this connection the asynchronous regime can be partially synchronous rotations. We find that the reverse transition from asynchronous to synchronous regime occurs via bifurcation of homoclinic orbit both of the saddle equilibrium point and of the saddle periodic orbit. In the case of homoclinic loop of the saddle point the synchrony appears only from asynchronous mode without partially synchronized rotations. In the case of the homoclinic curve of the saddle periodic orbit the system undergoes a chaotic rotation regime which results in a random return to synchrony. We establish that return transitions are hysteretic in the case of large inertia.

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The authors declare that there are no conflicts of interest present.