informИнформатикаInformatics1816-03012617-6963UIIP NASBinform-467Research ArticleМАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕMATHEMATICAL MODELINGСпектральный анализ хаотических колебаний в имитационной модели схемы Чжуа, разработанной на основе матричной декомпозицииA spectral analysis of chaotic oscillations in simulation model of Chua’s circuit developed with use of matrix decompositionКротА. М.KrotA. M.

доктор технических наук, профессор, заведующий лабораторией моделирования самоорганизующихся систем

Dr. Sci. (Eng.), Professor, Head of the Laboratory of Self-Organization Systems Modeling

alxkrot@newman.bas-net.byСычевВ. А.SychouU. A.

научный сотрудник лаборатории робототехнических систем

Researcher of the Laboratory of Robotic Systems

vsychyov@robotics.byОбъединённый институт проблем информатики НАН БеларусиThe United Institute of Informatics Problems of the National Academy of Sciences of BelarusОбъединенный институт проблем информатики Национальной академии наук БеларусиThe United Institute of Informatics Problems of the National Academy of Sciences of Belarus201914112018161723Copyright © Крот А.М., Сычев В.А., 20192019Крот А.М., Сычев В.А.Krot A.M., Sychou U.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://inf.grid.by/jour/article/view/467

Метод матричной декомпозиции А.М. Крота, предназначенный для анализа аттракторов сложных нелинейных динамических систем на основе матричного ряда в пространстве состояний, использован для нелинейного анализа такого генератора хаотических сигналов, как цепь Чжуа с кубическим полиномом в качестве нелинейной функции.  Показано, что исходная система дифференциальных уравнений Чжуа может быть представлена посредством линейного, квадратичного и кубического членов матричного ряда. Полученные члены ряда положены в основу имитационной модели, использованной для проведения вычислительных экспериментов. По результатам экспериментов определены значения управляющих параметров, при которых возникает хаотический режим, проведён бифуркационный и спектральный анализ генерируемых сигналов, позволяющий обосновать переход к хаосу через серию бифуркаций. Проведённые исследования позволили сделать вывод о том, что процесс возникновения хаотических колебаний в электрической схеме Чжуа соответствует модели начальной турбулентности Л.Д. Ландау и находится в полном согласии с теорией Рюэля-Такенса. Исследована корректность применения матричного разложения векторной функции в зависимости от величины возмущений (приращений) переменных в пространстве состояний.

The A. M. Krot’s matrix decomposition method developed for analysis of complex nonlinear dynamical system attractors based on matrix series in the state space has been used for nonlinear analysis of Chua’s circuit with cube non-linearity. It is shown that the operator of the system of Chua’s differential equations can be represented through the linear, quadratic and cubic terms of the matrix series. The obtained terms are the basis of the simulation model used for carrying out computational experiments. Using the results of the experiments, the values of the control parameters, leading to the chaotic regime, are determined, as well as bifurcation and spectral analysis of the generated signals are carried out. It allows to prove the transition to chaos through a series of bifurcations. The research allowed to draw a conclusion that the process of occurrence of chaotic oscillations in the Chua’s circuit corresponds to the L. D. Landau’s model of initial turbulence in full accordance with the theory of Ruelle – Takens. The correctness of application of the matrix expansion of a vector function depending on values of the perturbations (increments) of variables in the state space is investigated.

нелинейная динамическая системацепь Чжуахаотический аттракторметод матричной декомпозиции в пространстве состоянийимитационная модель схемы Чжуаспектральный анализ хаотических колебанийтеория Рюэля – Такенсаnonlinear dynamical systemChua’s circuitchaotic attractormatrix decomposition method in the state-spaceChua’s circuit computational modelspectral analysis of chaotic oscillationsthe theory of Ruelle – TakensРабота выполнена частично в рамках предоставленного гранта Президента Республики Беларусь в науке на 2019 г., а также при поддержке гранта БРФФИ № Ф18Р-229 «Исследование и разработка концепции мехатронных бортовых вычислительных и исполнительных систем групповых микророботов».This work has been supported partially by the grant of President of Republic of Belarus in science (2019) and the BRFFR grant № F18R-229 "Research and development of the mechatronic onboard computational and actuator systems concept for group-microrobots".ReferencesKrot, A. M. 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The authors declare that there are no conflicts of interest present.