Collective behaviors of globally coupled harmonic oscillators driven by different frequency fluctuations
Authors: Jiang Lei , Lai Li , Yu Tao , Luo Mao-Kang ,   College of Mathematics, Sichuan University, Chengdu 610064, China,   School of Aeronautics and Astronautics, Sichuan University, Chengdu 610064, China
Published: June 2021

For multi-particle coupled systems, the effects of environmental fluctuations on each particle are often different in actual situations. To this end, this paper studies the collective dynamic behaviors in globally coupled harmonic oscillators driven by different frequency fluctuations, including synchronization, stability and stochastic resonance (SR). The statistical synchronicity between particles' behaviors is derived by reasonably grouping variables and using random average method, and then the statistical equivalence between behaviors of mean field and behaviors of single particle is obtained. Therefore, the characteristics of mean field's behaviors (that is, collective behaviors) can be obtained by studying behaviors of any single particle. Moreover, the output amplitude gain and the necessary and sufficient condition for the system stability are obtained by using this synchronization. The former lays a theoretical foundation for analyzing the stochastic resonance behavior of the system, and the latter gives the scope of adaptation of the conclusions in this paper. In terms of numerical simulation, the research is mainly carried out through the stochastic Taylor expansion algorithm. Firstly, the influence of system size <i>N</i> and coupling strength <inline-formula><tex-math id="M3">\begin{document}$\varepsilon$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210157_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210157_M3.png"/></alternatives></inline-formula> on the stability area and synchronization time is analyzed. The results show that with the increase of the coupling strength <inline-formula><tex-math id="M4">\begin{document}$\varepsilon$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210157_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210157_M4.png"/></alternatives></inline-formula> or the increase of the system size <i>N</i>, the coupling force between particles increases, and the orderliness of the system increases, so that the stable region gradually increases and the synchronization time gradually decreases. Secondly, the stochastic resonance behavior of the system is studied. Noises provide randomness for the system, and coupling forces provide orderliness for the system. The two compete with each other, so that the system outputs about the noise intensity <inline-formula><tex-math id="M5">\begin{document}$\sigma$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210157_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210157_M5.png"/></alternatives></inline-formula>, the coupling strength <inline-formula><tex-math id="M6">\begin{document}$\varepsilon$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210157_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210157_M6.png"/></alternatives></inline-formula> and the system size <i>N</i> exhibit stochastic resonance behavior. As the coupling strength increases or the system size increases, the orderliness of the system increases, and greater noise intensity is required to provide stronger randomness to achieve optimal matching with it, so as to the resonance of the noise intensity <inline-formula><tex-math id="M7">\begin{document}$\sigma$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210157_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210157_M7.png"/></alternatives></inline-formula>, the peak gradually shifts to the right. Conversely, as the noise intensity <inline-formula><tex-math id="M8">\begin{document}$\sigma$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210157_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210157_M8.png"/></alternatives></inline-formula> increases, the resonance peak of the coupling strength <inline-formula><tex-math id="M9">\begin{document}$\varepsilon$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210157_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20210157_M9.png"/></alternatives></inline-formula> and the system size <i>N</i> will also shift to the right.