Scholarlink

<p>An incremental harmonic balance (IHB) method and an IHB method with two time scales are used in this work to calculate accurate periodic and quasi-periodic (QP) responses of two coupled van der Pol oscillators with strongly nonlinear coupling with three sets of linear natural frequencies, respectively. The three sets of linear natural frequencies are designed to reflect scenarios commonly encountered in practical systems: one set with values of two linear natural frequencies significantly separated, another set exhibiting a 1:3 internal resonance between two linear natural frequencies, and a final set exhibiting a 1:1 internal resonance between two linear natural frequencies. The QP responses obtained from the IHB method with two timescales are compared with those obtained from the method of multiple scales (MMS) and the fourth-order Runge-Kutta (RK) method. It is observed that the QP responses obtained from the MMS only have acceptable accuracy under conditions of weakly nonlinear coupling. Conversely, the QP responses obtained from the IHB method with two time scales maintain a high degree of concordance with those obtained from the fourth-order RK method, even in the presence of strongly nonlinear coupling. Multiple first-order harmonic response curves versus different system parameters and a two-parameter bifurcation diagram of the coupled van der Pol oscillators are obtained, and the Floquet theory and the extended Floquet theory are leveraged to determine stabilities of periodic and QP responses, respectively. In the study of the coupled van der Pol oscillators with strongly nonlinear coupling, two specific types of bifurcations are identified as catalysts for transition from periodic to QP responses: Neimark-Sacker bifurcations and saddle-node bifurcations. QP saddle-node bifurcations are identified for transition from QP responses to periodic responses. Furthermore, torus-doubling bifurcations of QP responses, which are analogous to period-doubling bifurcations of periodic responses, are identified for the coupled van der Pol oscillators.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Dynamics of two coupled van der Pol oscillators with strongly nonlinear coupling

  • 期刊论文

  • 发表日期：

    2025年6月14日

查看全文
  • Y. L. Li,
  • J. L. Huang,
  • W. D. Zhu

摘要

An incremental harmonic balance (IHB) method and an IHB method with two time scales are used in this work to calculate accurate periodic and quasi-periodic (QP) responses of two coupled van der Pol oscillators with strongly nonlinear coupling with three sets of linear natural frequencies, respectively. The three sets of linear natural frequencies are designed to reflect scenarios commonly encountered in practical systems: one set with values of two linear natural frequencies significantly separated, another set exhibiting a 1:3 internal resonance between two linear natural frequencies, and a final set exhibiting a 1:1 internal resonance between two linear natural frequencies. The QP responses obtained from the IHB method with two timescales are compared with those obtained from the method of multiple scales (MMS) and the fourth-order Runge-Kutta (RK) method. It is observed that the QP responses obtained from the MMS only have acceptable accuracy under conditions of weakly nonlinear coupling. Conversely, the QP responses obtained from the IHB method with two time scales maintain a high degree of concordance with those obtained from the fourth-order RK method, even in the presence of strongly nonlinear coupling. Multiple first-order harmonic response curves versus different system parameters and a two-parameter bifurcation diagram of the coupled van der Pol oscillators are obtained, and the Floquet theory and the extended Floquet theory are leveraged to determine stabilities of periodic and QP responses, respectively. In the study of the coupled van der Pol oscillators with strongly nonlinear coupling, two specific types of bifurcations are identified as catalysts for transition from periodic to QP responses: Neimark-Sacker bifurcations and saddle-node bifurcations. QP saddle-node bifurcations are identified for transition from QP responses to periodic responses. Furthermore, torus-doubling bifurcations of QP responses, which are analogous to period-doubling bifurcations of periodic responses, are identified for the coupled van der Pol oscillators.