This paper is concerned with the resonance manifestation of a stochastically driven system consisting of dual Duffing oscillators with coupled active-passive hybrid control (APHC), i.e., time-delayed feedback control and viscoelastic damping. Analytic solutions are successively determined in both deterministic and stochastic modes. Based on our simulations in the deterministic mode, changes to the rules and trends of the mean square response of the system, including coupling parameters such as the viscoelastic coefficient, displacement and velocity feedback gain coefficients variation in the stochastic mode, are depicted in detail. These are accompanied by (double) jump and bifurcation phenomena, the emergence of a time-delayed isle and its convergence with other branches. Our numerical simulation results confirm the effectiveness of the theoretical analyses method as well. In addition, the diversity of phase portraits in the two modes illustrates different effects of the coupled APHC parameters, the excitation amplitude and random noise intensity and also shows that the coupling parameters are significant factors in the alteration of dynamical behaviors.

本文关注的是由双Duffing振荡器与耦合的主动-被动混合控制（APHC）构成的随机驱动系统的共振表现，即时滞反馈控制和粘弹性阻尼。解析解决方案以确定性和随机方式相继确定。基于我们在确定性模式下的仿真，详细描述了系统均方响应的规则和趋势的变化，包括随机模式下的耦合参数，如粘弹性系数，位移和速度反馈增益系数变化。这些伴随着（两次）跳跃和分叉现象，一个延时岛的出现以及它与其他分支的融合。我们的数值模拟结果也证实了理论分析方法的有效性。此外，两种模式下相图的多样性说明了耦合的APHC参数，激励幅度和随机噪声强度的不同影响，并且还表明耦合参数是动力学行为改变的重要因素。