Resonance characteristics of stochastic dual Duffing oscillators with coupled APHC

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.