Complex dynamical behavior of Duffing oscillators under multiple time-delay coupling and fractional-order feedback

The widely existing multi time delay coupling and fractional order feedback effects in engineering vibration isolation systems cause Duffing oscillators to exhibit complex nonlinear dynamic behavior, especially multi steady states and chaos, which seriously threaten the reliability of vibration isolation and equipment safety. This article focuses on the multi branch parallel damping stiffness element under the background of viscoelastic vibration isolation, constructs a fractional order Duffing dynamic equation with multiple time delay coupling, uses the averaging method to solve the analytical solution of the main resonance, and combines stability theory and attraction domain visualization to analyze the multi steady state evolution law; Derive chaos threshold using Melnikov theory and characterize chaotic behavior through multidimensional numerical simulation; Quantify the synergistic control effect of multiple parameters using visualization methods for key parameters, and analyze the phase space evolution and attraction domain distortion mechanism. The study clarified the control mechanism of multiple time delays and fractional order feedback on the nonlinear vibration of the system, revealed the evolution law of multiple steady states and chaos under multi parameter coupling, and provided theoretical basis and quantitative support for the dynamic design and parameter optimization of nonlinear isolation systems.