Dynamics analysis of time-delayed nonlinear system with asymmetric stiffness

Time-delayed Duffing oscillators have been widely studied for their rich dynamic properties and their capacity to describe dynamic systems with delays and nonlinearities. However, due to the assumption of symmetry, these oscillators often fail to accurately represent systems influenced by asymmetric stiffness. Analyzing the dynamic characteristics of time-delayed nonlinear systems with asymmetric stiffness, as well as developing effective control strategies, remains particularly challenging. This paper introduces a quadratic stiffness term into the time-delayed Duffing oscillator, resulting in a Time-Delayed Nonlinear System with Asymmetric Stiffness (TD-ASNS). The TD-ASNS is designed to model dynamic systems that incorporate asymmetric stiffness and time delay. The Multiple Scales Method is used to solve the TD-ASNS, and numerical methods are employed to validate the analytical solution. This study examines the influence of time delay and excitation parameters on system response and stability. The time delay term functions like quasi-stiffness and quasi-excitation, shifting the amplitude-frequency response curve along the frequency axis and the resonance backbone, respectively. Similarly, the excitation term shifts the curve along the resonance backbone. This research highlights the critical roles of the delay and excitation parameters in TD-ASNS, which impact dynamic response, stability, and bifurcation behavior. It provides a theoretical foundation for analyzing and controlling the stability of dynamic systems characterized by both asymmetric stiffness and time delay.