Subharmonic resonance analysis of asymmetrical stiffness nonlinear systems with time delay

Incorporating asymmetric quadratic and cubic stiffnesses into a time-delayed Duffing oscillator provides a more accurate representation of practical systems, where the resulting nonlinearities critically influence subharmonic resonance phenomena, yet comprehensive investigations remain limited. This study employs the generalized harmonic balance (HB) method to conduct an analytical investigation of the subharmonic resonance behavior in asymmetric stiffness nonlinear systems with time delay. To further examine the switching behavior between primary and subharmonic resonances, a numerical continuation approach combining the shooting method and the parameter continuation algorithm is developed. The analytical and numerical continuation solutions are validated through direct numerical integration. Subsequently, the switching behavior and associated bifurcation points are analyzed by means of the numerical continuation results in conjunction with the Floquet theory. Finally, the effects of delay parameters on the existence range of subharmonic responses are discussed in detail, and the influence of initial conditions on system dynamics is explored with basin of attraction plots. This work establishes a comprehensive framework for the analytical and numerical study on time-delayed nonlinear systems with asymmetric stiffness, providing valuable theoretical insights into the stability management of such dynamic systems.