Nonlinear dynamics of duffing oscillator with time delayed term

The improved constrained optimization harmonic balance method(COHBM) is presented to solve the Duffing oscillator with time delayed term. Within the framework of the proposed method, the analytical gradients of the objective function and nonlinear quality constraints with respect to optimization variables are formulated and the sensitivity information of the Fourier coefficients can also obtained. The general formulas of the geometrically nonlinear and time delayed terms are analytically derived, which makes the calculations of nonlinear differential equations in the frequency domain easily. A stability analysis method based on the analytical formulation of the nonlinear equality constraints is presented for the nonlinear systems with time delayed. A hybrid method which combines the improved COHBM and the continuation technique is also presented to investigate the global dynamics of nonlinear delayed systems. Numerical results indicate that the Duffing system displays a wide variety of rich and interesting dynamical behaviors. It is found that the proposed method yields accurate prediction on the global dynamics of time-delayed systems than the traditional method of multiple scales.