Periodic Solution and Stability Behavior for Nonlinear Oscillator Having a Cubic Nonlinearity Time-Delayed

The current paper investigates the dynamics of the dissipative system with a cubic nonlinear time-delayed of the type of the damping Duffing equation. A coupling between the method of the multiple scales and the homotopy perturbation has been utilized to study the complicated dynamic problem. Through this approach, a cubic nonlinear amplitude equation resulted in at the first-order of perturbation; meanwhile, a quintic equation appears at the second-order of perturbation. These equations are combined into one nonlinear quintic Landau equation. The polar form solution is used, and linearized stability configuration is applied to the nonlinear amplitude equation. Also, a second-order approximate solution is achieved. The numerical illustrations showed that the damping, delay coefficient, and time delay play dual roles in the stability behavior. In addition, the nonlinear coefficient plays a destabilizing influence.