Abstract
This study is divided into two important axes; for the first one, a new symmetric analytical (approximate) solution to the Duffing-Helmholtz oscillatory equation in terms of elementary functions is derived. The obtained solution is compared with the numerical solution using 4th Range-Kutta (RK4) approach and with the exact analytical solution that is obtained using elliptic functions. As for the second axis, we consider the time-delayed version for the same oscillator taking the impact of both forcing and damping terms into consideration. Some analytical approximations for the time delayed Duffing-Helmholtz oscillator are derived using two different perturbation techniques, known as Krylov-Bogoliubov-Mitropolsky method (KBMM) and the multiple scales method (MSM). Moreover, these perturbed approximations are analyzed numerically and compared with the RK4 approximations.