On the Krýlov–Bogoliúbov-Mitropólsky and Multiple Scales Methods for Analyzing a Time Delay Duffing–Helmholtz Oscillator

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