Improving convergence of incremental harmonic balance method using homotopy analysis method

We have deduced an iteration scheme in theincremental harmonic balance (IHB) method using the harmonicbalance plus the Newton-Raphson method. Since theconvergence of the iteration is dependent upon the initial valuesin the iteration, the convergent region is greatly restrictedfor some cases. In this contribution, in order to enlarge theconvergent region of the IHB method, we constructed thezeroth-order deformation equation using the homotopy analysismethod, in which the IHB method is employed to solvethe deformation equation with an embedding parameter asthe active increment. Taking the Duffing and the van derPol equations as examples, we obtained the highly accuratesolutions. Importantly, the presented approach renders a convenientway to control and adjust the convergence.