Residue harmonic balance solution procedure to nonlinear delay differential systems

This paper develops the residue harmonic balance solution procedure to predict the bifurcated periodic solutions of some autonomous delay differential systems at and after Hopf bifurcation. In this solution procedure, the zeroth-order solution employs just one Fourier term. The unbalanced residues due to Fourier truncation are considered by solving linear equation iteratively to improve the accuracy. The number of Fourier terms is increased automatically. The well-known sunflower equation and van der Pol equation with unit delay are given as numerical examples. Their solutions are verified for a wide range of system parameters. Comparison with those available shows that the residue harmonic balance method is effective to solve the autonomous delay differential equations. Moreover, the present method works not only in determining the amplitude but also the frequency at bifurcation.