On the methods of non-linear analysis of dynamical systems

This paper is concerned with the methods of non-linear analysis of dynamical systems and the associated bifurcation and stability problems. Attention is focused on the intrinsic harmonic balancing (IHB) technique, and the interrelationship between this technique and the methods of normal forms and averaging. Recent improvements and a complex formulation of the technique, which facilitates comparisons with other methods, are described. Thus, it is demonstrated that the simplified equations of an autonomous system, obtained by both the IHB and averaging techniques are identical, and these equations are, in fact, normal forms. Hilbert's 16th problem is analyzed as an illustrative example. It is observed that the IHB technique lends itself to a symbolic computer language (MAPLE) more efficiently compared to other methods; furthermore, its efficiency increases with the complexity of the system analyzed.