On the Formal Equivalence of Normal Form Theory and the Method of Multiple Time Scales

Multiple time scales technique has long been an important method for the analysis of weakly nonlinear systems. In this technique, a set of multiple time scales is introduced that serve as independent variables. The evolution of state variables at slower time scales is then determined so as to make the expansions for solutions in a perturbation scheme uniform in natural and slower times. Normal form theory has also recently been used to approximate the dynamics of weakly nonlinear systems. This theory provides a way of finding a coordinate system in which the dynamical system takes the “simplest” form. This is achieved by constructing a series of near-identity nonlinear transformations that make the nonlinear terms as simple as possible. The simplest differential equations obtained by the normal form theory are topologically equivalent to the original systems. Both methods can be interpreted as nonlinear perturbations of linear differential equations. In this work, the formal equivalence of these two methods for constructing periodic solutions and amplitude evolution equations is proven for autonomous as well as harmonically excited nonlinear vibratory dynamical systems. The reasons as to why some studies have found the results obtained by the two techniques to be inconsistent are also pointed out.