 dominant slow-varying component. This can be interpreted as "averaging the fast- varying components out of the equations". The averaged system of ODEs that results serves as an excellent starting point for noise and stability analysis, for efficient simu- lation and for extraction of compact behavioral models. Technically, averaging involves the construction of a time-varying transformation of variables p -> p. Hereby, the values of the original vector p and the newly introduced vector p are related to each other by means of (5.84) Note that the difference p-p is a quantity that is O(e), i.e. small, in order of magnitude. Therefore, the transformation (5.84) is called a near-identity transformation. In this transformation, the term h(p,t) is chosen SO that 1. the behavior of the transformed state vector p(t) only contains slow-varying components. 2. the process eh(p(t),t) captures the fast-varying fluctuations of p(t) while re- maining bounded over the time interval of interest. The latter is, in our case, infinite. By construction, the process p(T) captures the slow-varying long-term characteristics of the state vector p(T). Therefore, it is called the averaged process corresponding to p(T). In many cases, this average is the quantity of main interest. The equations that govern this behavior are obtained by substituting the transformation (5.84) into the starting equation (5.82). Example (Harmonic oscillator): Fig. 5.17(a) shows a typical harmonic oscillator am- plitude startup characteristic. Here, A(T) is solved from a set of equations structured as (5.82). These equations are obtained by means of the perturbation analysis outlined in the previous section. Fig. 5.17(b) illustrates the decomposition of A(T) into a dominant slow-varying component A(T) and a small fast-varying fluctuation eh(A(t), T). Note that, for sake of a clear illustration, the magnitude of the latter component has been exaggerated. In most practical cases, we are mainly interested in the behavior of the averaged amplitude A(t). The equatio