expand this time-dependent vector field in a Taylor series (in powers of ) with remainder . We introduce the following notation: where is the -th derivative with . As we are concerned with averaging problems, in general is zero, so it turns out that we will be interested in vector fields given by Besides, we define the following initial value problem to be in the standard form: Theorem: averaging in the periodic case Consider for every connected and bounded and every there exist and such that the original system (a non-autonomous dynamical system) given by has solution , where is periodic with period and both with bounded on bounded sets. Then there exists a constant such that the solution of the averaged system (autonomous dynamical system) is is for and . Remarks There are two approximations in this what is called first approximation estimate: reduction to the average of the vector field and negligence of terms. Uniformity with respect to the initial condition : if we vary this affects the estimation of and . The proof and discussion of this can be found in J. Murdock's book. Reduction of regularity: there is a more general form of this theorem which requires only to be Lipschitz and continuous. It is a more recent proof and can be seen in Sanders et al.. The theorem statement presented here is due to the proof framework proposed by Krylov-Bogoliubov which is based on an introduction of a near-identity transformation. The advantage of this method is the extension to more general settings such as infinite-dimensional systems - partial differential equation or delay differential equations. J. Hale presents generalizations to almost periodic vector-fields. Strategy of the proof Krylov-Bogoliubov realized that the slow dynamics of the system determines the leading order of the asymptotic solution. In order to proof it, they proposed a near-identity transformation, which turned out to be a change of coordinates with its own time-scale transforming the original system to the averaged one. Sketch of the proof Determination of a near-identity transformation: the smooth mapping where is assumed to be regular enough and periodic. The proposed change of coordinates is given by . Choose an appropriate solving the homological equation of the averaging theory: . Change of coordinates carries the original system to Estimation of error due to truncation and comparison to the original variable. Non-autonomous class of systems: more examples Along the history