In mathematics, more specifically in dynamical systems, the method of averaging (also called averaging theory) exploits systems containing time-scales separation: a fast oscillation versus a slow drift. It suggests that we perform an averaging over a given amount of time in order to iron out the fast oscillations and observe the qualitative behavior from the resulting dynamics. The approximated solution holds under finite time inversely proportional to the parameter denoting the slow time scale. It turns out to be a customary problem where there exists the trade off between how good is the approximated solution balanced by how much time it holds to be close to the original solution.
More precisely, the system has the following form
of a phase space variable The fast oscillation is given by versus a slow drift of . The averaging method yields an autonomous dynamical system
which approximates the solution curves of inside a connected and compact region of the phase space and over time of .
Under the validity of this averaging technique, the asymptotic behavior of the original system is captured by the dynamical equation for . In this way, qualitative methods for autonomous dynamical systems may be employed to analyze the equilibria and more complex structures, such as slow manifold and invariant manifolds, as well as their stability in the phase space of the averaged system.
In addition, in a physical application it might be reasonable or natural to replace a mathematical model, which is given in the form of the differential equation for , with the corresponding averaged system , in order to use the averaged system to make a prediction and then test the prediction against the results of a physical experiment.
The averaging method has a long history, which is deeply rooted in perturbation problems that arose in celestial mechanics (see, for example in ).
First example
Consider a perturbed logistic growth
and the averaged equation
The purpose of the method of averaging is to tell us the qualitative behavior of the vector field when we average it over a period of time. It guarantees that the solution approximates for times Exceptionally: in this example the approximation is even better, it is valid for all times. We present it in a section below.
Definitions
We assume the vector field to be of differentiability class with (or even we will only say smooth), which we will denote . We 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.