By V.N. Bogaevski, A. Povzner

Many books have already been written in regards to the perturbation conception of differential equations with a small parameter. as a result, we wish to offer a few explanation why the reader should still hassle with nonetheless one other ebook in this subject. talking for the current in basic terms approximately usual differential equations and their purposes, we become aware of that tools of strategies are so quite a few and various that this a part of utilized arithmetic seems as an combination of poorly hooked up tools. nearly all of those tools require a few earlier guessing of a constitution of the specified asymptotics. The Poincare approach to general types and the Bogolyubov-Krylov Mitropolsky averaging equipment, renowned within the literature, will be pointed out in particular in reference to what is going to stick with. those tools don't suppose a right away look for options in a few detailed shape, yet utilize alterations of variables as regards to the identification transformation which carry the preliminary method to a undeniable basic shape. Applicability of those tools is particular by means of exact types of the preliminary systems.

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**Example text**

The system dzd dt = Y Zi, where i = 1, ... , n, may be extended trivially by considering how some additional functions Zn+1(z), ... , zn+p(z) which depend on Z = (ZI, . , zn) vary on its trajectories. For this we must adjoin the equations for zn+j, j = 1, ... ,p, to the initial equations. If we consider all new variables Z as independent ones, then the new system arises. , z(O) = ZO, zn+jlt=o = zn+j(ZO), we will not obtain anything new. Accordingly, let us adjoin to the basic system new functions zn+ 1 (z), ...

Thus the construction of Mcp, Srp is performed by the same scheme as that of MO and SO. )Pcp = 0 for any p is easily dealt with by induction. 2). 3 supplies us with a method of reducing operators to the normal form and is a reformulation of the wellknown Schroedinger perturbation theory. In examples we will show that this problem may often be solved by other means. 5 The existence of the Jordan basis and decompositions with respect to eigenfunctions of Xo doubtless imposes severe restrictions. But, as we have already mentioned, they are often satisfied.

Is ~~ = ( -iw + ~Eh(W)) cp, dw dt = Ewh(w). 1. 7) It describes the motion of the pendulum up to a quantity of order E during the time period ~ t;S liE and is easy to integrate. p and cp by values of order E2, and the third equation is exact. Starting from these facts it is easy to prove the validity of the above estimate. ° 50 3. 3 Recall that old and new variables are denoted by the same signs. , we do not take into account summands of order e, and therefore there is no need to distinguish between old and new variables.