By Professor Yu. A. Mitropolskii, Professor Nguyen Van Dao (auth.)

Many dynamical platforms are defined by means of differential equations that may be separated into one half, containing linear phrases with consistent coefficients, and a moment half, rather small in comparison with the 1st, containing nonlinear phrases. one of these method is related to be weakly nonlinear. The small phrases rendering the procedure nonlinear are known as perturbations. A weakly nonlinear procedure is named quasi-linear and is ruled by way of quasi-linear differential equations. we'll have an interest in platforms that lessen to harmonic oscillators within the absence of perturbations. This e-book is dedicated essentially to utilized asymptotic tools in nonlinear oscillations that are linked to the names of N. M. Krylov, N. N. Bogoli ubov and Yu. A. Mitropolskii. the benefits of the current equipment are their simplicity, in particular for computing greater approximations, and their applicability to a wide type of quasi-linear difficulties. during this ebook, we confine ourselves basi cally to the scheme proposed through Krylov, Bogoliubov as acknowledged within the monographs [6,211. We use those equipment, and likewise improve and increase them for fixing new difficulties and new sessions of nonlinear differential equations. even though those tools have many purposes in Mechanics, Physics and approach, we are going to illustrate them merely with examples which in actual fact express their power and that are themselves of serious curiosity. a specific amount of extra complicated fabric has additionally been incorporated, making the ebook compatible for a senior optional or a starting graduate path on nonlinear oscillations.

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33) 36 CHAPTER 1 ~; where ,p = w(r, a)z~(r,,p, a), = w(r,a)t+ ,po, r= d. " ". 34) with respect to ,p and a, we find Eliminating I; (1', z) from these equations, we find: Z' - w2z'" W2Z'" t/J3 a t/J'a Z''" - 2ww'az""', z''" =- 0 , or d [WZaZt/J' ,,, - Z'"' ( ' )'a] d,p WZt/J = o. 36) where 00 (1', a) does not depend on ,p. t the adiabatic invariant for this equation is the action integral J(r, a) = 2~ ! 2 .. w(r, a)z:;(r,,p, a)d,p. 37) o When r = const, a is a constant and therefore, J(r, a) is also a constant.

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41) We have w(r) = 0(;}, z(r,a,,p) = acos,p, z~(r,a,,p) = -asin,p. c(er} This formula shows that the amplitude of oscillation is inversely proportional to the fourth root of c(er}. The results obtained above can be generalized for systems, close to the Hamiltonian ones. 42) 39 FREE OSCILLATIONS OF QUASI-LINEAR SYSTEMS is known: z = z(r, a, tJ;). 42), the action integral 2 ... J(r, a) = mer) /w(r,a)z1 dtJ; 211" o is an adiabatic invariant. '( dt a r, a ) )dT. '(r a) ~ , 2ft' = em~(r) fwz, dtJ;+ em(r) f 211" '" o 2ft' - - em(r) f - 211" o d-I.