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Stability analysis: nonlinear mechanics equations. Transl. from the Russian by Svetlana Raschivalova. (English) Zbl 0840.34003

The “nonlinear mechanics equations” from the title are ordinary differential equations of the type (1) \(dy/dt= Y(t, y, \mu)\) with \(t\in {\mathcal T}_0\), \(y\in \Omega\subseteq \mathbb{R}^n\), and \(\mu\in M\subseteq [0, 1]\) a scalar parameter which is assumed to be small. A “motion” is a solution of (1) with initial value \(y(t_0)= y_0\), \(t_0\in {\mathcal T}_i\). Situations, where \({\mathcal T}_0\), \({\mathcal T}_i\) are finite intervals are obviously included; among others, various types of stability on a finite time interval are considered.
The aim of the book is to present stability results which in some sense hold uniformly for families of motions with parameter values \(\mu\) from some interval \(0\leq \mu\leq \mu_0\). So in Chapter 1 a great variety of notions of \(\mu\)-stability, \(\mu\)-attraction (including things like a “domain of \(t_0\)-uniform \(\mu\)-attraction with respect to \({\mathcal T}_i\)”) and \(\mu\)-asymptotic stability are introduced. These notions turn out to be of great use for estimating the deviations from certain “unperturbed motions” which, as a rule, are produced from limiting equations or via various asymptotic or averaging procedures going back to the classical approaches of N. M. Krylov, N. N. Bogolyubov and Yu. A. Mitropol’skij. These averaging techniques are combined with the comparison method and with Lyapunov’s direct method using scalar, vector or matrix Lyapunov functions.
The book consists of five chapters. In Chapter 1 results for general small-parameter systems of type (1) are presented, in the remaining four chapters systems with a more special structure are considered. Chapter 2 is devoted to standard systems in the terminology of Bogolyubov, i.e., to systems of the type \(dy/dt= \mu Y(t, y, \mu)\). Here, among others, conditions for stability on a finite interval \(J= [t_0, t_0+ L/\mu]\), \(L> 0\), are obtained. Chapter 3 deals with motions with slow drift and fast oscillations which are described by systems of the type \(dy/dt= Y(t, y, z, \mu)\), \(dz/dt= Z(t, y, z, \mu)\). In Chapter 4 systems with small perturbing forces, \(dy/dt= Y(t, y)+ \mu R(t, y, \mu)\), are considered, conditions for stability, for boundedness or for the existence of limit cycles and nested attractors are obtained. Chapter 5 deals with singularly perturbed systems, of the type \(dx/dt= f(t, x, y, \mu)\), \(\mu dy/dt= g(t, x, y, \mu)\), using the method of matrix Lyapunov functions. Among others, conditions for absolute stability are obtained for singularly perturbed Lur’e-Postnikov systems with one nonlinear element in the fast part and one in the slow part.
Throughout the book, an imposing collection of particular stability results is presented, among them a large number of new results of the author. The bibliography of about 130 titles obviously does not claim for completeness. The English is well readable, occasional misprints are easily corrected. The book can be warmly recommended to everybody interested in the subject.
Reviewer: W.Müller (Berlin)

MSC:

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
34C29 Averaging method for ordinary differential equations
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