Nonlinear equations with small parameter. Volume 1: Oscillations and resonances (Q2797804)

This book presents new methods for the construction of global asymptotic expansions of the solutions to nonlinear equations with a small parameter. The approach underlies modern asymptotic methods and gives a deep insight into crucial nonlinear phenomena of natural sciences. These are beginnings of chaos of dynamical systems, incipient solitary and shock waves, oscillatory processes in crystals, engineering constructions and quantum systems. The approximate solutions may serve as a foolproof basis for testing numerical algorithms.NEWLINENEWLINEThe book is divided in seven chapters. It contains an Introduction section, rich list of references and index terms.NEWLINENEWLINEChapter 1 explains basic ideas for constructions of asymptotic approximations. The main aims of this chapter are as follows: to show the naturalness of appearance of asymptotic series, to give definitions for basic statements of asymptotic analysis, to explain the regularity property of asymptotic series and combined asymptotic series. The chapter includes detailed examples but the technical calculations are not presented.NEWLINENEWLINEChapter 2, entitled ``Asymptotic methods for solving nonlinear equations'', provides some examples of using asymptotic methods for solving nonlinear equations. One discusses the following issues: construction of solutions for weak nonlinear equations by Wentzel, Kramers and Brillouin (WKB) method, boundary layer method, matching method. In this chapter, elliptic functions as solutions of nonlinear differential equations are considered: Weierstrass function and its periodicity and poles, Jacobi elliptic functions and properties for small oscillations and oscillations near a separatrix, Mathieu and Lame functions and their properties from the point of view of perturbation theory for nonlinear oscillations.NEWLINENEWLINEChapter 3 focuses on asymptotic solutions to nonlinear nonautonomous equations of order 2. The methods of perturbation theory are well motivated by celestial mechanics. Mainly, the approach contains two steps including appropriate choice of slow and fast variables and the study of oscillations first in the slow variable and then in the fast one. Basic features of the solutions of the considered nonlinear and nonautonomous equations are: the period depends on the energy of the oscillations, the phase space has three dimensions, the perturbed equations are non-integrable.NEWLINENEWLINEChapter 4 develops perturbation theory to describe the behaviour of trajectories in a small neighbourhood of separatrices for perturbed equations. In this approach, one constructs perturbation series with separatrix as a primary term of approximation. Then one shows ascending and descending discrete dynamics for parameters of the perturbed solution. NEWLINENEWLINEChapter 5 deals with autoresonances in nonlinear systems. The autoresonance means the growth of the amplitude of a nonlinear oscillator when this oscillator is subject to an external oscillating force. More specific, in this chapter the following issues are discussed: a threshold of amplitude for an existence of autoresonant pumping, an asymptotic solution for capture into autoresonance and a capture into a parametric autoresonance. NEWLINENEWLINEIn Chapter 6, entitled ``Asymptotics for loss of stability'', the loss of stability for equilibria is studied. Equations which are discussed here, have a small parameter with derivative and varying coefficients. So, slowly changing solutions exist. These solutions look like real equilibria, but due to the slowly changes these equilibria can loss their stability. In such a case the solutions begin to oscillate with high frequency.NEWLINENEWLINEThe last chapter, Chapter 7, is devoted to systems of coupled oscillators. The aim of this chapter is to study a system with two weakly coupled oscillators. It is shown that the external periodic perturbation can lead to the capture into resonance. A perturbed system of coupled nonlinear oscillators with small dissipation and outer periodic perturbation is studied.NEWLINENEWLINEPrerequisites for reading this book are some acquaintance with the elementary theory of ordinary differential equations and a good knowledge of classical analysis.NEWLINENEWLINEThe main concepts and methods are useful for graduate students of physics and engineering sciences.