scispace - formally typeset
Search or ask a question
Book

Asymptotic Methods in the Theory of Nonlinear Oscillations

TL;DR: In this article, a wide circle of engineering-technical and scientific workers who are concerned with oscillatory processes is devoted to the approximate asymptotic methods of solving the problems in the theory of nonlinear oscillations met in many fields of physics and engineering.
Abstract: : This book is devoted to the approximate asymptotic methods of solving the problems in the theory of nonlinear oscillations met in many fields of physics and engineering. It is intended for the wide circle of engineering-technical and scientific workers who are concerned with oscillatory processes. Contents include the following: Natural oscillations in quasi-linear systems; The method of the phase plane; The influence of external periodic forces; The method of the mean; Justification of the asymptotic methods.
Citations
More filters
Journal ArticleDOI
TL;DR: In this article, the authors demonstrate the mechanism for a universal instability, the Arnold diffusion, which occurs in the oscillating systems having more than two degrees of freedom, which results in an irregular, or stochastic, motion of the system as if the latter were influenced by a random perturbation even though, in fact, the motion is governed by purely dynamical equations.

3,527 citations

Journal ArticleDOI
Ji-Huan He1
TL;DR: In this paper, a survey of recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones, is presented.
Abstract: This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modied perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In this paper the following categories of asymptotic methods are emphasized: (1) variational approaches, (2) parameter-expanding methods, (3) parameterized perturbation method, (4) homotopy perturbation method (5) iteration perturbation method, and ancient Chinese methods. The emphasis of this article is put mainly on the developments in this eld in China so the references, therefore, are not exhaustive.

2,135 citations

Journal ArticleDOI
TL;DR: The cubic complex Ginzburg-Landau equation is one of the most-studied nonlinear equations in the physics community as mentioned in this paper, it describes a vast variety of phenomena from nonlinear waves to second-order phase transitions, from superconductivity, superfluidity, and Bose-Einstein condensation to liquid crystals and strings in field theory.
Abstract: The cubic complex Ginzburg-Landau equation is one of the most-studied nonlinear equations in the physics community. It describes a vast variety of phenomena from nonlinear waves to second-order phase transitions, from superconductivity, superfluidity, and Bose-Einstein condensation to liquid crystals and strings in field theory. The authors give an overview of various phenomena described by the complex Ginzburg-Landau equation in one, two, and three dimensions from the point of view of condensed-matter physicists. Their aim is to study the relevant solutions in order to gain insight into nonequilibrium phenomena in spatially extended systems.

1,557 citations

Journal ArticleDOI
01 Jun 1990
TL;DR: In this paper, a more general averaging procedure that encompasses state-space averaging and that is potentially applicable to a much broader class of circuits and systems is presented, including resonant type converters.
Abstract: A more general averaging procedure that encompasses state-space averaging and that is potentially applicable to a much broader class of circuits and systems is presented. Examples of its application in resonant and PWM power convertors are presented. The technique is shown to be effective on a number of examples. including resonant type converters. The approach offers refinements to the theory of state-space averaging, permitting a framework for analysis and design when small ripple conditions do not hold. The method may find applications in simulation and design since it is considerably easier to simulate an averaged model than a switched model. >

1,144 citations

Journal ArticleDOI
TL;DR: In this article, the basic ideas and current developments of the homotopy analysis method, an analytic approach to get convergent series solutions of strongly nonlinear problems, which recently attracts interests of more and more researchers, are described.

835 citations