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Asymptotology

About: Asymptotology is a research topic. Over the lifetime, 1319 publications have been published within this topic receiving 35831 citations.


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TL;DR: In this article, Kuzmak et al. extended the asymptotic method of G. B. Whitham to a more general class of nonlinear second order partial differential equations and obtained asymptic expansions of the solutions.
Abstract: By extending the asymptotic method of G. E. Kuzmak, J. C. Luke has shown how G. B. Whitham’s theory of nonlinear wave propagation can be derived directly from the partial differential equation without using the variational principle, in special cases. We apply the same method to a more general class of nonlinear second order partial differential equations or systems of first order equations containing a small parameter e and obtain asymptotic expansions of the solutions.

20 citations

Journal ArticleDOI
TL;DR: One of the purposes of this note is to construct suitable Lyapunov functions on the complete region of attraction instead of in just a small neighborhood of the given set, which has the advantage that the theorems on global asymptotic stability and ultimate boundedness become immediate corollaries of the results the authors obtain.
Abstract: 0. Introduction. The basic theorems of the Lyapunov's Direct or Second Method as applied to stability and boundedness problems in ordinary differential equations are well known (see, e.g., [1], [2], [3], [4], [5]). Krasovskii [5], Hale [6], [7], and others have extended this method to functional differential equations, and Zubov [8], Auslander and Seibert [9], Bhatia [ 10], and some others have developed the same in the setup of a dynamical system defined on a locally compact metric space. In [9] Auslander and Seibert formalized the long suspected duality between stability and boundedness by restricting themselves to a locally compact separable metric space. In proving the converse theorems of the direct method for stability and asymptotic stability of an equilibrium point or a compact set M, one invariably proves the existence of a Lyapunov function with desired properties in a sufficiently small neighborhood of the given set. For converse theorems on global asymptotic stability and boundedness or ultimate boundedness in which an added condition is needed on the Lyapunov function a separate proof is generally given. An exception is the use of the proven duality in [9] to obtain theorems on boundedness and ultimate boundedness from those on stability and asymptotic stability. Now every asymptotically stable set has a region of attraction (the smallest invariant neighborhood of the set) which is an open neighborhood of the given set (see, e.g., [10]). One of the purposes of this note is to construct suitable Lyapunov functions on the complete region of attraction instead of in just a small neighborhood of the given set. This has the advantage that the theorems on global asymptotic stability and ultimate boundedness become immediate corollaries of the results we obtain. Our method of proof indirectly exploits the situation that if M be a

20 citations

Journal ArticleDOI
TL;DR: In this paper, the authors examined the error in a general composite integration formula and determined a form of the error which permits a theory of asymptotic upper and lower bounds to be applied to a wide class of rules.
Abstract: In this paper we examine the error in a general composite integration formula. We determine a form of the error which permits a theory of asymptotic upper and lower bounds to be applied to a wide class of rules.

20 citations

Journal ArticleDOI
TL;DR: Systematic methods are given for simplification of the normal form of a perturbed dynamical system near a rest point which can be rigorously justified as a universal unfolding with respect to asymptotic equivalence.

20 citations

Journal ArticleDOI
TL;DR: In this article, an asymptotic expansion for the energy eigenvalues of anharmonic oscillators for potentials of the form V (x ) = κ x 2 q + ω x 2, q = 2, 3, etc.

20 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20231
20222
20181
201725
201626
201526