Topics in singular perturbations

O’Malley, Robert E.

doi:10.1016/0001-8708(68)90023-6

Cited by 115 publications

(59 citation statements)

References 98 publications

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“…The constant L that bounds the domain of validity in time is in general an O(1) quantity determined by the vector fields f and g. For proofs, see [12] and [17]. An intermediate step in the analysis is an expansion of the form…”

Section: Suppose That the Requirements Of Tikhonov's Theorem Have Beementioning

confidence: 99%

Singular perturbation methods for slow–fast dynamics

Verhulst

2007

Nonlinear Dyn

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Recently, geometric singular perturbation theory has been extended considerably while at the same time producing many new applications. We will review a number of aspects relevant to non-linear dynamics to apply this to periodic solutions within slow manifolds and to review a number of non-hyperbolic cases. The results are illustrated by examples.Keywords Singular perturbations . Slow manifolds . Periodic solutions . Nonhyperbolic This paper deals with slow-fast initial value problems that are of the forṁAs usual, ε is a small positive parameter, and an overdot denotes differentiation with respect to time. For a number of results, the vector fields f and g explicitly depending on time t present no obstruction. Part of the paper is a tutorial, but there are some new results. The Tikhonov theoremIn singular perturbations, certain attraction (or hyperbolicity) properties of the regular (outer) expansion play an essential part in the construction of the formal approximation. Remarkably enough, this hyperbolicity does not include the behaviour of the slow equation.In the constructions, the following theorem provides a basic boundary layer property of the solution. Theorem 1.1. (Tikhonov, 1952, see [15]) Consider the initial value probleṁ For f and g, we take sufficiently smooth vector functions in x, y and t; the dots represent (smooth) higherorder terms in ε.a. We assume that a unique solution of the initial value problem exists and suppose this holds also for the reduced probleṁ x = f (x, y, t), x(0) = x 0 , = g(x, y, t), with solutionsx(t),ȳ(t).Springer

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Section: Suppose That the Requirements Of Tikhonov's Theorem Have Beementioning

confidence: 99%

Singular perturbation methods for slow–fast dynamics

Verhulst

2007

Nonlinear Dyn

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“…The method of additive decomposition which works for ordinary differential equations and integral equations with continuous kernel (cf. [1,9,12,13,16]) is used here. Problems of the type (1.1) do not exhibit an exponential decay in the initial layer and therefore, the methodology developed by Angell and Olmostead, and Lange and Smith, can be improved.…”

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confidence: 99%

Singularly perturbed Volterra integral equations with weakly singular kernels

Bijura

2002

International Journal of Mathematics and Mathematical Sciences

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We consider finding asymptotic solutions of the singularly perturbed linear Volterra integral equations with weakly singular kernels. An interesting aspect of these problems is that the discontinuity of the kernel causes layer solutions to decay algebraically rather than exponentially within the initial (boundary) layer. To analyse this phenomenon, the paper demonstrates the similarity that these solutions have to a special function called the Mittag-Leffler function.

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“…In the classical singular perturbation theory, β is so called an intermediate variable [21,10]. Note that in the x variable, the length of inner layers goes to zero as → 0, but in the stretched variable ξ = (x − x i 0 )/ , the length O( β−1 ) goes to infinity as → 0.…”

Section: Existence Of True Eigenvalue/eigenfunctionsmentioning

confidence: 99%

Slow Eigenvalues of Self-similar Solutions of the Dafermos Regularization of a System of Conservation Laws: an Analytic Approach

Lin

2006

J Dyn Diff Equat

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Abstract. The Dafermos regularization of a system of n hyperbolic conservation laws in one space dimension has, near a Riemann solution consisting of n Lax shock waves, a self-similar solution u = u (X/T ). In [19] it is shown that the linearized Dafermos operator at such a solution may have two kinds of eigenvalues: fast eigenvalues of order 1/ and slow eigenvalues of order one. The fast eigenvalues represent motion in an initial time layer, where near the shock waves solutions quickly converge to traveling-wave-like motion. The slow eigenvalues represent motion after the initial time layer, where motion between the shock waves is dominant.In this paper we use tools from dynamical systems and singular perturbation theory to study the slow eigenvalues. We show how to construct asymptotic expansions of eigenvalue-eigenfunction pairs to any order in . We also prove the existence of true eigenvalue-eigenfunction pairs near the asymptotic expansions.

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Topics in singular perturbations

Cited by 115 publications

References 98 publications

Singular perturbation methods for slow–fast dynamics

Singular perturbation methods for slow–fast dynamics

Singularly perturbed Volterra integral equations with weakly singular kernels

Slow Eigenvalues of Self-similar Solutions of the Dafermos Regularization of a System of Conservation Laws: an Analytic Approach

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