Classification of resonant equations

[摘要] Consider a singularly perturbed equation of the form epsilony - f (x, epsilon) y', + g (x, epsilon) y = 0 where x, y is an element of C epsilon > 0 is a small parameter, and f and g are two analytic functions in a neighborhood of (0, 0), real for real values of x, epsilon, f (0, 0) = 0, f' (0, 0) > 0; this means that x = 0 is a turning point. Eq. (1) is called resonant in the sense of Ackerberg-O'Malley, if there is a solution, analytic for x in some neighborhood of 0 and epsilon in some sector, which tends to a non-trivial solution of the reduced equation f (x, 0) y' = g (x, 0) y as epsilon --> 0. The article presents a classification of such resonant equations with respect to analytic transformations (y) over bar = a (x, epsilon) y + b (x, epsilon)epsilony'. First of all, f(0)(x) = f (x, 0) is a formal invariant considered fixed below. Furthermore, to each resonant equation are associated three formal series in;, which are Gevrey of order I and invariant under analytic transformations. It is shown that this correspondence between equivalence classes of resonant equations and triples of Gevrey series is essentially bijective, and that each equivalence class contains an equation of a particular form: f (x, epsilon) = f(0) (x) and g (x, epsilon) = f(1) (x) + epsilonf(2) (x) with f(1) (0) = 0. (C) 2004 Elsevier Inc. All rights reserved.

[发布日期] 2004-12-15

[关键词] Ackerberg-O ' Malley resonance;Gevrey asymptotic singular perturbation