Asymptotic analysis of the peeling-off point of a French duck

Abstract

The asymptotic theory of a relaxation oscillator displaying a duck (or canard) trajectory is studied. In a previous paper [18], the Poincare part of the asymptotic approximation was constructed. In the present paper, the associated terms containing exponentially small factors are constructed, thereby giving a formal complete asymptotic approximation. All formulas are explicit and are compared with numerical experiments using Mathematica.

Asymptotic solution of a laminar flow in a porous channel with large suction: A nonlinear turning point problem

“…An interesting and, at the same time, essential feature of the asymptotic analysis is the introduction of a transition approximation which covers the region of the turning point. The construction of this intermediate expansion, and its matching with the outer and boundary-layer approximations, incorporates and exploits some ideas about exponential terms, which may have application to other problems and, in fact, have already been applied to one problem outside of fluid dynamics [9].…”

“…Some of the ideas introduced in the present problem have found application in a completely unrelated area, namely nonlinear oscillation theory, [9].…”

Stability Loss Delay in Harvesting Competing Populations

“…Canards solutions were first rigorously studied in 1978 by a group of French mathematicians, namely E. Beno@^t, J. L. Callot, F. Diener, and M. Diener (see [4,17,22] for complete references), using Nonstandard Analysis. We have also to our disposal standard studies of the French canards [5,7,14]. Rigorous proofs (using standard analysis) in a two dimensional system were given also by Schecter [18] in 1985.…”