THE DYNAMICS OF SLOW MANIFOLDS

Autor: Ferdinand Verhulst, Taoufik Bakri
Jazyk: angličtina
Rok vydání: 2012
Předmět:
Zdroj: Journal of the Indonesian Mathematical Society, Vol 13, Iss 1, Pp 73-90 (2012)
Journal of the Indonesian Mathematical Society, 1. MIHMI
STARTPAGE=1;ISSN=0854-1380;TITLE=Journal of the Indonesian Mathematical Society
ISSN: 2460-0245
2086-8952
Popis: After reviewing a number of results from geometric singular perturbation theory, we discuss several approaches to obtain periodic solutions in a slow manifold. Regarding nonhyperbolic transitions we consider relaxation oscillations and canard-like solutions. The results are illustrated by prey-predator systems. x = 1; x(0) = 1; "˙ y = iy + "f(x); y(0) = 1; with f(x) a smooth scalar function. Putting " = 0 we have 0 = iy; ˙ x = 1 with solution x(t) = 1 + t;y(t) = 0. The 'unperturbed solution' does not satisfy the initial condition for y, but in the theory of singular perturbations, techniques have been developed to handle such cases. In this example, the solution y(t) changes quickly in a neighborhood of t = 0, a so-called boundary layer in time. For a recent survey of methods see (27). In this paper we will review a number of the theorems available for singularly perturbed initial value problems of ordinary dierential equations, while adding results on periodic solutions and examples for simple looking but surprisingly rich prey-predator systems. The numerics for autonomous two-dimensional systems is carried out by pplane, using MATLAB. The nonautonomous systems were integrated using Runge-Kutta 7(8). In the actual constructions of asymptotic approximations, the Tikhonov theorem is basic for providing a boundary layer property of the solution. This leads naturally to a number of qualitative and quantitative results. Also certain attraction (or hyperbolicity) properties of the 'unperturbed solution' play an essential part in the construction of the asymptotic approximation, adding a geometric flavour to the analysis that is essential. In the case of our nearly-trivial example, as we shall see, the 'unperturbed solution' x(t) = 0;y(t) = 1 + t is associated with the existence of a so-called slow manifold. ⁄ Invited lecture at Konferensi Nasional Matematika XIII, Semarang, 24-27 juli, 2006; to be publ. in J. Indones. Math. Soc. (2007)
Databáze: OpenAIRE
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