Perturbation singuliere en dimension trois: canards en un point pseudo-singulier nud

摘要

We study singularly perturbed system of differential equations like {x = f(x,y,z,ε), y = g(x,y,z,ε), εz = h(x,y,z,ε), where f, g and h are analytic functions. In known papers, regular points of the slow surface h = 0 are studied. At this point, the fast flow (vertical) is tranverse to the slow surface. The Tikhonov's theorem can be applied here. In other papers, fold points and cusps of the slow surface were studied. The list of generic singularities contains also the pseudo-singular points which are connected to the turning points in lower dimension. They are (generically) saddle, focus or node. In the neighborhood of focus points, nothing happens, the saddle were studied in their papers, but the node points were never studied in the litterature. In this paper, whe prove that generally, there exist two overstable (i.e. regular with respect ε) solutions. When the ratio between two eigenvalues is an integer, a resonance appears, and one of the two overstable solutions disappears. Technically, we transform first the system into a more canonical equation. After that, we prove the existence of formal solutions, and, using the implicit function theorme on Banach spaces of Gevrey series, we can prove that the formal solution is Gevrey. The theory of summation of Gevrey series gives the over-stable solutions.