Invariant curves of an analytic slow-fast mapping and bifurcation delay. (Courbes invariantes d’une application lente-rapide analytique et retard à la bifurcation.) (French. Abridged English version) Zbl 0802.58047

Baesens, Claude

C. R. Acad. Sci., Paris, Sér. I 317, No. 12, 1109-1114 (1993).

Let \(F_ v : \mathbb{C}^ 2 \to \mathbb{C}^ 2\) \((v \in \mathbb{R}^ +)\) be a holomorphic map with the form: \(F_ v (x, \lambda) = (x', \lambda')\), \(x' = f(x, \lambda)\), \(\lambda' = \lambda + v\), where the map \(f(\cdot, \lambda)\) admits a fixed point \(y(\lambda)\). In order to investigate the orbits of the system \((x, \lambda) \to (x', \lambda')\), the author considers the solutions \(x = U (\lambda, v)\) of the functional equation \[ U (\lambda + v,v) - f \bigl( U (\lambda,v), \lambda \bigr) = 0. \tag{*} \] Suppose that there is an open set \(L \subset \mathbb{C}\) such that \(y:L \to \mathbb{C}\) is analytic and that \(dy/d \lambda \neq 0\) on \(L\). We consider a formal solution \(\widehat U (\lambda,v) = y (\lambda) + \widehat u(\lambda,v)\) of \((*)\) where \(\widehat u (\lambda,v) = \sum^ \infty_{n = 1} v^ n \cdot u_ n (\lambda)\), \(u_ n (\lambda)\) being analytic for \(\lambda \in L\). Let \(D (\lambda^*, \rho)\) be the disc in \(L\) with center \(\lambda^*\) and radius \(\rho\). The author states the following main theorem: the formal series \(\widehat u (\lambda,v)\) has Gevrey-1 property on a family of concentric discs \(D(\lambda^*,t)\) \((0<t<1)\) in \(L\). More precisely, there exist constants \(\mu>0\) and \(C>0\) such that, for \(0<t<1\) and \(n \geq 1\), \(| u_ n (\lambda) | \leq n! R^{-n}B\) holds for \(\lambda \in D (\lambda^*,t \rho)\), where \(B=C \mu \rho (1 - t)\) and \(R = \mu \rho (1 - t)\).

Reviewer: A.Morimoto (Nagoya)

MSC:

37F99

Dynamical systems over complex numbers

Keywords:

formal solutions; Gevrey-1 property