## 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

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 $$R = \mu \rho (1 - t)$$.

### MSC:

 37F99 Dynamical systems over complex numbers

### Keywords:

formal solutions; Gevrey-1 property